Internet DRAFT - draft-irtf-cfrg-cpace
draft-irtf-cfrg-cpace
Network Working Group M. Abdalla
Internet-Draft DFINITY - Zurich
Intended status: Informational B. Haase
Expires: 28 March 2024 Endress + Hauser Liquid Analysis - Gerlingen
J. Hesse
IBM Research Europe - Zurich
25 September 2023
CPace, a balanced composable PAKE
draft-irtf-cfrg-cpace-10
Abstract
This document describes CPace which is a protocol that allows two
parties that share a low-entropy secret (password) to derive a strong
shared key without disclosing the secret to offline dictionary
attacks. The CPace protocol was tailored for constrained devices and
can be used on groups of prime- and non-prime order.
Discussion Venues
This note is to be removed before publishing as an RFC.
Discussion of this document takes place on the Crypto Forum Research
Group mailing list (cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg.
Source for this draft and an issue tracker can be found at
https://github.com/cfrg/draft-irtf-cfrg-cpace.
Status of This Memo
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provisions of BCP 78 and BCP 79.
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This Internet-Draft will expire on 28 March 2024.
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Copyright Notice
Copyright (c) 2023 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
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Please review these documents carefully, as they describe your rights
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1. Outline of this document . . . . . . . . . . . . . . . . 6
2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 6
3. High-level application perspective . . . . . . . . . . . . . 7
3.1. Optional CPace inputs . . . . . . . . . . . . . . . . . . 7
3.2. Responsibilities of the application layer . . . . . . . . 8
4. CPace cipher suites . . . . . . . . . . . . . . . . . . . . . 9
5. Definitions and notation . . . . . . . . . . . . . . . . . . 10
5.1. Hash function H . . . . . . . . . . . . . . . . . . . . . 10
5.2. Group environment G . . . . . . . . . . . . . . . . . . . 11
5.3. Notation for string operations . . . . . . . . . . . . . 11
5.4. Notation for group operations . . . . . . . . . . . . . . 13
6. The CPace protocol . . . . . . . . . . . . . . . . . . . . . 13
6.1. Protocol flow . . . . . . . . . . . . . . . . . . . . . . 13
6.2. CPace protocol instructions . . . . . . . . . . . . . . . 13
7. Implementation of recommended CPace cipher suites . . . . . . 14
7.1. Common function for computing generators . . . . . . . . 14
7.2. CPace group objects G_X25519 and G_X448 for
single-coordinate Ladders on Montgomery curves . . . . . 15
7.2.1. Verification tests . . . . . . . . . . . . . . . . . 16
7.3. CPace group objects G_Ristretto255 and G_Decaf448 for
prime-order group abstractions . . . . . . . . . . . . . 17
7.3.1. Verification tests . . . . . . . . . . . . . . . . . 19
7.4. CPace group objects for curves in Short-Weierstrass
representation . . . . . . . . . . . . . . . . . . . . . 19
7.4.1. Curves and associated functions . . . . . . . . . . . 19
7.4.2. Suitable encode_to_curve methods . . . . . . . . . . 20
7.4.3. Definition of the group environment G for
Short-Weierstrass curves . . . . . . . . . . . . . . 20
7.4.4. Verification tests . . . . . . . . . . . . . . . . . 22
8. Implementation verification . . . . . . . . . . . . . . . . . 22
9. Security Considerations . . . . . . . . . . . . . . . . . . . 22
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9.1. Party identifiers and relay attacks . . . . . . . . . . . 22
9.2. Network message encoding and hashing protocol
transcripts . . . . . . . . . . . . . . . . . . . . . . . 23
9.3. Key derivation . . . . . . . . . . . . . . . . . . . . . 23
9.4. Key confirmation . . . . . . . . . . . . . . . . . . . . 23
9.5. Sampling of scalars . . . . . . . . . . . . . . . . . . . 24
9.6. Single-coordinate CPace on Montgomery curves . . . . . . 25
9.7. Nonce values . . . . . . . . . . . . . . . . . . . . . . 25
9.8. Side channel attacks . . . . . . . . . . . . . . . . . . 25
9.9. Quantum computers . . . . . . . . . . . . . . . . . . . . 26
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 26
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 26
12. References . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.1. Normative References . . . . . . . . . . . . . . . . . . 26
12.2. Informative References . . . . . . . . . . . . . . . . . 27
Appendix A. CPace function definitions . . . . . . . . . . . . . 29
A.1. Definition and test vectors for string utility
functions . . . . . . . . . . . . . . . . . . . . . . . . 29
A.1.1. prepend_len function . . . . . . . . . . . . . . . . 29
A.1.2. prepend_len test vectors . . . . . . . . . . . . . . 29
A.1.3. lv_cat function . . . . . . . . . . . . . . . . . . . 29
A.1.4. Testvector for lv_cat() . . . . . . . . . . . . . . . 30
A.1.5. Examples for messages not obtained from a lv_cat-based
encoding . . . . . . . . . . . . . . . . . . . . . . 30
A.2. Definition of generator_string function. . . . . . . . . 30
A.3. Definitions and test vector ordered concatenation . . . . 30
A.3.1. Definitions for lexiographical ordering . . . . . . . 30
A.3.2. Definitions for ordered concatenation . . . . . . . . 31
A.3.3. Test vectors ordered concatenation . . . . . . . . . 31
A.4. Decoding and Encoding functions according to RFC7748 . . 31
A.5. Elligator 2 reference implementation . . . . . . . . . . 32
Appendix B. Test vectors . . . . . . . . . . . . . . . . . . . . 33
B.1. Test vector for CPace using group X25519 and hash
SHA-512 . . . . . . . . . . . . . . . . . . . . . . . . . 33
B.1.1. Test vectors for calculate_generator with group
X25519 . . . . . . . . . . . . . . . . . . . . . . . 33
B.1.2. Test vector for MSGa . . . . . . . . . . . . . . . . 33
B.1.3. Test vector for MSGb . . . . . . . . . . . . . . . . 34
B.1.4. Test vector for secret points K . . . . . . . . . . . 34
B.1.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 34
B.1.6. Test vector for ISK calculation parallel execution . 35
B.1.7. Corresponding ANSI-C initializers . . . . . . . . . . 35
B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order
points . . . . . . . . . . . . . . . . . . . . . . . 37
B.2. Test vector for CPace using group X448 and hash
SHAKE-256 . . . . . . . . . . . . . . . . . . . . . . . . 38
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B.2.1. Test vectors for calculate_generator with group
X448 . . . . . . . . . . . . . . . . . . . . . . . . 38
B.2.2. Test vector for MSGa . . . . . . . . . . . . . . . . 38
B.2.3. Test vector for MSGb . . . . . . . . . . . . . . . . 38
B.2.4. Test vector for secret points K . . . . . . . . . . . 39
B.2.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 39
B.2.6. Test vector for ISK calculation parallel execution . 40
B.2.7. Corresponding ANSI-C initializers . . . . . . . . . . 40
B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order
points . . . . . . . . . . . . . . . . . . . . . . . 42
B.3. Test vector for CPace using group ristretto255 and hash
SHA-512 . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.3.1. Test vectors for calculate_generator with group
ristretto255 . . . . . . . . . . . . . . . . . . . . 43
B.3.2. Test vector for MSGa . . . . . . . . . . . . . . . . 44
B.3.3. Test vector for MSGb . . . . . . . . . . . . . . . . 44
B.3.4. Test vector for secret points K . . . . . . . . . . . 45
B.3.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 45
B.3.6. Test vector for ISK calculation parallel execution . 45
B.3.7. Corresponding ANSI-C initializers . . . . . . . . . . 46
B.3.8. Test case for scalar_mult with valid inputs . . . . . 47
B.3.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 48
B.4. Test vector for CPace using group decaf448 and hash
SHAKE-256 . . . . . . . . . . . . . . . . . . . . . . . . 48
B.4.1. Test vectors for calculate_generator with group
decaf448 . . . . . . . . . . . . . . . . . . . . . . 48
B.4.2. Test vector for MSGa . . . . . . . . . . . . . . . . 49
B.4.3. Test vector for MSGb . . . . . . . . . . . . . . . . 49
B.4.4. Test vector for secret points K . . . . . . . . . . . 50
B.4.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 50
B.4.6. Test vector for ISK calculation parallel execution . 51
B.4.7. Corresponding ANSI-C initializers . . . . . . . . . . 51
B.4.8. Test case for scalar_mult with valid inputs . . . . . 53
B.4.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 53
B.5. Test vector for CPace using group NIST P-256 and hash
SHA-256 . . . . . . . . . . . . . . . . . . . . . . . . . 53
B.5.1. Test vectors for calculate_generator with group NIST
P-256 . . . . . . . . . . . . . . . . . . . . . . . . 53
B.5.2. Test vector for MSGa . . . . . . . . . . . . . . . . 54
B.5.3. Test vector for MSGb . . . . . . . . . . . . . . . . 54
B.5.4. Test vector for secret points K . . . . . . . . . . . 55
B.5.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 55
B.5.6. Test vector for ISK calculation parallel execution . 56
B.5.7. Corresponding ANSI-C initializers . . . . . . . . . . 57
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B.5.8. Test case for scalar_mult_vfy with correct inputs . . 58
B.5.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 59
B.6. Test vector for CPace using group NIST P-384 and hash
SHA-384 . . . . . . . . . . . . . . . . . . . . . . . . . 59
B.6.1. Test vectors for calculate_generator with group NIST
P-384 . . . . . . . . . . . . . . . . . . . . . . . . 59
B.6.2. Test vector for MSGa . . . . . . . . . . . . . . . . 60
B.6.3. Test vector for MSGb . . . . . . . . . . . . . . . . 61
B.6.4. Test vector for secret points K . . . . . . . . . . . 61
B.6.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 61
B.6.6. Test vector for ISK calculation parallel execution . 62
B.6.7. Corresponding ANSI-C initializers . . . . . . . . . . 63
B.6.8. Test case for scalar_mult_vfy with correct inputs . . 65
B.6.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 65
B.7. Test vector for CPace using group NIST P-521 and hash
SHA-512 . . . . . . . . . . . . . . . . . . . . . . . . . 66
B.7.1. Test vectors for calculate_generator with group NIST
P-521 . . . . . . . . . . . . . . . . . . . . . . . . 66
B.7.2. Test vector for MSGa . . . . . . . . . . . . . . . . 66
B.7.3. Test vector for MSGb . . . . . . . . . . . . . . . . 67
B.7.4. Test vector for secret points K . . . . . . . . . . . 68
B.7.5. Test vector for ISK calculation initiator/
responder . . . . . . . . . . . . . . . . . . . . . . 68
B.7.6. Test vector for ISK calculation parallel execution . 69
B.7.7. Corresponding ANSI-C initializers . . . . . . . . . . 70
B.7.8. Test case for scalar_mult_vfy with correct inputs . . 72
B.7.9. Invalid inputs for scalar_mult_vfy . . . . . . . . . 73
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 73
1. Introduction
This document describes CPace which is a balanced Password-
Authenticated-Key-Establishment (PAKE) protocol for two parties where
both parties derive a cryptographic key of high entropy from a shared
secret of low-entropy. CPace protects the passwords against offline
dictionary attacks by requiring adversaries to actively interact with
a protocol party and by allowing for at most one single password
guess per active interaction.
The CPace design was tailored considering the following main
objectives:
* Efficiency: Deployment of CPace is feasible on resource-
constrained devices.
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* Versatility: CPace supports different application scenarios via
versatile input formats, and by supporting applications with and
without clear initiator and responder roles.
* Implementation error resistance: CPace aims at avoiding common
implementation pitfalls already by-design, such as avoiding
incentives for insecure execution-time speed optimizations. For
smooth integration into different cryptographic library
ecosystems, this document provides a variety of cipher suites.
* Post-quantum annoyance: CPace comes with mitigations with respect
to adversaries that become capable of breaking the discrete
logarithm problem on elliptic curves.
1.1. Outline of this document
* Section 3 describes the expected properties of an application
using CPace, and discusses in particular which application-level
aspects are relevant for CPace's security.
* Section 4 gives an overview over the recommended cipher suites for
CPace which were optimized for different types of cryptographic
library ecosystems.
* Section 5 introduces the notation used throughout this document.
* Section 6 specifies the CPace protocol.
* The final section provides explicit reference implementations and
test vectors of all of the functions defined for CPace in the
appendix.
As this document is primarily written for implementers and
application designers, we would like to refer the theory-inclined
reader to the scientific paper [AHH21] which covers the detailed
security analysis of the different CPace instantiations as defined in
this document via the cipher suites.
2. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
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3. High-level application perspective
CPace enables balanced password-authenticated key establishment.
CPace requires a shared secret octet string, the password-related
string (PRS), is available for both parties A and B. PRS can be a
low-entropy secret itself, for instance a clear-text password encoded
according to [RFC8265], or any string derived from a common secret,
for instance by use of a password-based key derivation function.
Applications with clients and servers where the server side is
storing account and password information in its persistent memory are
recommended to use augmented PAKE protocols such as OPAQUE
[I-D.irtf-cfrg-opaque].
In the course of the CPace protocol, A sends one message MSGa to B
and B sends one message MSGb to A. CPace does not mandate any
ordering of these two messages. We use the term "initiator-
responder" for CPace where A always speaks first, and the term
"symmetric" setting where anyone can speak first.
CPace's output is an intermediate session key (ISK), but any party
might abort in case of an invalid received message. A and B will
produce the same ISK value only if both sides did initiate the
protocol using the same protocol inputs, specifically the same PRS
string and the same value for the optional input parameters CI, ADa,
ADb and sid that will be specified in the upcoming sections.
The naming of ISK key as "intermediate" session key highlights the
fact that it is RECOMMENDED that applications process ISK by use of a
suitable strong key derivation function KDF (such as defined in
[RFC5869]) before using the key in a higher-level protocol.
3.1. Optional CPace inputs
For accomodating different application settings, CPace offers the
following OPTIONAL inputs, i.e. inputs which MAY also be the empty
string:
* Channel identifier (CI). CI can be used to bind a session key
exchanged with CPace to a specific networking channel which
interconnects the protocol parties. Both parties are required to
have the same view of CI. CI will not be publicly sent on the
wire and may also include confidential information.
* Associated data fields (ADa and ADb). These fields can be used to
authenticate public associated data alongside the CPace protocol.
The values ADa (and ADb, respectively) are guaranteed to be
authenticated in case both parties agree on a key.
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ADa and ADb can for instance include party identities or protocol
version information of an application protocol (e.g. to avoid
downgrade attacks).
If party identities are not encoded as part of CI, party
identities SHOULD be included in ADa and ADb (see Section 9.1).
In a setting with clear initiator and responder roles, identity
information in ADa sent by the initiator can be used by the
responder for choosing the right PRS string (respectively
password) for this identity.
* Session identifier (sid). CPace comes with a security analysis
[AHH21] in the framework of universal composability. This
framework allows for modular analysis of a larger application
protocol which uses CPace as a building block. For such analysis
the CPace protocol is bound to a specific session of the larger
protocol by use of a sid string that is globally unique. As a
result, when used with a unique sid, CPace instances remain secure
when running concurrently with other CPace instances, and even
arbitrary other protocols.
For this reason, it is RECOMMENDED that applications establish a
unique session identifier sid prior to running the CPace protocol.
This can be implemented by concatenating random bytes produced by
A with random bytes produced by B. If such preceding round is not
an option but parties are assigned clear initiator-responder
roles, it is RECOMMENDED to let the initiator A choose a fresh
random sid and send it to B together with the first message. If a
sid string is used it SHOULD HAVE a length of at least 8 bytes.
3.2. Responsibilities of the application layer
The following tasks are out of the scope of this document and left to
the application layer
* Setup phase:
- The application layer is responsible for the handshake that
makes parties agree on a common CPace cipher suite.
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- The application layer needs to specify how to encode the CPace
byte strings Ya / Yb and ADa / ADb defined in section Section 6
for transfer over the network. For CPace it is RECOMMENDED to
encode network messages by using MSGa = lv_cat(Ya,ADa) and MSGb
= lv_cat(Yb,ADb) using the length-value concatenation function
lv_cat speficied in Section 5.3. This document provides test
vectors for lv_cat-encoded messages. Alternative network
encodings, e.g., the encoding method used for the client hello
and server hello messages of the TLS protocol, MAY be used when
considering the guidance given in Section 9.
* This document does not specify which encodings applications use
for the mandatory PRS input and the optional inputs CI, sid, ADa
and ADb. If PRS is a clear-text password or an octet string
derived from a clear-text password, e.g. by use of a key-
derivation function, the clear-text password SHOULD BE encoded
according to [RFC8265].
* The application needs to settle whether CPace is used in the
initiator-responder or the symmetric setting, as in the symmetric
setting transcripts must be generated using ordered string
concatenation. In this document we will provide test vectors for
both, initiator-responder and symmetric settings.
4. CPace cipher suites
In the setup phase of CPace, both communication partners need to
agree on a common cipher suite. Cipher suites consist of a
combination of a hash function H and an elliptic curve environment G.
For naming cipher suites we use the convention "CPACE-G-H". We
RECOMMEND the following cipher suites:
* CPACE-X25519-SHA512. This suite uses the group environment
G_X25519 defined in Section 7.2 and SHA-512 as hash function.
This cipher suite comes with the smallest messages on the wire and
a low computational cost.
* CPACE-P256_XMD:SHA-256_SSWU_NU_-SHA256. This suite instantiates
the group environment G as specified in Section 7.4 using the
encode_to_curve function P256_XMD:SHA-256_SSWU_NU_ from [RFC9380]
on curve NIST-P256, and hash function SHA-256.
The following RECOMMENDED cipher suites provide higher security
margins.
* CPACE-X448-SHAKE256. This suite uses the group environment G_X448
defined in Section 7.2 and SHAKE-256 as hash function.
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* CPACE-P384_XMD:SHA-384_SSWU_NU_-SHA384. This suite instantiates G
as specified in Section 7.4 using the encode_to_curve function
P384_XMD:SHA-384_SSWU_NU_ from [RFC9380] on curve NIST-P384 with H
= SHA-384.
* CPACE-P521_XMD:SHA-512_SSWU_NU_-SHA512. This suite instantiates G
as specified in Section 7.4 using the encode_to_curve function
P521_XMD:SHA-512_SSWU_NU_ from [RFC9380] on curve NIST-P521 with H
= SHA-512.
CPace can also securely be implemented using the cipher suites CPACE-
RISTR255-SHA512 and CPACE-DECAF448-SHAKE256 defined in Section 7.3.
Section 9 gives guidance on how to implement CPace on further
elliptic curves.
5. Definitions and notation
5.1. Hash function H
Common choices for H are SHA-512 [RFC6234] or SHAKE-256 [FIPS202].
(I.e. the hash function outputs octet strings, and not group
elements.) For considering both variable-output-length hashes and
fixed-output-length hashes, we use the following convention. In case
that the hash function is specified for a fixed-size output, we
define H.hash(m,l) such that it returns the first l octets of the
output.
We use the following notation for referring to the specific
properties of a hash function H:
* H.hash(m,l) is a function that operates on an input octet string m
and returns a hashing result of l octets.
* H.b_in_bytes denotes the default output size in bytes
corresponding to the symmetric security level of the hash
function. E.g. H.b_in_bytes = 64 for SHA-512 and SHAKE-256 and
H.b_in_bytes = 32 for SHA-256 and SHAKE-128. We use the notation
H.hash(m) = H.hash(m, H.b_in_bytes) and let the hash operation
output the default length if no explicit length parameter is
given.
* H.bmax_in_bytes denotes the _maximum_ output size in octets
supported by the hash function. In case of fixed-size hashes such
as SHA-256, this is the same as H.b_in_bytes, while there is no
such limit for hash functions such as SHAKE-256.
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* H.s_in_bytes denotes the _input block size_ used by H. For
instance, for SHA-512 the input block size s_in_bytes is 128,
while for SHAKE-256 the input block size amounts to 136 bytes.
5.2. Group environment G
The group environment G specifies an elliptic curve group (also
denoted G for convenience) and associated constants and functions as
detailed below. In this document we use multiplicative notation for
the group operation.
* G.calculate_generator(H,PRS,CI,sid) denotes a function that
outputs a representation of a generator (referred to as
"generator" from now on) of the group which is derived from input
octet strings PRS, CI, and sid and with the help of the hash
function H.
* G.sample_scalar() is a function returning a representation of a
scalar (referred to as "scalar" from now on) appropriate as a
private Diffie-Hellman key for the group.
* G.scalar_mult(y,g) is a function operating on a scalar y and a
group element g. It returns an octet string representation of the
group element Y = g^y.
* G.I denotes a unique octet string representation of the neutral
element of the group. G.I is used for detecting and signaling
certain error conditions.
* G.scalar_mult_vfy(y,g) is a function operating on a scalar y and a
group element g. It returns an octet string representation of the
group element g^y. Additionally, scalar_mult_vfy specifies
validity conditions for y,g and g^y and outputs G.I in case they
are not met.
* G.DSI denotes a domain-separation identifier string which SHALL be
uniquely identifying the group environment G.
5.3. Notation for string operations
* bytes1 || bytes2 and denotes concatenation of octet strings.
* len(S) denotes the number of octets in a string S.
* nil denotes an empty octet string, i.e., len(nil) = 0.
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* prepend_len(octet_string) denotes the octet sequence that is
obtained from prepending the length of the octet string to the
string itself. The length shall be prepended by using an LEB128
encoding of the length. This will result in a single-byte
encoding for values below 128. (Test vectors and reference
implementations for prepend_len and the LEB128 encodings are given
in the appendix.)
* lv_cat(a0,a1, ...) is the "length-value" encoding function which
returns the concatenation of the input strings with an encoding of
their respective length prepended. E.g. lv_cat(a0,a1) returns
prepend_len(a0) || prepend_len(a1). The detailed specification of
lv_cat and a reference implementations are given in the appendix.
* network_encode(Y,AD) denotes the function specified by the
application layer that outputs an octet string encoding of the
input octet strings Y and AD for transfer on the network. The
implementation of MSG = network_encode(Y,AD) SHALL allow the
receiver party to parse MSG for the individual subcomponents Y and
AD. For CPace we RECOMMEND to implement network_encode(Y,AD) as
network_encode(Y,AD) = lv_cat(Y,AD).
Other encodings, such as the network encoding used for the client-
hello and server-hello messages in TLS MAY also be used when
following the guidance given in the security consideration
section.
* sample_random_bytes(n) denotes a function that returns n octets
uniformly distributed between 0 and 255.
* zero_bytes(n) denotes a function that returns n octets with value
0.
* oCat(bytes1,bytes2) denotes ordered concatenation of octet
strings, which places the lexiographically larger octet string
first. (Explicit reference code for this function is given in the
appendix.)
* transcript(MSGa,MSGb) denotes function outputing a string for the
protocol transcript with messages MSGa and MSGb. In applications
where CPace is used without clear initiator and responder roles,
i.e. where the ordering of messages is not enforced by the
protocol flow, transcript(MSGa,MSGb) = oCat(MSGa,MSGb) SHOULD be
used. In the initiator-responder setting transcript(MSGa,MSGb)
SHOULD BE implemented such that the later message is appended to
the earlier message, i.e., transcript(MSGa,MSGb) = MSGa||MSGb if
MSGa is sent first.
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5.4. Notation for group operations
We use multiplicative notation for the group, i.e., X^2 denotes the
element that is obtained by computing X*X, for group element X and
group operation *.
6. The CPace protocol
CPace is a one round protocol between two parties, A and B. At
invocation, A and B are provisioned with PRS,G,H and OPTIONAL
CI,sid,ADa (for A) and CI,sid,ADb (for B). A sends a message MSGa to
B. MSGa contains the public share Ya and OPTIONAL associated data
ADa (i.e. an ADa field that MAY have a length of 0 bytes). Likewise,
B sends a message MSGb to A. MSGb contains the public share Yb and
OPTIONAL associated data ADb (i.e. an ADb field that MAY have a
length of 0 bytes). Both A and B use the received messages for
deriving a shared intermediate session key, ISK.
6.1. Protocol flow
Optional parameters and messages are denoted with [].
public: G, H, [CI], [sid]
A: PRS,[ADa] B: PRS,[ADb]
---------------------------------------
compute Ya | Ya,[ADa] | compute Yb
|----------------->|
| Yb,[ADb] |
verify inputs |<-----------------| verify inputs
derive ISK | | derive ISK
---------------------------------------
output ISK output ISK
6.2. CPace protocol instructions
A computes a generator g = G.calculate_generator(H,PRS,CI,sid),
scalar ya = G.sample_scalar() and group element Ya = G.scalar_mult
(ya,g). A then transmits MSGa = network_encode(Ya, ADa) with
optional associated data ADa to B.
B computes a generator g = G.calculate_generator(H,PRS,CI,sid),
scalar yb = G.sample_scalar() and group element Yb =
G.scalar_mult(yb,g). B sends MSGb = network_encode(Yb, ADb) with
optional associated data ADb to A.
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Upon reception of MSGa, B checks that MSGa was properly generated
conform with the chosen encoding of network messages (notably correct
length fields). If this parsing fails, then B MUST abort.
(Testvectors of examples for invalid messages when using lv_cat() as
network_encode function for CPace are given in the appendix.) B then
computes K = G.scalar_mult_vfy(yb,Ya). B MUST abort if K=G.I.
Otherwise B returns ISK = H.hash(lv_cat(G.DSI || "_ISK", sid,
K)||transcript(MSGa, MSGb)). B returns ISK and terminates.
Likewise upon reception of MSGb, A parses MSGb for Yb and ADb and
checks for a valid encoding. If this parsing fails, then A MUST
abort. A then computes K = G.scalar_mult_vfy(ya,Yb). A MUST abort
if K=G.I. Otherwise A returns ISK = H.hash(lv_cat(G.DSI || "_ISK",
sid, K) || transcript(MSGa, MSGb)). A returns ISK and terminates.
The session key ISK returned by A and B is identical if and only if
the supplied input parameters PRS, CI and sid match on both sides and
transcript view (containing of MSGa and MSGb) of both parties match.
(Note that in case of a symmetric protocol execution without clear
initiator/responder roles, transcript(MSGa, MSGb) needs to be
implemented using ordered concatenation for generating a matching
view by both parties.)
7. Implementation of recommended CPace cipher suites
7.1. Common function for computing generators
The different cipher suites for CPace defined in the upcoming
sections share the same method for deterministically combining the
individual strings PRS, CI, sid and the domain-separation identifier
DSI to a generator string that we describe here.
* generator_string(DSI, PRS, CI, sid, s_in_bytes) denotes a function
that returns the string lv_cat(DSI, PRS, zero_bytes(len_zpad), CI,
sid).
* len_zpad = MAX(0, s_in_bytes - len(prepend_len(PRS)) -
len(prepend_len(G.DSI)) - 1)
The zero padding of length len_zpad is designed such that the
encoding of DSI and PRS together with the zero padding field
completely fills the first input block (of length s_in_bytes) of the
hash. As a result for the common case of short PRS the number of
bytes to hash becomes independent of the actual length of the
password (PRS). (A reference implementation and test vectors are
provided in the appendix.)
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The introduction of a zero-padding within the generator string also
helps mitigating attacks of a side-channel adversary that analyzes
correlations between publicly known variable information with the
low-entropy PRS string. Note that the hash of the first block is
intentionally made independent of session-specific inputs, such as
sid or CI.
7.2. CPace group objects G_X25519 and G_X448 for single-coordinate
Ladders on Montgomery curves
In this section we consider the case of CPace when using the X25519
and X448 Diffie-Hellman functions from [RFC7748] operating on the
Montgomery curves Curve25519 and Curve448 [RFC7748]. CPace
implementations using single-coordinate ladders on further Montgomery
curves SHALL use the definitions in line with the specifications for
X25519 and X448 and review the guidance given in Section 9.
For the group environment G_X25519 the following definitions apply:
* G_X25519.field_size_bytes = 32
* G_X25519.field_size_bits = 255
* G_X25519.sample_scalar() = sample_random_bytes(G.field_size_bytes)
* G_X25519.scalar_mult(y,g) = G.scalar_mult_vfy(y,g) = X25519(y,g)
* G_X25519.I = zero_bytes(G.field_size_bytes)
* G_X25519.DSI = "CPace255"
CPace cipher suites using G_X25519 MUST use a hash function producing
at least H.b_max_in_bytes >= 32 bytes of output. It is RECOMMENDED
to use G_X25519 in combination with SHA-512.
For X448 the following definitions apply:
* G_X448.field_size_bytes = 56
* G_X448.field_size_bits = 448
* G_X448.sample_scalar() = sample_random_bytes(G.field_size_bytes)
* G_X448.scalar_mult(y,g) = G.scalar_mult_vfy(y,g) = X448(y,g)
* G_X448.I = zero_bytes(G.field_size_bytes)
* G_X448.DSI = "CPace448"
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CPace cipher suites using G_X448 MUST use a hash function producing
at least H.b_max_in_bytes >= 56 bytes of output. It is RECOMMENDED
to use G_X448 in combination with SHAKE-256.
For both G_X448 and G_X25519 the G.calculate_generator(H, PRS,sid,CI)
function shall be implemented as follows.
* First gen_str = generator_string(G.DSI,PRS,CI,sid, H.s_in_bytes)
SHALL BE calculated using the input block size of the chosen hash
function.
* This string SHALL then BE hashed to the required length
gen_str_hash = H.hash(gen_str, G.field_size_bytes). Note that
this implies that the permissible output length H.maxb_in_bytes
MUST BE larger or equal to the field size of the group G for
making a hashing function suitable.
* This result is then considered as a field coordinate using the u =
decodeUCoordinate(gen_str_hash, G.field_size_bits) function from
[RFC7748] which we repeat in the appendix for convenience.
* The result point g is then calculated as (g,v) =
map_to_curve_elligator2(u) using the function from [RFC9380].
Note that the v coordinate produced by the map_to_curve_elligator2
function is not required for CPace and discarded. The appendix
repeats the definitions from [RFC9380] for convenience.
In the appendix we show sage code that can be used as reference
implementation.
7.2.1. Verification tests
For single-coordinate Montgomery ladders on Montgomery curves
verification tests according to Section 8 SHALL check for proper
handling of the abort conditions, when a party is receiving u
coordinate values that encode a low-order point on either, the curve
or the quadratic twist.
In addition to that in case of G_X25519 the tests SHALL also verify
that the implementation of G.scalar_mult_vfy(y,g) produces the
expected results for non-canonical u coordinate values with bit #255
set, which may also encode low-order points.
Corresponding test vectors are provided in the appendix.
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7.3. CPace group objects G_Ristretto255 and G_Decaf448 for prime-order
group abstractions
In this section we consider the case of CPace using the Ristretto255
and Decaf448 group abstractions
[I-D.draft-irtf-cfrg-ristretto255-decaf448]. These abstractions
define an encode and decode function, group operations using an
internal encoding and an element-derivation function that maps a byte
string to a group element. With the group abstractions there is a
distinction between an internal representation of group elements and
an external encoding of the same group element. In order to
distinguish between these different representations, we prepend an
underscore before values using the internal representation within
this section.
For Ristretto255 the following definitions apply:
* G_Ristretto255.DSI = "CPaceRistretto255"
* G_Ristretto255.field_size_bytes = 32
* G_Ristretto255.group_size_bits = 252
* G_Ristretto255.group_order = 2^252 +
27742317777372353535851937790883648493
CPace cipher suites using G_Ristretto255 MUST use a hash function
producing at least H.b_max_in_bytes >= 64 bytes of output. It is
RECOMMENDED to use G_Ristretto255 in combination with SHA-512.
For decaf448 the following definitions apply:
* G_Decaf448.DSI = "CPaceDecaf448"
* G_Decaf448.field_size_bytes = 56
* G_Decaf448.group_size_bits = 445
* G_Decaf448.group_order = l = 2^446 -
1381806680989511535200738674851542
6880336692474882178609894547503885
CPace cipher suites using G_Decaf448 MUST use a hash function
producing at least H.b_max_in_bytes >= 112 bytes of output. It is
RECOMMENDED to use G_Decaf448 in combination with SHAKE-256.
For both abstractions the following definitions apply:
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* It is RECOMMENDED to implement G.sample_scalar() as follows.
- Set scalar = sample_random_bytes(G.group_size_bytes).
- Then clear the most significant bits larger than
G.group_size_bits.
- Interpret the result as the little-endian encoding of an
integer value and return the result.
* Alternatively, if G.sample_scalar() is not implemented according
to the above recommendation, it SHALL be implemented using uniform
sampling between 1 and (G.group_order - 1). Note that the more
complex uniform sampling process can provide a larger side-channel
attack surface for embedded systems in hostile environments.
* G.scalar_mult(y,_g) SHALL operate on a scalar y and a group
element _g in the internal representation of the group abstraction
environment. It returns the value Y = encode((_g)^y), i.e. it
returns a value using the public encoding.
* G.I = is the public encoding representation of the identity
element.
* G.scalar_mult_vfy(y,X) operates on a value using the public
encoding and a scalar and is implemented as follows. If the
decode(X) function fails, it returns G.I. Otherwise it returns
encode( decode(X)^y ).
* The G.calculate_generator(H, PRS,sid,CI) function SHALL return a
decoded point and SHALL BE implemented as follows.
- First gen_str = generator_string(G.DSI,PRS,CI,sid,
H.s_in_bytes) is calculated using the input block size of the
chosen hash function.
- This string is then hashed to the required length gen_str_hash
= H.hash(gen_str, 2 * G.field_size_bytes). Note that this
implies that the permissible output length H.maxb_in_bytes MUST
BE larger or equal to twice the field size of the group G for
making a hash function suitable.
- Finally the internal representation of the generator _g is
calculated as _g = element_derivation(gen_str_hash) using the
element derivation function from the abstraction.
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Note that with these definitions the scalar_mult function operates on
a decoded point _g and returns an encoded point, while the
scalar_mult_vfy(y,X) function operates on an encoded point X (and
also returns an encoded point).
7.3.1. Verification tests
For group abstractions verification tests according to Section 8
SHALL check for proper handling of the abort conditions, when a party
is receiving encodings of the neutral element or receives an octet
string that does not decode to a valid group element.
7.4. CPace group objects for curves in Short-Weierstrass representation
The group environment objects G defined in this section for use with
Short-Weierstrass curves, are parametrized by the choice of an
elliptic curve and by choice of a suitable encode_to_curve function.
encode_to_curve must map an octet string to a point on the curve.
7.4.1. Curves and associated functions
Elliptic curves in Short-Weierstrass form are considered in
[IEEE1363]. [IEEE1363] allows for both, curves of prime and non-
prime order. However, for the procedures described in this section
any suitable group MUST BE of prime order.
The specification for the group environment objects specified in this
section closely follow the ECKAS-DH1 method from [IEEE1363]. I.e. we
use the same methods and encodings and protocol substeps as employed
in the TLS [RFC5246] [RFC8446] protocol family.
For CPace only the uncompressed full-coordinate encodings from [SEC1]
(x and y coordinate) SHOULD be used. Commonly used curve groups are
specified in [SEC2] and [RFC5639]. A typical representative of such
a Short-Weierstrass curve is NIST-P256. Point verification as used
in ECKAS-DH1 is described in Annex A.16.10. of [IEEE1363].
For deriving Diffie-Hellman shared secrets ECKAS-DH1 from [IEEE1363]
specifies the use of an ECSVDP-DH method. We use ECSVDP-DH in
combination with the identy map such that it either returns "error"
or the x-coordinate of the Diffie-Hellman result point as shared
secret in big endian format (fixed length output by FE2OSP without
truncating leading zeros).
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7.4.2. Suitable encode_to_curve methods
All the encode_to_curve methods specified in [RFC9380] are suitable
for CPace. For Short-Weierstrass curves it is RECOMMENDED to use the
non-uniform variant of the SSWU mapping primitive from [RFC9380] if a
SSWU mapping is available for the chosen curve. (We recommend non-
uniform maps in order to give implementations the flexibility to opt
for x-coordinate-only scalar multiplication algorithms.)
7.4.3. Definition of the group environment G for Short-Weierstrass
curves
In this paragraph we use the following notation for defining the
group object G for a selected curve and encode_to_curve method:
* With G.group_order we denote the order of the elliptic curve which
MUST BE a prime.
* With is_valid(X) we denote a method which operates on an octet
stream according to [SEC1] of a point on the group and returns
true if the point is valid and returns false otherwise. This
is_valid(X) method SHALL be implemented according to Annex
A.16.10. of [IEEE1363]. I.e. it shall return false if X encodes
either the neutral element on the group or does not form a valid
encoding of a point on the group.
* With encode_to_curve(str,DST) we denote a mapping function from
[RFC9380]. I.e. a function that maps octet string str to a point
on the group using the domain separation tag DST. [RFC9380]
considers both, uniform and non-uniform mappings based on several
different strategies. It is RECOMMENDED to use the nonuniform
variant of the SSWU mapping primitive within [RFC9380].
* G.DSI denotes a domain-separation identifier string. G.DSI which
SHALL BE obtained by the concatenation of "CPace" and the
associated name of the cipher suite used for the encode_to_curve
function as specified in [RFC9380]. E.g. when using the map with
the name "P384_XMD:SHA-384_SSWU_NU_" on curve NIST-P384 the
resulting value SHALL BE G.DSI = "CPaceP384_XMD:SHA-384_SSWU_NU_".
Using the above definitions, the CPace functions required for the
group object G are defined as follows.
* G.DST denotes the domain-separation tag value to use in
conjunction with the encode_to_curve function from [RFC9380].
G.DST shall be obtained by concatenating G.DSI and "_DST".
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* G.sample_scalar() SHALL return a value between 1 and
(G.group_order - 1). The value sampling MUST BE uniformly random.
It is RECOMMENDED to use rejection sampling for converting a
uniform bitstring to a uniform value between 1 and (G.group_order
- 1).
* G.calculate_generator(H, PRS,sid,CI) function SHALL be implemented
as follows.
- First gen_str = generator_string(G.DSI,PRS,CI,sid,
H.s_in_bytes) is calculated.
- Then the output of a call to encode_to_curve(gen_str, G.DST) is
returned, using the selected suite from [RFC9380].
* G.scalar_mult(s,X) is a function that operates on a scalar s and
an input point X. The input X shall use the same encoding as
produced by the G.calculate_generator method above.
G.scalar_mult(s,X) SHALL return an encoding of either the point
X^s or the point X^(-s) according to [SEC1]. Implementations
SHOULD use the full-coordinate format without compression, as
important protocols such as TLS 1.3 removed support for
compression. Implementations of scalar_mult(s,X) MAY output
either X^s or X^(-s) as both points X^s and X^(-s) have the same
x-coordinate and result in the same Diffie-Hellman shared secrets
K. (This allows implementations to opt for x-coordinate-only
scalar multiplication algorithms.)
* G.scalar_mult_vfy(s,X) merges verification of point X according to
[IEEE1363] A.16.10. and the the ECSVDP-DH procedure from
[IEEE1363]. It SHALL BE implemented as follows:
- If is_valid(X) = False then G.scalar_mult_vfy(s,X) SHALL return
"error" as specified in [IEEE1363] A.16.10 and 7.2.1.
- Otherwise G.scalar_mult_vfy(s,X) SHALL return the result of the
ECSVDP-DH procedure from [IEEE1363] (section 7.2.1). I.e. it
shall either return "error" (in case that X^s is the neutral
element) or the secret shared value "z" (otherwise). "z" SHALL
be encoded by using the big-endian encoding of the x-coordinate
of the result point X^s according to [SEC1].
* We represent the neutral element G.I by using the representation
of the "error" result case from [IEEE1363] as used in the
G.scalar_mult_vfy method above.
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7.4.4. Verification tests
For Short-Weierstrass curves verification tests according to
Section 8 SHALL check for proper handling of the abort conditions,
when a party is receiving an encoding of the point at infinity and an
encoding of a point not on the group.
8. Implementation verification
Any CPace implementation MUST be tested against invalid or weak point
attacks. Implementation MUST be verified to abort upon conditions
where G.scalar_mult_vfy functions outputs G.I. For testing an
implementation it is RECOMMENDED to include weak or invalid point
encodings within MSGa and MSGb and introduce this in a protocol run.
It SHALL be verified that the abort condition is properly handled.
Moreover regarding the network format any implementation MUST be
tested with respect invalid encodings of MSGa and MSGb. E.g. when
lv_cat is used as network format for encoding MSGa and MSGb, the sum
of the prepended lengths of the fields must be verified to match the
actual length of the message. Tests SHALL verify that a party aborts
in case that incorrectly encoded messages are recieved.
Corresponding test vectors are given in the appendix for all
recommended cipher suites.
9. Security Considerations
A security proof of CPace is found in [AHH21]. This proof covers all
recommended cipher suites included in this document. In the
following sections we describe how to protect CPace against several
attack families, such as relay-, length extension- or side channel
attacks. We also describe aspects to consider when deviating from
recommended cipher suites.
9.1. Party identifiers and relay attacks
If unique strings identifying the protocol partners are included
either as part of the channel identifier CI, the session id sid or
the associated data fields ADa, ADb, the ISK will provide implicit
authentication also regarding the party identities. Incorporating
party identifier strings is important for fending off relay attacks.
Such attacks become relevant in a setting where several parties, say,
A, B and C, share the same password PRS. An adversary might relay
messages from a honest user A, who aims at interacting with user B,
to a party C instead. If no party identifier strings are used, and B
and C use the same PRS value, A might be establishing a common ISK
key with C while assuming to interact with party B. Including and
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checking party identifiers can fend off such relay attacks.
9.2. Network message encoding and hashing protocol transcripts
It is RECOMMENDED to encode the (Ya,ADa) and (Yb,ADb) fields on the
network by using network_encode(Y,AD) = lv_cat(Y,AD). I.e. we
RECOMMEND to prepend an encoding of the length of the subfields.
Prepending the length of of all variable-size input strings results
in a so-called prefix-free encoding of transcript strings, using
terminology introduced in [CDMP05]. This property allows for
disregarding length-extension imperfections that come with the
commonly used Merkle-Damgard hash function constructions such as
SHA256 and SHA512.
Other alternative network encoding formats which prepend an encoding
of the length of variable-size data fields in the protocol messages
are equally suitable. This includes, e.g., the type-length-value
format specified in the DER encoding standard (X.690) or the protocol
message encoding used in the TLS protocol family for the TLS client-
hello or server-hello messages.
In case that an application uses another form of network message
encoding which is not prefix-free, the guidance given in [CDMP05]
SHOULD BE considered (e.g. by replacing hash functions with the HMAC
constructions from[RFC2104]).
9.3. Key derivation
Although already K is a shared value, it MUST NOT itself be used as
an application key. Instead, ISK MUST BE used. Leakage of K to an
adversary can lead to offline dictionary attacks.
As noted already in Section 6 it is RECOMMENDED to process ISK by use
of a suitable strong key derivation function KDF (such as defined in
[RFC5869]) first, before using the key in a higher-level protocol.
9.4. Key confirmation
In many applications it is advisable to add an explicit key
confirmation round after the CPace protocol flow. However, as some
applications might only require implicit authentication and as
explicit authentication messages are already a built-in feature in
many higher-level protocols (e.g. TLS 1.3) the CPace protocol
described here does not mandate use of a key confirmation on the
level of the CPace sub-protocol.
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Already without explicit key confirmation, CPace enjoys weak forward
security under the sCDH and sSDH assumptions [AHH21]. With added
explicit confirmation, CPace enjoys perfect forward security also
under the strong sCDH and sSDH assumptions [AHH21].
Note that in [ABKLX21] it was shown that an idealized variant of
CPace also enjoys perfect forward security without explicit key
confirmation. However this proof does not explicitly cover the
recommended cipher suites in this document and requires the stronger
assumption of an algebraic adversary model. For this reason, we
recommend adding explicit key confirmation if perfect forward
security is required.
When implementing explicit key confirmation, it is recommended to use
an appropriate message-authentication code (MAC) such as HMAC
[RFC2104] or CMAC [RFC4493] using a key mac_key derived from ISK.
One suitable option that works also in the parallel setting without
message ordering is to proceed as follows.
* First calculate mac_key as as mac_key = H.hash(b"CPaceMac" ||
ISK).
* Then let each party send an authenticator tag Ta, Tb that is
calculated over the protocol message that it has sent previously.
I.e. let party A calculate its transmitted authentication code Ta
as Ta = MAC(mac_key, MSGa) and let party B calculate its
transmitted authentication code Tb as Tb = MAC(mac_key, MSGb).
* Let the receiving party check the remote authentication tag for
the correct value and abort in case that it's incorrect.
9.5. Sampling of scalars
For curves over fields F_p where p is a prime close to a power of
two, we recommend sampling scalars as a uniform bit string of length
field_size_bits. We do so in order to reduce both, complexity of the
implementation and reducing the attack surface with respect to side-
channels for embedded systems in hostile environments. The effect of
non-uniform sampling on security was demonstrated to be begning in
[AHH21] for the case of Curve25519 and Curve448. This analysis
however does not transfer to most curves in Short-Weierstrass form.
As a result, we recommend rejection sampling if G is as in
Section 7.4.
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9.6. Single-coordinate CPace on Montgomery curves
The recommended cipher suites for the Montgomery curves Curve25519
and Curve448 in Section 7.2 rely on the following properties [AHH21]:
* The curve has order (p * c) with p prime and c a small cofactor.
Also the curve's quadratic twist must be of order (p' * c') with
p' prime and c' a cofactor.
* The cofactor c' of the twist MUST BE EQUAL to or an integer
multiple of the cofactor c of the curve.
* Both field order q and group order p MUST BE close to a power of
two along the lines of [AHH21], Appendix E.
* The representation of the neutral element G.I MUST BE the same for
both, the curve and its twist.
* The implementation of G.scalar_mult_vfy(y,X) MUST map all c low-
order points on the curve and all c' low-order points on the twist
to G.I.
Montgomery curves other than the ones recommended here can use the
specifications given in Section 7.2, given that the above properties
hold.
9.7. Nonce values
Secret scalars ya and yb MUST NOT be reused. Values for sid SHOULD
NOT be reused since the composability guarantees established by the
simulation-based proof rely on the uniqueness of session ids [AHH21].
If CPace is used in a concurrent system, it is RECOMMENDED that a
unique sid is generated by the higher-level protocol and passed to
CPace. One suitable option is that sid is generated by concatenating
ephemeral random strings contributed by both parties.
9.8. Side channel attacks
All state-of-the art methods for realizing constant-time execution
SHOULD be used. In case that side channel attacks are to be
considered practical for a given application, it is RECOMMENDED to
pay special attention on computing the secret generator
G.calculate_generator(PRS,CI,sid). The most critical substep to
consider might be the processing of the first block of the hash that
includes the PRS string. The zero-padding introduced when hashing
the sensitive PRS string can be expected to make the task for a side-
channel attack somewhat more complex. Still this feature alone is
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not sufficient for ruling out power analysis attacks.
9.9. Quantum computers
CPace is proven secure under the hardness of the strong computational
Simultaneous Diffie-Hellmann (sSDH) and strong computational Diffie-
Hellmann (sCDH) assumptions in the group G (as defined in [AHH21]).
These assumptions are not expected to hold any longer when large-
scale quantum computers (LSQC) are available. Still, even in case
that LSQC emerge, it is reasonable to assume that discrete-logarithm
computations will remain costly. CPace with ephemeral session id
values sid forces the adversary to solve one computational Diffie-
Hellman problem per password guess [ES21]. In this sense, using the
wording suggested by Steve Thomas on the CFRG mailing list, CPace is
"quantum-annoying".
10. IANA Considerations
No IANA action is required.
11. Acknowledgements
We would like to thank the participants on the CFRG list for comments
and advice. Any comment and advice is appreciated.
12. References
12.1. Normative References
[I-D.draft-irtf-cfrg-ristretto255-decaf448]
de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I.,
Tankersley, G., and F. Valsorda, "The ristretto255 and
decaf448 Groups", Work in Progress, Internet-Draft, draft-
irtf-cfrg-ristretto255-decaf448-08, 5 September 2023,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
ristretto255-decaf448-08>.
[I-D.irtf-cfrg-opaque]
Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The
OPAQUE Asymmetric PAKE Protocol", Work in Progress,
Internet-Draft, draft-irtf-cfrg-opaque-11, 8 June 2023,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
opaque-11>.
[IEEE1363] "Standard Specifications for Public Key Cryptography, IEEE
1363", 2000.
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[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/rfc/rfc2119>.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, <https://www.rfc-editor.org/rfc/rfc7748>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.
[SEC1] Standards for Efficient Cryptography Group (SECG), "SEC 1:
Elliptic Curve Cryptography", May 2009,
<http://www.secg.org/sec1-v2.pdf>.
12.2. Informative References
[ABKLX21] Abdalla, M., Barbosa, M., Katz, J., Loss, J., and J. Xu,
"Algebraic Adversaries in the Universal Composability
Framework.", n.d., <https://eprint.iacr.org/2021/1218>.
[AHH21] Abdalla, M., Haase, B., and J. Hesse, "Security analysis
of CPace", n.d., <https://eprint.iacr.org/2021/114>.
[CDMP05] Coron, J.-S., Dodis, Y., Malinaud, C., and P. Puniya,
"Merkle-Damgaard Revisited: How to Construct a Hash
Function", In Advances in Cryptology - CRYPTO 2005,
pages 430-448, DOI 10.1007/11535218_26, 2005,
<https://doi.org/10.1007/11535218_26>.
[ES21] Eaton, E. and D. Stebila, "The 'quantum annoying' property
of password-authenticated key exchange protocols.", n.d.,
<https://eprint.iacr.org/2021/696>.
[FIPS202] National Institute of Standards and Technology (NIST),
"SHA-3 Standard: Permutation-Based Hash and Extendable-
Output Functions", August 2015,
<https://nvlpubs.nist.gov/nistpubs/FIPS/
NIST.FIPS.202.pdf>.
[RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
Hashing for Message Authentication", RFC 2104,
DOI 10.17487/RFC2104, February 1997,
<https://www.rfc-editor.org/rfc/rfc2104>.
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[RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
2006, <https://www.rfc-editor.org/rfc/rfc4493>.
[RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security
(TLS) Protocol Version 1.2", RFC 5246,
DOI 10.17487/RFC5246, August 2008,
<https://www.rfc-editor.org/rfc/rfc5246>.
[RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
(ECC) Brainpool Standard Curves and Curve Generation",
RFC 5639, DOI 10.17487/RFC5639, March 2010,
<https://www.rfc-editor.org/rfc/rfc5639>.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
Key Derivation Function (HKDF)", RFC 5869,
DOI 10.17487/RFC5869, May 2010,
<https://www.rfc-editor.org/rfc/rfc5869>.
[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011,
<https://www.rfc-editor.org/rfc/rfc6234>.
[RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation,
Enforcement, and Comparison of Internationalized Strings
Representing Usernames and Passwords", RFC 8265,
DOI 10.17487/RFC8265, October 2017,
<https://www.rfc-editor.org/rfc/rfc8265>.
[RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
<https://www.rfc-editor.org/rfc/rfc8446>.
[RFC9380] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380,
DOI 10.17487/RFC9380, August 2023,
<https://www.rfc-editor.org/rfc/rfc9380>.
[SEC2] Standards for Efficient Cryptography Group (SECG), "SEC 2:
Recommended Elliptic Curve Domain Parameters", January
2010, <http://www.secg.org/sec2-v2.pdf>.
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Appendix A. CPace function definitions
A.1. Definition and test vectors for string utility functions
A.1.1. prepend_len function
def prepend_len(data):
"prepend LEB128 encoding of length"
length = len(data)
length_encoded = b""
while True:
if length < 128:
length_encoded += bytes([length])
else:
length_encoded += bytes([(length & 0x7f) + 0x80])
length = int(length >> 7)
if length == 0:
break;
return length_encoded + data
A.1.2. prepend_len test vectors
prepend_len(b""): (length: 1 bytes)
00
prepend_len(b"1234"): (length: 5 bytes)
0431323334
prepend_len(bytes(range(127))): (length: 128 bytes)
7f000102030405060708090a0b0c0d0e0f101112131415161718191a1b
1c1d1e1f202122232425262728292a2b2c2d2e2f303132333435363738
393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f505152535455
565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f707172
737475767778797a7b7c7d7e
prepend_len(bytes(range(128))): (length: 130 bytes)
8001000102030405060708090a0b0c0d0e0f101112131415161718191a
1b1c1d1e1f202122232425262728292a2b2c2d2e2f3031323334353637
38393a3b3c3d3e3f404142434445464748494a4b4c4d4e4f5051525354
55565758595a5b5c5d5e5f606162636465666768696a6b6c6d6e6f7071
72737475767778797a7b7c7d7e7f
A.1.3. lv_cat function
def lv_cat(*args):
result = b""
for arg in args:
result += prepend_len(arg)
return result
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A.1.4. Testvector for lv_cat()
lv_cat(b"1234",b"5",b"",b"6789"): (length: 13 bytes)
04313233340135000436373839
A.1.5. Examples for messages not obtained from a lv_cat-based encoding
The following messages are examples which have invalid encoded length
fields. I.e. they are examples where parsing for the sum of the
length of subfields as expected for a message generated using
lv_cat(Y,AD) does not give the correct length of the message.
Parties MUST abort upon reception of such invalid messages as MSGa or
MSGb.
Inv_MSG1 not encoded by lv_cat: (length: 3 bytes)
ffffff
Inv_MSG2 not encoded by lv_cat: (length: 3 bytes)
ffff03
Inv_MSG3 not encoded by lv_cat: (length: 4 bytes)
00ffff03
Inv_MSG4 not encoded by lv_cat: (length: 4 bytes)
00ffffff
A.2. Definition of generator_string function.
def generator_string(DSI,PRS,CI,sid,s_in_bytes):
# Concat all input fields with prepended length information.
# Add zero padding in the first hash block after DSI and PRS.
len_zpad = max(0,s_in_bytes - 1 - len(prepend_len(PRS))
- len(prepend_len(DSI)))
return lv_cat(DSI, PRS, zero_bytes(len_zpad),
CI, sid)
A.3. Definitions and test vector ordered concatenation
A.3.1. Definitions for lexiographical ordering
For ordered concatenation lexiographical ordering of byte sequences
is used:
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def lexiographically_larger(bytes1,bytes2):
"Returns True if bytes1 > bytes2 using lexiographical ordering."
min_len = min (len(bytes1), len(bytes2))
for m in range(min_len):
if bytes1[m] > bytes2[m]:
return True;
elif bytes1[m] < bytes2[m]:
return False;
return len(bytes1) > len(bytes2)
A.3.2. Definitions for ordered concatenation
With the above definition of lexiographical ordering ordered
concatenation is specified as follows.
def oCAT(bytes1,bytes2):
if lexiographically_larger(bytes1,bytes2):
return bytes1 + bytes2
else:
return bytes2 + bytes1
A.3.3. Test vectors ordered concatenation
string comparison for oCAT:
lexiographically_larger(b"\0", b"\0\0") == False
lexiographically_larger(b"\1", b"\0\0") == True
lexiographically_larger(b"\0\0", b"\0") == True
lexiographically_larger(b"\0\0", b"\1") == False
lexiographically_larger(b"\0\1", b"\1") == False
lexiographically_larger(b"ABCD", b"BCD") == False
oCAT(b"ABCD",b"BCD"): (length: 7 bytes)
42434441424344
oCAT(b"BCD",b"ABCDE"): (length: 8 bytes)
4243444142434445
A.4. Decoding and Encoding functions according to RFC7748
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def decodeLittleEndian(b, bits):
return sum([b[i] << 8*i for i in range((bits+7)/8)])
def decodeUCoordinate(u, bits):
u_list = [ord(b) for b in u]
# Ignore any unused bits.
if bits % 8:
u_list[-1] &= (1<<(bits%8))-1
return decodeLittleEndian(u_list, bits)
def encodeUCoordinate(u, bits):
return ''.join([chr((u >> 8*i) & 0xff)
for i in range((bits+7)/8)])
A.5. Elligator 2 reference implementation
The Elligator 2 map requires a non-square field element Z which shall
be calculated as follows.
def find_z_ell2(F):
# Find nonsquare for Elligator2
# Argument: F, a field object, e.g., F = GF(2^255 - 19)
ctr = F.gen()
while True:
for Z_cand in (F(ctr), F(-ctr)):
# Z must be a non-square in F.
if is_square(Z_cand):
continue
return Z_cand
ctr += 1
The values of the non-square Z only depend on the curve. The
algorithm above results in a value of Z = 2 for Curve25519 and Z=-1
for Ed448.
The following code maps a field element r to an encoded field element
which is a valid u-coordinate of a Montgomery curve with curve
parameter A.
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def elligator2(r, q, A, field_size_bits):
# Inputs: field element r, field order q,
# curve parameter A and field size in bits
Fq = GF(q); A = Fq(A); B = Fq(1);
# get non-square z as specified in the hash2curve draft.
z = Fq(find_z_ell2(Fq))
powerForLegendreSymbol = floor((q-1)/2)
v = - A / (1 + z * r^2)
epsilon = (v^3 + A * v^2 + B * v)^powerForLegendreSymbol
x = epsilon * v - (1 - epsilon) * A/2
return encodeUCoordinate(Integer(x), field_size_bits)
Appendix B. Test vectors
B.1. Test vector for CPace using group X25519 and hash SHA-512
B.1.1. Test vectors for calculate_generator with group X25519
Inputs
H = SHA-512 with input block size 128 bytes.
PRS = b'Password' ; ZPAD length: 109 ; DSI = b'CPace255'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 7e4b4791d6a8ef019b936c79fb7f2c57
Outputs
generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
(length: 168 bytes)
0843506163653235350850617373776f72646d000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000160a41696e69746961746f72
0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57
hash generator string: (length: 32 bytes)
10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863
94e821fc
decoded field element of 255 bits: (length: 32 bytes)
10047198e8c4cacf0ab8a6d0ac337b8ae497209d042f7f3a50945863
94e8217c
generator g: (length: 32 bytes)
4e6098733061c0e8486611a904fe5edb049804d26130a44131a6229e
55c5c321
B.1.2. Test vector for MSGa
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Inputs
ADa = b'ADa'
ya (little endian): (length: 32 bytes)
45acf93116ae5d3dae995a7c627df2924321a8e857d9a200807131e3
8839b0c2
Outputs
Ya: (length: 32 bytes)
6f7fd31863b18b0cc9830fc842c60dea80120ccf2fd375498225e45a
52065361
MSGa = lv_cat(Ya,ADa): (length: 37 bytes)
206f7fd31863b18b0cc9830fc842c60dea80120ccf2fd375498225e4
5a5206536103414461
B.1.3. Test vector for MSGb
Inputs
ADb = b'ADb'
yb (little endian): (length: 32 bytes)
a145e914b347002d298ce2051394f0ed68cf3623dfe5db082c78ffa5
a667acdc
Outputs
Yb: (length: 32 bytes)
e1b730a4956c0f853d96c5d125cebeeea46952c07c6f66da65bd9ffd
2f71a462
MSGb = lv_cat(Yb,ADb): (length: 37 bytes)
20e1b730a4956c0f853d96c5d125cebeeea46952c07c6f66da65bd9f
fd2f71a46203414462
B.1.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 32 bytes)
2a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86d9e199
befa6024
scalar_mult_vfy(yb,Ya): (length: 32 bytes)
2a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86d9e199
befa6024
B.1.5. Test vector for ISK calculation initiator/responder
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unordered cat of transcript : (length: 74 bytes)
206f7fd31863b18b0cc9830fc842c60dea80120ccf2fd375498225e4
5a520653610341446120e1b730a4956c0f853d96c5d125cebeeea469
52c07c6f66da65bd9ffd2f71a46203414462
DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes)
43506163653235355f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 137 bytes)
0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f
2c57202a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86
d9e199befa6024206f7fd31863b18b0cc9830fc842c60dea80120ccf
2fd375498225e45a520653610341446120e1b730a4956c0f853d96c5
d125cebeeea46952c07c6f66da65bd9ffd2f71a46203414462
ISK result: (length: 64 bytes)
99a9e0ff35acb94ad8af1cd6b32ac409dc7d00557ccd9a7d19d3b462
9e5f1f084f9332096162438c7ecc78331b4eda17e1a229a47182eccc
9ea58cd9cdcd8e9a
B.1.6. Test vector for ISK calculation parallel execution
ordered cat of transcript : (length: 74 bytes)
20e1b730a4956c0f853d96c5d125cebeeea46952c07c6f66da65bd9f
fd2f71a46203414462206f7fd31863b18b0cc9830fc842c60dea8012
0ccf2fd375498225e45a5206536103414461
DSI = G.DSI_ISK, b'CPace255_ISK': (length: 12 bytes)
43506163653235355f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 137 bytes)
0c43506163653235355f49534b107e4b4791d6a8ef019b936c79fb7f
2c57202a905bc5f0b93ee72ac4b6ea8723520941adfc892935bf6f86
d9e199befa602420e1b730a4956c0f853d96c5d125cebeeea46952c0
7c6f66da65bd9ffd2f71a46203414462206f7fd31863b18b0cc9830f
c842c60dea80120ccf2fd375498225e45a5206536103414461
ISK result: (length: 64 bytes)
3cd6a9670fa3ff211d829b845baa0f5ba9ad580c3ba0ee790bd0e9cd
556290a8ffce44419fbf94e4cb8e7fe9f454fd25dc13e689e4d6ab0a
c2211c70a8ac0062
B.1.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79,
0xfb,0x7f,0x2c,0x57,
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};
const uint8_t tc_g[] = {
0x4e,0x60,0x98,0x73,0x30,0x61,0xc0,0xe8,0x48,0x66,0x11,0xa9,
0x04,0xfe,0x5e,0xdb,0x04,0x98,0x04,0xd2,0x61,0x30,0xa4,0x41,
0x31,0xa6,0x22,0x9e,0x55,0xc5,0xc3,0x21,
};
const uint8_t tc_ya[] = {
0x45,0xac,0xf9,0x31,0x16,0xae,0x5d,0x3d,0xae,0x99,0x5a,0x7c,
0x62,0x7d,0xf2,0x92,0x43,0x21,0xa8,0xe8,0x57,0xd9,0xa2,0x00,
0x80,0x71,0x31,0xe3,0x88,0x39,0xb0,0xc2,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0x6f,0x7f,0xd3,0x18,0x63,0xb1,0x8b,0x0c,0xc9,0x83,0x0f,0xc8,
0x42,0xc6,0x0d,0xea,0x80,0x12,0x0c,0xcf,0x2f,0xd3,0x75,0x49,
0x82,0x25,0xe4,0x5a,0x52,0x06,0x53,0x61,
};
const uint8_t tc_yb[] = {
0xa1,0x45,0xe9,0x14,0xb3,0x47,0x00,0x2d,0x29,0x8c,0xe2,0x05,
0x13,0x94,0xf0,0xed,0x68,0xcf,0x36,0x23,0xdf,0xe5,0xdb,0x08,
0x2c,0x78,0xff,0xa5,0xa6,0x67,0xac,0xdc,
};
const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0xe1,0xb7,0x30,0xa4,0x95,0x6c,0x0f,0x85,0x3d,0x96,0xc5,0xd1,
0x25,0xce,0xbe,0xee,0xa4,0x69,0x52,0xc0,0x7c,0x6f,0x66,0xda,
0x65,0xbd,0x9f,0xfd,0x2f,0x71,0xa4,0x62,
};
const uint8_t tc_K[] = {
0x2a,0x90,0x5b,0xc5,0xf0,0xb9,0x3e,0xe7,0x2a,0xc4,0xb6,0xea,
0x87,0x23,0x52,0x09,0x41,0xad,0xfc,0x89,0x29,0x35,0xbf,0x6f,
0x86,0xd9,0xe1,0x99,0xbe,0xfa,0x60,0x24,
};
const uint8_t tc_ISK_IR[] = {
0x99,0xa9,0xe0,0xff,0x35,0xac,0xb9,0x4a,0xd8,0xaf,0x1c,0xd6,
0xb3,0x2a,0xc4,0x09,0xdc,0x7d,0x00,0x55,0x7c,0xcd,0x9a,0x7d,
0x19,0xd3,0xb4,0x62,0x9e,0x5f,0x1f,0x08,0x4f,0x93,0x32,0x09,
0x61,0x62,0x43,0x8c,0x7e,0xcc,0x78,0x33,0x1b,0x4e,0xda,0x17,
0xe1,0xa2,0x29,0xa4,0x71,0x82,0xec,0xcc,0x9e,0xa5,0x8c,0xd9,
0xcd,0xcd,0x8e,0x9a,
};
const uint8_t tc_ISK_SY[] = {
0x3c,0xd6,0xa9,0x67,0x0f,0xa3,0xff,0x21,0x1d,0x82,0x9b,0x84,
0x5b,0xaa,0x0f,0x5b,0xa9,0xad,0x58,0x0c,0x3b,0xa0,0xee,0x79,
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0x0b,0xd0,0xe9,0xcd,0x55,0x62,0x90,0xa8,0xff,0xce,0x44,0x41,
0x9f,0xbf,0x94,0xe4,0xcb,0x8e,0x7f,0xe9,0xf4,0x54,0xfd,0x25,
0xdc,0x13,0xe6,0x89,0xe4,0xd6,0xab,0x0a,0xc2,0x21,0x1c,0x70,
0xa8,0xac,0x00,0x62,
};
B.1.8. Test vectors for G_X25519.scalar_mult_vfy: low order points
Test vectors for which G_X25519.scalar_mult_vfy(s_in,ux) must return
the neutral element or would return the neutral element if bit #255
of field element representation was not correctly cleared. (The
decodeUCoordinate function from RFC7748 mandates clearing bit #255
for field element representations for use in the X25519 function.).
u0: 0000000000000000000000000000000000000000000000000000000000000000
u1: 0100000000000000000000000000000000000000000000000000000000000000
u2: ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f
u3: e0eb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b800
u4: 5f9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f1157
u5: edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f
u6: daffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
u7: eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f
u8: dbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
u9: d9ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ua: cdeb7a7c3b41b8ae1656e3faf19fc46ada098deb9c32b1fd866205165f49b880
ub: 4c9c95bca3508c24b1d0b1559c83ef5b04445cc4581c8e86d8224eddd09f11d7
u0 ... ub MUST be verified to produce the correct results q0 ... qb:
Additionally, u0,u1,u2,u3,u4,u5 and u7 MUST trigger the abort case
when included in MSGa or MSGb.
s = af46e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449aff
qN = G_X25519.scalar_mult_vfy(s, uX)
q0: 0000000000000000000000000000000000000000000000000000000000000000
q1: 0000000000000000000000000000000000000000000000000000000000000000
q2: 0000000000000000000000000000000000000000000000000000000000000000
q3: 0000000000000000000000000000000000000000000000000000000000000000
q4: 0000000000000000000000000000000000000000000000000000000000000000
q5: 0000000000000000000000000000000000000000000000000000000000000000
q6: d8e2c776bbacd510d09fd9278b7edcd25fc5ae9adfba3b6e040e8d3b71b21806
q7: 0000000000000000000000000000000000000000000000000000000000000000
q8: c85c655ebe8be44ba9c0ffde69f2fe10194458d137f09bbff725ce58803cdb38
q9: db64dafa9b8fdd136914e61461935fe92aa372cb056314e1231bc4ec12417456
qa: e062dcd5376d58297be2618c7498f55baa07d7e03184e8aada20bca28888bf7a
qb: 993c6ad11c4c29da9a56f7691fd0ff8d732e49de6250b6c2e80003ff4629a175
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B.2. Test vector for CPace using group X448 and hash SHAKE-256
B.2.1. Test vectors for calculate_generator with group X448
Inputs
H = SHAKE-256 with input block size 136 bytes.
PRS = b'Password' ; ZPAD length: 117 ; DSI = b'CPace448'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 5223e0cdc45d6575668d64c552004124
Outputs
generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
(length: 176 bytes)
0843506163653434380850617373776f726475000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000160a4169
6e69746961746f720a42726573706f6e646572105223e0cdc45d6575
668d64c552004124
hash generator string: (length: 56 bytes)
769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221
d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293
decoded field element of 448 bits: (length: 56 bytes)
769e06d6c41c8cf1c87aa3df8e687167f6d0a2e41821e856276a0221
d88272359d0b43204b546174c9179c83c107b707f296eafaa1c5a293
generator g: (length: 56 bytes)
6fdae14718eb7506dd96e3f7797896efdb8db9ec0797485c9c48a192
2e44961da097f2908b084a5de33ab671630660d27d79ffd6ee8ec846
B.2.2. Test vector for MSGa
Inputs
ADa = b'ADa'
ya (little endian): (length: 56 bytes)
21b4f4bd9e64ed355c3eb676a28ebedaf6d8f17bdc365995b3190971
53044080516bd083bfcce66121a3072646994c8430cc382b8dc543e8
Outputs
Ya: (length: 56 bytes)
396bd11daf223711e575cac6021e3fa31558012048a1cec7876292b9
6c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4b5
MSGa = lv_cat(Ya,ADa): (length: 61 bytes)
38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292
b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4
b503414461
B.2.3. Test vector for MSGb
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Inputs
ADb = b'ADb'
yb (little endian): (length: 56 bytes)
848b0779ff415f0af4ea14df9dd1d3c29ac41d836c7808896c4eba19
c51ac40a439caf5e61ec88c307c7d619195229412eaa73fb2a5ea20d
Outputs
Yb: (length: 56 bytes)
53c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9c6
0422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d58
MSGb = lv_cat(Yb,ADb): (length: 61 bytes)
3853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9
c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d
5803414462
B.2.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 56 bytes)
e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1
e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997
scalar_mult_vfy(yb,Ya): (length: 56 bytes)
e00af217556a40ccbc9822cc27a43542e45166a653aa4df746d5f8e1
e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a659997
B.2.5. Test vector for ISK calculation initiator/responder
unordered cat of transcript : (length: 122 bytes)
38396bd11daf223711e575cac6021e3fa31558012048a1cec7876292
b96c61eda353fe04f33028d2352779668a934084da776c1c51a58ce4
b5034144613853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0
ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e
4beb6af86d5803414462
DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes)
43506163653434385f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 209 bytes)
0c43506163653434385f49534b105223e0cdc45d6575668d64c55200
412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746
d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a
65999738396bd11daf223711e575cac6021e3fa31558012048a1cec7
876292b96c61eda353fe04f33028d2352779668a934084da776c1c51
a58ce4b5034144613853c519fb490fde5a04bda8c18b327d0fc1a939
1d19e0ac00c59df9c60422284e593d6b092eac94f5aa644ed883f39b
d4f04e4beb6af86d5803414462
ISK result: (length: 64 bytes)
4030297722c1914711da6b2a224a44b53b30c05ab02c2a3d3ccc7272
a3333ce3a4564c17031b634e89f65681f52d5c3d1df7baeb88523d2e
481b3858aed86315
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B.2.6. Test vector for ISK calculation parallel execution
ordered cat of transcript : (length: 122 bytes)
3853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00c59df9
c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb6af86d
580341446238396bd11daf223711e575cac6021e3fa31558012048a1
cec7876292b96c61eda353fe04f33028d2352779668a934084da776c
1c51a58ce4b503414461
DSI = G.DSI_ISK, b'CPace448_ISK': (length: 12 bytes)
43506163653434385f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 209 bytes)
0c43506163653434385f49534b105223e0cdc45d6575668d64c55200
412438e00af217556a40ccbc9822cc27a43542e45166a653aa4df746
d5f8e1e8df483e9baff71c9eb03ee20a688ad4e4d359f70ac9ec3f6a
6599973853c519fb490fde5a04bda8c18b327d0fc1a9391d19e0ac00
c59df9c60422284e593d6b092eac94f5aa644ed883f39bd4f04e4beb
6af86d580341446238396bd11daf223711e575cac6021e3fa3155801
2048a1cec7876292b96c61eda353fe04f33028d2352779668a934084
da776c1c51a58ce4b503414461
ISK result: (length: 64 bytes)
925e95d1095dad1af6378d5ef8b9a998bd3855bfc7d36cb5ca05b0a7
a93346abcb8cef04bceb28c38fdaf0cc608fd1dcd462ab523f3b7f75
2c77c411be3ac8fb
B.2.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5,
0x52,0x00,0x41,0x24,
};
const uint8_t tc_g[] = {
0x6f,0xda,0xe1,0x47,0x18,0xeb,0x75,0x06,0xdd,0x96,0xe3,0xf7,
0x79,0x78,0x96,0xef,0xdb,0x8d,0xb9,0xec,0x07,0x97,0x48,0x5c,
0x9c,0x48,0xa1,0x92,0x2e,0x44,0x96,0x1d,0xa0,0x97,0xf2,0x90,
0x8b,0x08,0x4a,0x5d,0xe3,0x3a,0xb6,0x71,0x63,0x06,0x60,0xd2,
0x7d,0x79,0xff,0xd6,0xee,0x8e,0xc8,0x46,
};
const uint8_t tc_ya[] = {
0x21,0xb4,0xf4,0xbd,0x9e,0x64,0xed,0x35,0x5c,0x3e,0xb6,0x76,
0xa2,0x8e,0xbe,0xda,0xf6,0xd8,0xf1,0x7b,0xdc,0x36,0x59,0x95,
0xb3,0x19,0x09,0x71,0x53,0x04,0x40,0x80,0x51,0x6b,0xd0,0x83,
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0xbf,0xcc,0xe6,0x61,0x21,0xa3,0x07,0x26,0x46,0x99,0x4c,0x84,
0x30,0xcc,0x38,0x2b,0x8d,0xc5,0x43,0xe8,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0x39,0x6b,0xd1,0x1d,0xaf,0x22,0x37,0x11,0xe5,0x75,0xca,0xc6,
0x02,0x1e,0x3f,0xa3,0x15,0x58,0x01,0x20,0x48,0xa1,0xce,0xc7,
0x87,0x62,0x92,0xb9,0x6c,0x61,0xed,0xa3,0x53,0xfe,0x04,0xf3,
0x30,0x28,0xd2,0x35,0x27,0x79,0x66,0x8a,0x93,0x40,0x84,0xda,
0x77,0x6c,0x1c,0x51,0xa5,0x8c,0xe4,0xb5,
};
const uint8_t tc_yb[] = {
0x84,0x8b,0x07,0x79,0xff,0x41,0x5f,0x0a,0xf4,0xea,0x14,0xdf,
0x9d,0xd1,0xd3,0xc2,0x9a,0xc4,0x1d,0x83,0x6c,0x78,0x08,0x89,
0x6c,0x4e,0xba,0x19,0xc5,0x1a,0xc4,0x0a,0x43,0x9c,0xaf,0x5e,
0x61,0xec,0x88,0xc3,0x07,0xc7,0xd6,0x19,0x19,0x52,0x29,0x41,
0x2e,0xaa,0x73,0xfb,0x2a,0x5e,0xa2,0x0d,
};
const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0x53,0xc5,0x19,0xfb,0x49,0x0f,0xde,0x5a,0x04,0xbd,0xa8,0xc1,
0x8b,0x32,0x7d,0x0f,0xc1,0xa9,0x39,0x1d,0x19,0xe0,0xac,0x00,
0xc5,0x9d,0xf9,0xc6,0x04,0x22,0x28,0x4e,0x59,0x3d,0x6b,0x09,
0x2e,0xac,0x94,0xf5,0xaa,0x64,0x4e,0xd8,0x83,0xf3,0x9b,0xd4,
0xf0,0x4e,0x4b,0xeb,0x6a,0xf8,0x6d,0x58,
};
const uint8_t tc_K[] = {
0xe0,0x0a,0xf2,0x17,0x55,0x6a,0x40,0xcc,0xbc,0x98,0x22,0xcc,
0x27,0xa4,0x35,0x42,0xe4,0x51,0x66,0xa6,0x53,0xaa,0x4d,0xf7,
0x46,0xd5,0xf8,0xe1,0xe8,0xdf,0x48,0x3e,0x9b,0xaf,0xf7,0x1c,
0x9e,0xb0,0x3e,0xe2,0x0a,0x68,0x8a,0xd4,0xe4,0xd3,0x59,0xf7,
0x0a,0xc9,0xec,0x3f,0x6a,0x65,0x99,0x97,
};
const uint8_t tc_ISK_IR[] = {
0x40,0x30,0x29,0x77,0x22,0xc1,0x91,0x47,0x11,0xda,0x6b,0x2a,
0x22,0x4a,0x44,0xb5,0x3b,0x30,0xc0,0x5a,0xb0,0x2c,0x2a,0x3d,
0x3c,0xcc,0x72,0x72,0xa3,0x33,0x3c,0xe3,0xa4,0x56,0x4c,0x17,
0x03,0x1b,0x63,0x4e,0x89,0xf6,0x56,0x81,0xf5,0x2d,0x5c,0x3d,
0x1d,0xf7,0xba,0xeb,0x88,0x52,0x3d,0x2e,0x48,0x1b,0x38,0x58,
0xae,0xd8,0x63,0x15,
};
const uint8_t tc_ISK_SY[] = {
0x92,0x5e,0x95,0xd1,0x09,0x5d,0xad,0x1a,0xf6,0x37,0x8d,0x5e,
0xf8,0xb9,0xa9,0x98,0xbd,0x38,0x55,0xbf,0xc7,0xd3,0x6c,0xb5,
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0xca,0x05,0xb0,0xa7,0xa9,0x33,0x46,0xab,0xcb,0x8c,0xef,0x04,
0xbc,0xeb,0x28,0xc3,0x8f,0xda,0xf0,0xcc,0x60,0x8f,0xd1,0xdc,
0xd4,0x62,0xab,0x52,0x3f,0x3b,0x7f,0x75,0x2c,0x77,0xc4,0x11,
0xbe,0x3a,0xc8,0xfb,
};
B.2.8. Test vectors for G_X448.scalar_mult_vfy: low order points
Test vectors for which G_X448.scalar_mult_vfy(s_in,ux) must return
the neutral element. This includes points that are non-canonicaly
encoded, i.e. have coordinate values larger than the field prime.
Weak points for X448 smaller than the field prime (canonical)
u0: (length: 56 bytes)
0000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
u1: (length: 56 bytes)
0100000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
u2: (length: 56 bytes)
fefffffffffffffffffffffffffffffffffffffffffffffffffffffffe
ffffffffffffffffffffffffffffffffffffffffffffffffffffff
Weak points for X448 larger or equal to the field prime (non-
canonical)
u3: (length: 56 bytes)
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffe
ffffffffffffffffffffffffffffffffffffffffffffffffffffff
u4: (length: 56 bytes)
00000000000000000000000000000000000000000000000000000000ff
ffffffffffffffffffffffffffffffffffffffffffffffffffffff
All of the above points u0 ... u4 MUST trigger the abort case
when included in the protocol messages MSGa or MSGb.
Expected results for X448 resp. G_X448.scalar_mult_vfy
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scalar s: (length: 56 bytes)
af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4
523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd
G_X448.scalar_mult_vfy(s,u0): (length: 56 bytes)
0000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
G_X448.scalar_mult_vfy(s,u1): (length: 56 bytes)
0000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
G_X448.scalar_mult_vfy(s,u2): (length: 56 bytes)
0000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
G_X448.scalar_mult_vfy(s,u3): (length: 56 bytes)
0000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
G_X448.scalar_mult_vfy(s,u4): (length: 56 bytes)
0000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000
Test vectors for scalar_mult with nonzero outputs
scalar s: (length: 56 bytes)
af8a14218bf2a2062926d2ea9b8fe4e8b6817349b6ed2feb1e5d64d7a4
523f15fceec70fb111e870dc58d191e66a14d3e9d482d04432cadd
point coordinate u_curve on the curve: (length: 56 bytes)
ab0c68d772ec2eb9de25c49700e46d6325e66d6aa39d7b65eb84a68c55
69d47bd71b41f3e0d210f44e146dec8926b174acb3f940a0b82cab
G_X448.scalar_mult_vfy(s,u_curve): (length: 56 bytes)
3b0fa9bc40a6fdc78c9e06ff7a54c143c5d52f365607053bf0656f5142
0496295f910a101b38edc1acd3bd240fd55dcb7a360553b8a7627e
point coordinate u_twist on the twist: (length: 56 bytes)
c981cd1e1f72d9c35c7d7cf6be426757c0dc8206a2fcfa564a8e7618c0
3c0e61f9a2eb1c3e0dd97d6e9b1010f5edd03397a83f5a914cb3ff
G_X448.scalar_mult_vfy(s,u_twist): (length: 56 bytes)
d0a2bb7e9c5c2c627793d8342f23b759fe7d9e3320a85ca4fd61376331
50ffd9a9148a9b75c349fac43d64bec49a6e126cc92cbfbf353961
B.3. Test vector for CPace using group ristretto255 and hash SHA-512
B.3.1. Test vectors for calculate_generator with group ristretto255
Abdalla, et al. Expires 28 March 2024 [Page 43]
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Inputs
H = SHA-512 with input block size 128 bytes.
PRS = b'Password' ; ZPAD length: 100 ;
DSI = b'CPaceRistretto255'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 7e4b4791d6a8ef019b936c79fb7f2c57
Outputs
generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
(length: 168 bytes)
11435061636552697374726574746f3235350850617373776f726464
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000160a41696e69746961746f72
0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57
hash result: (length: 64 bytes)
a5ce446f63a1ae6d1fee80fa67d0b4004a4b1283ec5549a462bf33a6
c1ae06a0871f9bf48545f49b2a792eed255ac04f52758c9c60448306
810b44e986e3dcbb
encoded generator g: (length: 32 bytes)
5e25411ca1ad7c9debfd0b33ad987a95cefef2d3f15dcc8bd26415a5
dfe2e15a
B.3.2. Test vector for MSGa
Inputs
ADa = b'ADa'
ya (little endian): (length: 32 bytes)
da3d23700a9e5699258aef94dc060dfda5ebb61f02a5ea77fad53f4f
f0976d08
Outputs
Ya: (length: 32 bytes)
383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040f
f77ca82d
MSGa = lv_cat(Ya,ADa): (length: 37 bytes)
20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04
0ff77ca82d03414461
B.3.3. Test vector for MSGb
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Inputs
ADb = b'ADb'
yb (little endian): (length: 32 bytes)
d2316b454718c35362d83d69df6320f38578ed5984651435e2949762
d900b80d
Outputs
Yb: (length: 32 bytes)
a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e69
8fa1383c
MSGb = lv_cat(Yb,ADb): (length: 37 bytes)
20a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e
698fa1383c03414462
B.3.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 32 bytes)
fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9
8893853f
scalar_mult_vfy(yb,Ya): (length: 32 bytes)
fa1d0318864e2cacb26875f1b791c9ae83204fe8359addb53e95a2e9
8893853f
B.3.5. Test vector for ISK calculation initiator/responder
unordered cat of transcript : (length: 74 bytes)
20383a85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce04
0ff77ca82d0341446120a6206309c0e8e5f579295e35997ac4300ab3
fecec3c17f7b604f3e698fa1383c03414462
DSI = G.DSI_ISK, b'CPaceRistretto255_ISK':
(length: 21 bytes)
435061636552697374726574746f3235355f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 146 bytes)
15435061636552697374726574746f3235355f49534b107e4b4791d6
a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae
83204fe8359addb53e95a2e98893853f20383a85dd236978f17f8c85
45b50dabc52a39fcdab2cf8bc531ce040ff77ca82d0341446120a620
6309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e698fa1
383c03414462
ISK result: (length: 64 bytes)
e91ccb2c0f5e0d0993a33956e3be59754f3f2b07db57631f5394452e
a2e7b4354674eb1f5686c078462bf83bec72e8743df440108e638f35
26d9b90e85be096f
B.3.6. Test vector for ISK calculation parallel execution
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ordered cat of transcript : (length: 74 bytes)
20a6206309c0e8e5f579295e35997ac4300ab3fecec3c17f7b604f3e
698fa1383c0341446220383a85dd236978f17f8c8545b50dabc52a39
fcdab2cf8bc531ce040ff77ca82d03414461
DSI = G.DSI_ISK, b'CPaceRistretto255_ISK':
(length: 21 bytes)
435061636552697374726574746f3235355f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 146 bytes)
15435061636552697374726574746f3235355f49534b107e4b4791d6
a8ef019b936c79fb7f2c5720fa1d0318864e2cacb26875f1b791c9ae
83204fe8359addb53e95a2e98893853f20a6206309c0e8e5f579295e
35997ac4300ab3fecec3c17f7b604f3e698fa1383c0341446220383a
85dd236978f17f8c8545b50dabc52a39fcdab2cf8bc531ce040ff77c
a82d03414461
ISK result: (length: 64 bytes)
2472dedbff868bfc12b4c256f790539af0e2bab7efc28d1a995d18a1
a58e5bec639273d4604512669ab7953153d437eb90314dcba7539724
02b0d9c5ec5283f8
B.3.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79,
0xfb,0x7f,0x2c,0x57,
};
const uint8_t tc_g[] = {
0x5e,0x25,0x41,0x1c,0xa1,0xad,0x7c,0x9d,0xeb,0xfd,0x0b,0x33,
0xad,0x98,0x7a,0x95,0xce,0xfe,0xf2,0xd3,0xf1,0x5d,0xcc,0x8b,
0xd2,0x64,0x15,0xa5,0xdf,0xe2,0xe1,0x5a,
};
const uint8_t tc_ya[] = {
0xda,0x3d,0x23,0x70,0x0a,0x9e,0x56,0x99,0x25,0x8a,0xef,0x94,
0xdc,0x06,0x0d,0xfd,0xa5,0xeb,0xb6,0x1f,0x02,0xa5,0xea,0x77,
0xfa,0xd5,0x3f,0x4f,0xf0,0x97,0x6d,0x08,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0x38,0x3a,0x85,0xdd,0x23,0x69,0x78,0xf1,0x7f,0x8c,0x85,0x45,
0xb5,0x0d,0xab,0xc5,0x2a,0x39,0xfc,0xda,0xb2,0xcf,0x8b,0xc5,
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0x31,0xce,0x04,0x0f,0xf7,0x7c,0xa8,0x2d,
};
const uint8_t tc_yb[] = {
0xd2,0x31,0x6b,0x45,0x47,0x18,0xc3,0x53,0x62,0xd8,0x3d,0x69,
0xdf,0x63,0x20,0xf3,0x85,0x78,0xed,0x59,0x84,0x65,0x14,0x35,
0xe2,0x94,0x97,0x62,0xd9,0x00,0xb8,0x0d,
};
const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0xa6,0x20,0x63,0x09,0xc0,0xe8,0xe5,0xf5,0x79,0x29,0x5e,0x35,
0x99,0x7a,0xc4,0x30,0x0a,0xb3,0xfe,0xce,0xc3,0xc1,0x7f,0x7b,
0x60,0x4f,0x3e,0x69,0x8f,0xa1,0x38,0x3c,
};
const uint8_t tc_K[] = {
0xfa,0x1d,0x03,0x18,0x86,0x4e,0x2c,0xac,0xb2,0x68,0x75,0xf1,
0xb7,0x91,0xc9,0xae,0x83,0x20,0x4f,0xe8,0x35,0x9a,0xdd,0xb5,
0x3e,0x95,0xa2,0xe9,0x88,0x93,0x85,0x3f,
};
const uint8_t tc_ISK_IR[] = {
0xe9,0x1c,0xcb,0x2c,0x0f,0x5e,0x0d,0x09,0x93,0xa3,0x39,0x56,
0xe3,0xbe,0x59,0x75,0x4f,0x3f,0x2b,0x07,0xdb,0x57,0x63,0x1f,
0x53,0x94,0x45,0x2e,0xa2,0xe7,0xb4,0x35,0x46,0x74,0xeb,0x1f,
0x56,0x86,0xc0,0x78,0x46,0x2b,0xf8,0x3b,0xec,0x72,0xe8,0x74,
0x3d,0xf4,0x40,0x10,0x8e,0x63,0x8f,0x35,0x26,0xd9,0xb9,0x0e,
0x85,0xbe,0x09,0x6f,
};
const uint8_t tc_ISK_SY[] = {
0x24,0x72,0xde,0xdb,0xff,0x86,0x8b,0xfc,0x12,0xb4,0xc2,0x56,
0xf7,0x90,0x53,0x9a,0xf0,0xe2,0xba,0xb7,0xef,0xc2,0x8d,0x1a,
0x99,0x5d,0x18,0xa1,0xa5,0x8e,0x5b,0xec,0x63,0x92,0x73,0xd4,
0x60,0x45,0x12,0x66,0x9a,0xb7,0x95,0x31,0x53,0xd4,0x37,0xeb,
0x90,0x31,0x4d,0xcb,0xa7,0x53,0x97,0x24,0x02,0xb0,0xd9,0xc5,
0xec,0x52,0x83,0xf8,
};
B.3.8. Test case for scalar_mult with valid inputs
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s: (length: 32 bytes)
7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88
cf538a05
X: (length: 32 bytes)
2c3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6
54b4a51c
G.scalar_mult(s,decode(X)): (length: 32 bytes)
7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e
7f6b600e
G.scalar_mult_vfy(s,X): (length: 32 bytes)
7c13645fe790a468f62c39beb7388e541d8405d1ade69d1778c5fe3e
7f6b600e
B.3.9. Invalid inputs for scalar_mult_vfy
For these test cases scalar_mult_vfy(y,.) MUST return the
representation of the neutral element G.I. When points Y_i1 or Y_i2
are included in MSGa or MSGb the protocol MUST abort.
s: (length: 32 bytes)
7cd0e075fa7955ba52c02759a6c90dbbfc10e6d40aea8d283e407d88
cf538a05
Y_i1: (length: 32 bytes)
2b3c6b8c4f3800e7aef6864025b4ed79bd599117e427c41bd47d93d6
54b4a51c
Y_i2 == G.I: (length: 32 bytes)
00000000000000000000000000000000000000000000000000000000
00000000
G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
B.4. Test vector for CPace using group decaf448 and hash SHAKE-256
B.4.1. Test vectors for calculate_generator with group decaf448
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Inputs
H = SHAKE-256 with input block size 136 bytes.
PRS = b'Password' ; ZPAD length: 112 ;
DSI = b'CPaceDecaf448'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 5223e0cdc45d6575668d64c552004124
Outputs
generator_string(G.DSI,PRS,CI,sid,H.s_in_bytes):
(length: 176 bytes)
0d435061636544656361663434380850617373776f72647000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000160a4169
6e69746961746f720a42726573706f6e646572105223e0cdc45d6575
668d64c552004124
hash result: (length: 112 bytes)
8955b426ff1d3a22032d21c013cf94134cee9a4235e93261a4911edb
f68f2945f0267c983954262c7f59badb9caf468ebe21b7e9885657af
b8f1a3b783c2047ba519e113ecf81b2b580dd481f499beabd401cc77
1d28915fb750011209040f5f03b2ceb5e5eb259c96b478382d5a5c57
encoded generator g: (length: 56 bytes)
682d1a4f49fc2a4834356ae4d7f58636bc9481521c845e66e6fb0b29
69341df45fbaeaea9e2221b3f5babc54c5f8ce456988ffc519defaeb
B.4.2. Test vector for MSGa
Inputs
ADa = b'ADa'
ya (little endian): (length: 56 bytes)
d8d2e26c821a12d7f59a8dee023d3f6155976152e16c73cbf68c303d
f0404399f0a7b614a65df50a9788f00b410586b443f738ad7ff03930
Outputs
Ya: (length: 56 bytes)
d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f84a71199bfd3
dc8d09d2b823038f579f517591474be366968e2fb599bf14e55704f4
MSGa = lv_cat(Ya,ADa): (length: 61 bytes)
38d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f84a71199bf
d3dc8d09d2b823038f579f517591474be366968e2fb599bf14e55704
f403414461
B.4.3. Test vector for MSGb
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Inputs
ADb = b'ADb'
yb (little endian): (length: 56 bytes)
91bae9793f4a8aceb1b5c54375a7ed1858a79a6e72dab959c8bdf3a7
5ac9bb4de2a25af4d4a9a5c5bc5441d19b8e3f6fcce7196c6afc2236
Outputs
Yb: (length: 56 bytes)
d61c6c039c01560e8b19b8299fb39513f39302eebd4c462694a33155
a3a387e44aa613647fcf6303f918bad598aaab53bea849b9fd14da74
MSGb = lv_cat(Yb,ADb): (length: 61 bytes)
38d61c6c039c01560e8b19b8299fb39513f39302eebd4c462694a331
55a3a387e44aa613647fcf6303f918bad598aaab53bea849b9fd14da
7403414462
B.4.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 56 bytes)
e434cda1783ddaaef08fc1d5f2201f1540fbc295fe2dd7cc38f20385
64824c98dbbe1978f121bdfead8e1a638913a6952cbec54867eb770a
scalar_mult_vfy(yb,Ya): (length: 56 bytes)
e434cda1783ddaaef08fc1d5f2201f1540fbc295fe2dd7cc38f20385
64824c98dbbe1978f121bdfead8e1a638913a6952cbec54867eb770a
B.4.5. Test vector for ISK calculation initiator/responder
unordered cat of transcript : (length: 122 bytes)
38d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f84a71199bf
d3dc8d09d2b823038f579f517591474be366968e2fb599bf14e55704
f40341446138d61c6c039c01560e8b19b8299fb39513f39302eebd4c
462694a33155a3a387e44aa613647fcf6303f918bad598aaab53bea8
49b9fd14da7403414462
DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes)
435061636544656361663434385f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 214 bytes)
11435061636544656361663434385f49534b105223e0cdc45d657566
8d64c55200412438e434cda1783ddaaef08fc1d5f2201f1540fbc295
fe2dd7cc38f2038564824c98dbbe1978f121bdfead8e1a638913a695
2cbec54867eb770a38d4b87d2fcdcac1096dba1898361f27e29dc1e0
19f74f84a71199bfd3dc8d09d2b823038f579f517591474be366968e
2fb599bf14e55704f40341446138d61c6c039c01560e8b19b8299fb3
9513f39302eebd4c462694a33155a3a387e44aa613647fcf6303f918
bad598aaab53bea849b9fd14da7403414462
ISK result: (length: 64 bytes)
13636dc9b7d233ac24a2d5c4a85a72fe20145f7a47ad51cab40e087c
057831b69ee59b9c828732bde171cfca99afda4852bcaf04fe9f0a97
592cdf5e2c9a5948
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B.4.6. Test vector for ISK calculation parallel execution
ordered cat of transcript : (length: 122 bytes)
38d61c6c039c01560e8b19b8299fb39513f39302eebd4c462694a331
55a3a387e44aa613647fcf6303f918bad598aaab53bea849b9fd14da
740341446238d4b87d2fcdcac1096dba1898361f27e29dc1e019f74f
84a71199bfd3dc8d09d2b823038f579f517591474be366968e2fb599
bf14e55704f403414461
DSI = G.DSI_ISK, b'CPaceDecaf448_ISK': (length: 17 bytes)
435061636544656361663434385f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 214 bytes)
11435061636544656361663434385f49534b105223e0cdc45d657566
8d64c55200412438e434cda1783ddaaef08fc1d5f2201f1540fbc295
fe2dd7cc38f2038564824c98dbbe1978f121bdfead8e1a638913a695
2cbec54867eb770a38d61c6c039c01560e8b19b8299fb39513f39302
eebd4c462694a33155a3a387e44aa613647fcf6303f918bad598aaab
53bea849b9fd14da740341446238d4b87d2fcdcac1096dba1898361f
27e29dc1e019f74f84a71199bfd3dc8d09d2b823038f579f51759147
4be366968e2fb599bf14e55704f403414461
ISK result: (length: 64 bytes)
999e8f8486670bc1bf874a4d8f1496b9ebd8909eb01cf46b275ec942
2f22593064b272ba9e9e201a4a34a18729e48859a2d038c7c8cf0a0f
e8a90ddcbdde1126
B.4.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x52,0x23,0xe0,0xcd,0xc4,0x5d,0x65,0x75,0x66,0x8d,0x64,0xc5,
0x52,0x00,0x41,0x24,
};
const uint8_t tc_g[] = {
0x68,0x2d,0x1a,0x4f,0x49,0xfc,0x2a,0x48,0x34,0x35,0x6a,0xe4,
0xd7,0xf5,0x86,0x36,0xbc,0x94,0x81,0x52,0x1c,0x84,0x5e,0x66,
0xe6,0xfb,0x0b,0x29,0x69,0x34,0x1d,0xf4,0x5f,0xba,0xea,0xea,
0x9e,0x22,0x21,0xb3,0xf5,0xba,0xbc,0x54,0xc5,0xf8,0xce,0x45,
0x69,0x88,0xff,0xc5,0x19,0xde,0xfa,0xeb,
};
const uint8_t tc_ya[] = {
0xd8,0xd2,0xe2,0x6c,0x82,0x1a,0x12,0xd7,0xf5,0x9a,0x8d,0xee,
0x02,0x3d,0x3f,0x61,0x55,0x97,0x61,0x52,0xe1,0x6c,0x73,0xcb,
0xf6,0x8c,0x30,0x3d,0xf0,0x40,0x43,0x99,0xf0,0xa7,0xb6,0x14,
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0xa6,0x5d,0xf5,0x0a,0x97,0x88,0xf0,0x0b,0x41,0x05,0x86,0xb4,
0x43,0xf7,0x38,0xad,0x7f,0xf0,0x39,0x30,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0xd4,0xb8,0x7d,0x2f,0xcd,0xca,0xc1,0x09,0x6d,0xba,0x18,0x98,
0x36,0x1f,0x27,0xe2,0x9d,0xc1,0xe0,0x19,0xf7,0x4f,0x84,0xa7,
0x11,0x99,0xbf,0xd3,0xdc,0x8d,0x09,0xd2,0xb8,0x23,0x03,0x8f,
0x57,0x9f,0x51,0x75,0x91,0x47,0x4b,0xe3,0x66,0x96,0x8e,0x2f,
0xb5,0x99,0xbf,0x14,0xe5,0x57,0x04,0xf4,
};
const uint8_t tc_yb[] = {
0x91,0xba,0xe9,0x79,0x3f,0x4a,0x8a,0xce,0xb1,0xb5,0xc5,0x43,
0x75,0xa7,0xed,0x18,0x58,0xa7,0x9a,0x6e,0x72,0xda,0xb9,0x59,
0xc8,0xbd,0xf3,0xa7,0x5a,0xc9,0xbb,0x4d,0xe2,0xa2,0x5a,0xf4,
0xd4,0xa9,0xa5,0xc5,0xbc,0x54,0x41,0xd1,0x9b,0x8e,0x3f,0x6f,
0xcc,0xe7,0x19,0x6c,0x6a,0xfc,0x22,0x36,
};
const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0xd6,0x1c,0x6c,0x03,0x9c,0x01,0x56,0x0e,0x8b,0x19,0xb8,0x29,
0x9f,0xb3,0x95,0x13,0xf3,0x93,0x02,0xee,0xbd,0x4c,0x46,0x26,
0x94,0xa3,0x31,0x55,0xa3,0xa3,0x87,0xe4,0x4a,0xa6,0x13,0x64,
0x7f,0xcf,0x63,0x03,0xf9,0x18,0xba,0xd5,0x98,0xaa,0xab,0x53,
0xbe,0xa8,0x49,0xb9,0xfd,0x14,0xda,0x74,
};
const uint8_t tc_K[] = {
0xe4,0x34,0xcd,0xa1,0x78,0x3d,0xda,0xae,0xf0,0x8f,0xc1,0xd5,
0xf2,0x20,0x1f,0x15,0x40,0xfb,0xc2,0x95,0xfe,0x2d,0xd7,0xcc,
0x38,0xf2,0x03,0x85,0x64,0x82,0x4c,0x98,0xdb,0xbe,0x19,0x78,
0xf1,0x21,0xbd,0xfe,0xad,0x8e,0x1a,0x63,0x89,0x13,0xa6,0x95,
0x2c,0xbe,0xc5,0x48,0x67,0xeb,0x77,0x0a,
};
const uint8_t tc_ISK_IR[] = {
0x13,0x63,0x6d,0xc9,0xb7,0xd2,0x33,0xac,0x24,0xa2,0xd5,0xc4,
0xa8,0x5a,0x72,0xfe,0x20,0x14,0x5f,0x7a,0x47,0xad,0x51,0xca,
0xb4,0x0e,0x08,0x7c,0x05,0x78,0x31,0xb6,0x9e,0xe5,0x9b,0x9c,
0x82,0x87,0x32,0xbd,0xe1,0x71,0xcf,0xca,0x99,0xaf,0xda,0x48,
0x52,0xbc,0xaf,0x04,0xfe,0x9f,0x0a,0x97,0x59,0x2c,0xdf,0x5e,
0x2c,0x9a,0x59,0x48,
};
const uint8_t tc_ISK_SY[] = {
0x99,0x9e,0x8f,0x84,0x86,0x67,0x0b,0xc1,0xbf,0x87,0x4a,0x4d,
0x8f,0x14,0x96,0xb9,0xeb,0xd8,0x90,0x9e,0xb0,0x1c,0xf4,0x6b,
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0x27,0x5e,0xc9,0x42,0x2f,0x22,0x59,0x30,0x64,0xb2,0x72,0xba,
0x9e,0x9e,0x20,0x1a,0x4a,0x34,0xa1,0x87,0x29,0xe4,0x88,0x59,
0xa2,0xd0,0x38,0xc7,0xc8,0xcf,0x0a,0x0f,0xe8,0xa9,0x0d,0xdc,
0xbd,0xde,0x11,0x26,
};
B.4.8. Test case for scalar_mult with valid inputs
s: (length: 56 bytes)
dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61
34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819
X: (length: 56 bytes)
601431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea
ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21
G.scalar_mult(s,decode(X)): (length: 56 bytes)
388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f
21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4
G.scalar_mult_vfy(s,X): (length: 56 bytes)
388b35c60eb41b66085a2118316218681d78979d667702de105fdc1f
21ffe884a577d795f45691781390a229a3bd7b527e831380f2f585a4
B.4.9. Invalid inputs for scalar_mult_vfy
For these test cases scalar_mult_vfy(y,.) MUST return the
representation of the neutral element G.I. When points Y_i1 or Y_i2
are included in MSGa or MSGb the protocol MUST abort.
s: (length: 56 bytes)
dd1bc7015daabb7672129cc35a3ba815486b139deff9bdeca7a4fc61
34323d34658761e90ff079972a7ca8aa5606498f4f4f0ebc0933a819
Y_i1: (length: 56 bytes)
5f1431d5e51f43d422a92d3fb2373bde28217aab42524c341aa404ea
ba5aa5541f7042dbb3253ce4c90f772b038a413dcb3a0f6bf3ae9e21
Y_i2 == G.I: (length: 56 bytes)
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
B.5. Test vector for CPace using group NIST P-256 and hash SHA-256
B.5.1. Test vectors for calculate_generator with group NIST P-256
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Inputs
H = SHA-256 with input block size 64 bytes.
PRS = b'Password' ; ZPAD length: 23 ;
DSI = b'CPaceP256_XMD:SHA-256_SSWU_NU_'
DST = b'CPaceP256_XMD:SHA-256_SSWU_NU__DST'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 34b36454cab2e7842c389f7d88ecb7df
Outputs
generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes):
(length: 104 bytes)
1e4350616365503235365f584d443a5348412d3235365f535357555f
4e555f0850617373776f726417000000000000000000000000000000
0000000000000000160a41696e69746961746f720a42726573706f6e
6465721034b36454cab2e7842c389f7d88ecb7df
generator g: (length: 65 bytes)
041b51433114e096c9d595f0955f5717a75169afb95557f4a6f51155
035dee19c76887bce5c7c054fa1fe48a4a62c7fb96dc75e34259d2f7
2b8d41f31b8e586bcd
B.5.2. Test vector for MSGa
Inputs
ADa = b'ADa'
ya (big endian): (length: 32 bytes)
37574cfbf1b95ff6a8e2d7be462d4d01e6dde2618f34f4de9df869b2
4f532c5d
Outputs
Ya: (length: 65 bytes)
04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d
81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610b4
021488279e3b989d52
Alternative correct value for Ya: g^(-ya):
(length: 65 bytes)
04b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb320d
81df0146493d5396e5da031f1415382438a135da195eaa7f9a59ef4b
fdeb77d861c46762ad
MSGa = lv_cat(Ya,ADa): (length: 70 bytes)
4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32
0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610
b4021488279e3b989d5203414461
B.5.3. Test vector for MSGb
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Inputs
ADb = b'ADb'
yb (big endian): (length: 32 bytes)
e5672fc9eb4e721f41d80181ec4c9fd9886668acc48024d33c82bb10
2aecba52
Outputs
Yb: (length: 65 bytes)
04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64
777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb65
562b78c793947dcada
Alternative correct value for Yb: g^(-yb):
(length: 65 bytes)
04bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e64
777ed27a9017f566bb71d0e3f9db9d0d76a392520e4bc79847d0049a
a9d487386c6b823525
MSGb = lv_cat(Yb,ADb): (length: 70 bytes)
4104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e
64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb
65562b78c793947dcada03414462
B.5.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 32 bytes)
8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2
bce12c39
scalar_mult_vfy(yb,Ya): (length: 32 bytes)
8fd12b283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2
bce12c39
B.5.5. Test vector for ISK calculation initiator/responder
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unordered cat of transcript : (length: 140 bytes)
4104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb32
0d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a610
b4021488279e3b989d52034144614104bb2783a57337e74671f76452
876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c07
2462f2895c6dadf1b43867b92ffb65562b78c793947dcada03414462
DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK':
(length: 34 bytes)
4350616365503235365f584d443a5348412d3235365f535357555f4e
555f5f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 225 bytes)
224350616365503235365f584d443a5348412d3235365f535357555f
4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b
283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c
394104b75c1bcda84a0f324aabb7f25cf853ed7fb327c33f23db6aeb
320d81df014649c2ac691925fce0eceac7dbc75eca25e6a1558066a6
10b4021488279e3b989d52034144614104bb2783a57337e74671f764
52876b27839c0ea9e044e3aadaad2e64777ed27a90e80a99438e2f1c
072462f2895c6dadf1b43867b92ffb65562b78c793947dcada034144
62
ISK result: (length: 32 bytes)
7ae1e916606e44652e3c0d7231198af6519226339c241e546afd0bbf
48e1c96a
B.5.6. Test vector for ISK calculation parallel execution
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ordered cat of transcript : (length: 140 bytes)
4104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad2e
64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92ffb
65562b78c793947dcada034144624104b75c1bcda84a0f324aabb7f2
5cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925fce0ec
eac7dbc75eca25e6a1558066a610b4021488279e3b989d5203414461
DSI = G.DSI_ISK, b'CPaceP256_XMD:SHA-256_SSWU_NU__ISK':
(length: 34 bytes)
4350616365503235365f584d443a5348412d3235365f535357555f4e
555f5f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 225 bytes)
224350616365503235365f584d443a5348412d3235365f535357555f
4e555f5f49534b1034b36454cab2e7842c389f7d88ecb7df208fd12b
283805750aeee6151bcd4211a6b71019e8fc416293ade24ed2bce12c
394104bb2783a57337e74671f76452876b27839c0ea9e044e3aadaad
2e64777ed27a90e80a99438e2f1c072462f2895c6dadf1b43867b92f
fb65562b78c793947dcada034144624104b75c1bcda84a0f324aabb7
f25cf853ed7fb327c33f23db6aeb320d81df014649c2ac691925fce0
eceac7dbc75eca25e6a1558066a610b4021488279e3b989d52034144
61
ISK result: (length: 32 bytes)
c5b4e6d44f5bbb7637a77ec67afd768a1343c410f7e1f76f6549eb00
2623c0f1
B.5.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x34,0xb3,0x64,0x54,0xca,0xb2,0xe7,0x84,0x2c,0x38,0x9f,0x7d,
0x88,0xec,0xb7,0xdf,
};
const uint8_t tc_g[] = {
0x04,0x1b,0x51,0x43,0x31,0x14,0xe0,0x96,0xc9,0xd5,0x95,0xf0,
0x95,0x5f,0x57,0x17,0xa7,0x51,0x69,0xaf,0xb9,0x55,0x57,0xf4,
0xa6,0xf5,0x11,0x55,0x03,0x5d,0xee,0x19,0xc7,0x68,0x87,0xbc,
0xe5,0xc7,0xc0,0x54,0xfa,0x1f,0xe4,0x8a,0x4a,0x62,0xc7,0xfb,
0x96,0xdc,0x75,0xe3,0x42,0x59,0xd2,0xf7,0x2b,0x8d,0x41,0xf3,
0x1b,0x8e,0x58,0x6b,0xcd,
};
const uint8_t tc_ya[] = {
0x37,0x57,0x4c,0xfb,0xf1,0xb9,0x5f,0xf6,0xa8,0xe2,0xd7,0xbe,
0x46,0x2d,0x4d,0x01,0xe6,0xdd,0xe2,0x61,0x8f,0x34,0xf4,0xde,
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0x9d,0xf8,0x69,0xb2,0x4f,0x53,0x2c,0x5d,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0x04,0xb7,0x5c,0x1b,0xcd,0xa8,0x4a,0x0f,0x32,0x4a,0xab,0xb7,
0xf2,0x5c,0xf8,0x53,0xed,0x7f,0xb3,0x27,0xc3,0x3f,0x23,0xdb,
0x6a,0xeb,0x32,0x0d,0x81,0xdf,0x01,0x46,0x49,0xc2,0xac,0x69,
0x19,0x25,0xfc,0xe0,0xec,0xea,0xc7,0xdb,0xc7,0x5e,0xca,0x25,
0xe6,0xa1,0x55,0x80,0x66,0xa6,0x10,0xb4,0x02,0x14,0x88,0x27,
0x9e,0x3b,0x98,0x9d,0x52,
};
const uint8_t tc_yb[] = {
0xe5,0x67,0x2f,0xc9,0xeb,0x4e,0x72,0x1f,0x41,0xd8,0x01,0x81,
0xec,0x4c,0x9f,0xd9,0x88,0x66,0x68,0xac,0xc4,0x80,0x24,0xd3,
0x3c,0x82,0xbb,0x10,0x2a,0xec,0xba,0x52,
};
const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0x04,0xbb,0x27,0x83,0xa5,0x73,0x37,0xe7,0x46,0x71,0xf7,0x64,
0x52,0x87,0x6b,0x27,0x83,0x9c,0x0e,0xa9,0xe0,0x44,0xe3,0xaa,
0xda,0xad,0x2e,0x64,0x77,0x7e,0xd2,0x7a,0x90,0xe8,0x0a,0x99,
0x43,0x8e,0x2f,0x1c,0x07,0x24,0x62,0xf2,0x89,0x5c,0x6d,0xad,
0xf1,0xb4,0x38,0x67,0xb9,0x2f,0xfb,0x65,0x56,0x2b,0x78,0xc7,
0x93,0x94,0x7d,0xca,0xda,
};
const uint8_t tc_K[] = {
0x8f,0xd1,0x2b,0x28,0x38,0x05,0x75,0x0a,0xee,0xe6,0x15,0x1b,
0xcd,0x42,0x11,0xa6,0xb7,0x10,0x19,0xe8,0xfc,0x41,0x62,0x93,
0xad,0xe2,0x4e,0xd2,0xbc,0xe1,0x2c,0x39,
};
const uint8_t tc_ISK_IR[] = {
0x7a,0xe1,0xe9,0x16,0x60,0x6e,0x44,0x65,0x2e,0x3c,0x0d,0x72,
0x31,0x19,0x8a,0xf6,0x51,0x92,0x26,0x33,0x9c,0x24,0x1e,0x54,
0x6a,0xfd,0x0b,0xbf,0x48,0xe1,0xc9,0x6a,
};
const uint8_t tc_ISK_SY[] = {
0xc5,0xb4,0xe6,0xd4,0x4f,0x5b,0xbb,0x76,0x37,0xa7,0x7e,0xc6,
0x7a,0xfd,0x76,0x8a,0x13,0x43,0xc4,0x10,0xf7,0xe1,0xf7,0x6f,
0x65,0x49,0xeb,0x00,0x26,0x23,0xc0,0xf1,
};
B.5.8. Test case for scalar_mult_vfy with correct inputs
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s: (length: 32 bytes)
f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399
0835643b
X: (length: 65 bytes)
0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1
fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d
10a4ab2514faef5831
G.scalar_mult(s,X) (full coordinates): (length: 65 bytes)
04f5a191f078c87c36633b78c701751159d56c59f3fe9105b5720673
470f303ab925b6a7fd1cdd8f649a21cf36b68d9e9c4a11919a951892
519786104b27033757
G.scalar_mult_vfy(s,X) (only X-coordinate):
(length: 32 bytes)
f5a191f078c87c36633b78c701751159d56c59f3fe9105b572067347
0f303ab9
B.5.9. Invalid inputs for scalar_mult_vfy
For these test cases scalar_mult_vfy(y,.) MUST return the
representation of the neutral element G.I. When including Y_i1 or
Y_i2 in MSGa or MSGb the protocol MUST abort.
s: (length: 32 bytes)
f012501c091ff9b99a123fffe571d8bc01e8077ee581362e1bd21399
0835643b
Y_i1: (length: 65 bytes)
0424648eb986c2be0af636455cef0550671d6bcd8aa26e0d72ffa1b1
fd12ba4e0f78da2b6d2184f31af39e566aef127014b6936c9a37346d
10a4ab2514faef5857
Y_i2: (length: 1 bytes)
00
G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
B.6. Test vector for CPace using group NIST P-384 and hash SHA-384
B.6.1. Test vectors for calculate_generator with group NIST P-384
Abdalla, et al. Expires 28 March 2024 [Page 59]
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Inputs
H = SHA-384 with input block size 128 bytes.
PRS = b'Password' ; ZPAD length: 87 ;
DSI = b'CPaceP384_XMD:SHA-384_SSWU_NU_'
DST = b'CPaceP384_XMD:SHA-384_SSWU_NU__DST'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 5b3773aa90e8f23c61563a4b645b276c
Outputs
generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes):
(length: 168 bytes)
1e4350616365503338345f584d443a5348412d3338345f535357555f
4e555f0850617373776f726457000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000160a41696e69746961746f72
0a42726573706f6e646572105b3773aa90e8f23c61563a4b645b276c
generator g: (length: 97 bytes)
04f35a925fe82e54350e80b084a8013b1960cb3f73c49b0c2ae9b523
997846ddd14c66f24f62223112cf35b866065f91ad86674cce2a2876
84904e49f01287b54666bb518df2ea53cec627fa6e1283f14c6ed4bc
d11b33fbb962da3e2e4ff1345c
B.6.2. Test vector for MSGa
Inputs
ADa = b'ADa'
ya (big endian): (length: 48 bytes)
7d5bc6a8959f9db2655b8b6642e393dc13d25150d69c6675fb3efd41
ae6255bf54202b960f9aacd97fd6d2841b461f18
Outputs
Ya: (length: 97 bytes)
048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c974
eceb71dfb6d36e989addf2ae8c4e338f204b2cd754e1c43b43a12692
8d8d81ce2e6edbc22a99ed478ad3487b87e1052bce2d94b6464a2228
eab73c01f79d6b290af6b218cf
Alternative correct value for Ya: g^(-ya):
(length: 97 bytes)
048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c974
eceb71dfb6d36e989addf2ae8c4e338f204b2cd7541e3bc4bc5ed96d
72727e31d191243dd56612b8752cb784781efad431d26b49b8b5ddd7
1448c3fe086294d6f6094de730
MSGa = lv_cat(Ya,ADa): (length: 102 bytes)
61048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c9
74eceb71dfb6d36e989addf2ae8c4e338f204b2cd754e1c43b43a126
928d8d81ce2e6edbc22a99ed478ad3487b87e1052bce2d94b6464a22
28eab73c01f79d6b290af6b218cf03414461
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B.6.3. Test vector for MSGb
Inputs
ADb = b'ADb'
yb (big endian): (length: 48 bytes)
5cc9465bdb3ae626b77521ea36218fc93a9693c36ff126899e3d8777
c126ef05483e34c05576c9e8c64b1a0b6f5b53d1
Outputs
Yb: (length: 97 bytes)
04cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a083
63f5e458938d6fe634ed6393bc8440ec9b9f8a30841ffcc7cd65d9cf
9617155e129bccded9888ac738f78e940f9887f9089ab6275d36c3ab
1bccbd048b088b1b80a0f56f27
Alternative correct value for Yb: g^(-yb):
(length: 97 bytes)
04cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a083
63f5e458938d6fe634ed6393bc8440ec9b9f8a3084e00338329a2630
69e8eaa1ed64332126777538c708716bf0677806f76549d8a1c93c54
e33342fb74f774e4805f0a90d8
MSGb = lv_cat(Yb,ADb): (length: 102 bytes)
6104cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a0
8363f5e458938d6fe634ed6393bc8440ec9b9f8a30841ffcc7cd65d9
cf9617155e129bccded9888ac738f78e940f9887f9089ab6275d36c3
ab1bccbd048b088b1b80a0f56f2703414462
B.6.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 48 bytes)
c862709d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1
8631361ed7d8cd97b12931844b7ac61b2f31d332
scalar_mult_vfy(yb,Ya): (length: 48 bytes)
c862709d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1
8631361ed7d8cd97b12931844b7ac61b2f31d332
B.6.5. Test vector for ISK calculation initiator/responder
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unordered cat of transcript : (length: 204 bytes)
61048b65b9ef4c5726664391ceeae241834b275960a6f9316799f5c9
74eceb71dfb6d36e989addf2ae8c4e338f204b2cd754e1c43b43a126
928d8d81ce2e6edbc22a99ed478ad3487b87e1052bce2d94b6464a22
28eab73c01f79d6b290af6b218cf034144616104cb68451813699abd
a3dc0ed9d521baf9108bc2c4b2a1dbcd90a08363f5e458938d6fe634
ed6393bc8440ec9b9f8a30841ffcc7cd65d9cf9617155e129bccded9
888ac738f78e940f9887f9089ab6275d36c3ab1bccbd048b088b1b80
a0f56f2703414462
DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK':
(length: 34 bytes)
4350616365503338345f584d443a5348412d3338345f535357555f4e
555f5f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 305 bytes)
224350616365503338345f584d443a5348412d3338345f535357555f
4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30c86270
9d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1863136
1ed7d8cd97b12931844b7ac61b2f31d33261048b65b9ef4c57266643
91ceeae241834b275960a6f9316799f5c974eceb71dfb6d36e989add
f2ae8c4e338f204b2cd754e1c43b43a126928d8d81ce2e6edbc22a99
ed478ad3487b87e1052bce2d94b6464a2228eab73c01f79d6b290af6
b218cf034144616104cb68451813699abda3dc0ed9d521baf9108bc2
c4b2a1dbcd90a08363f5e458938d6fe634ed6393bc8440ec9b9f8a30
841ffcc7cd65d9cf9617155e129bccded9888ac738f78e940f9887f9
089ab6275d36c3ab1bccbd048b088b1b80a0f56f2703414462
ISK result: (length: 48 bytes)
db1e8133be8359b9aa8cd563043ee784344f26580876862e28b3f98b
51b2f611a65362c1d77db66c879de466f5b6148a
B.6.6. Test vector for ISK calculation parallel execution
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ordered cat of transcript : (length: 204 bytes)
6104cb68451813699abda3dc0ed9d521baf9108bc2c4b2a1dbcd90a0
8363f5e458938d6fe634ed6393bc8440ec9b9f8a30841ffcc7cd65d9
cf9617155e129bccded9888ac738f78e940f9887f9089ab6275d36c3
ab1bccbd048b088b1b80a0f56f270341446261048b65b9ef4c572666
4391ceeae241834b275960a6f9316799f5c974eceb71dfb6d36e989a
ddf2ae8c4e338f204b2cd754e1c43b43a126928d8d81ce2e6edbc22a
99ed478ad3487b87e1052bce2d94b6464a2228eab73c01f79d6b290a
f6b218cf03414461
DSI = G.DSI_ISK, b'CPaceP384_XMD:SHA-384_SSWU_NU__ISK':
(length: 34 bytes)
4350616365503338345f584d443a5348412d3338345f535357555f4e
555f5f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 305 bytes)
224350616365503338345f584d443a5348412d3338345f535357555f
4e555f5f49534b105b3773aa90e8f23c61563a4b645b276c30c86270
9d6bfe7cc02f0c11dafdbf4ef8db1c5e4cb13a22985a83bef1863136
1ed7d8cd97b12931844b7ac61b2f31d3326104cb68451813699abda3
dc0ed9d521baf9108bc2c4b2a1dbcd90a08363f5e458938d6fe634ed
6393bc8440ec9b9f8a30841ffcc7cd65d9cf9617155e129bccded988
8ac738f78e940f9887f9089ab6275d36c3ab1bccbd048b088b1b80a0
f56f270341446261048b65b9ef4c5726664391ceeae241834b275960
a6f9316799f5c974eceb71dfb6d36e989addf2ae8c4e338f204b2cd7
54e1c43b43a126928d8d81ce2e6edbc22a99ed478ad3487b87e1052b
ce2d94b6464a2228eab73c01f79d6b290af6b218cf03414461
ISK result: (length: 48 bytes)
519bfbb1477652e8ed1b4ec5774e310c4f44da46f3c36be91b0dd6b4
e3a3245942cf4d9db8f79023dad6e1b57aed4891
B.6.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x5b,0x37,0x73,0xaa,0x90,0xe8,0xf2,0x3c,0x61,0x56,0x3a,0x4b,
0x64,0x5b,0x27,0x6c,
};
const uint8_t tc_g[] = {
0x04,0xf3,0x5a,0x92,0x5f,0xe8,0x2e,0x54,0x35,0x0e,0x80,0xb0,
0x84,0xa8,0x01,0x3b,0x19,0x60,0xcb,0x3f,0x73,0xc4,0x9b,0x0c,
0x2a,0xe9,0xb5,0x23,0x99,0x78,0x46,0xdd,0xd1,0x4c,0x66,0xf2,
0x4f,0x62,0x22,0x31,0x12,0xcf,0x35,0xb8,0x66,0x06,0x5f,0x91,
0xad,0x86,0x67,0x4c,0xce,0x2a,0x28,0x76,0x84,0x90,0x4e,0x49,
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0xf0,0x12,0x87,0xb5,0x46,0x66,0xbb,0x51,0x8d,0xf2,0xea,0x53,
0xce,0xc6,0x27,0xfa,0x6e,0x12,0x83,0xf1,0x4c,0x6e,0xd4,0xbc,
0xd1,0x1b,0x33,0xfb,0xb9,0x62,0xda,0x3e,0x2e,0x4f,0xf1,0x34,
0x5c,
};
const uint8_t tc_ya[] = {
0x7d,0x5b,0xc6,0xa8,0x95,0x9f,0x9d,0xb2,0x65,0x5b,0x8b,0x66,
0x42,0xe3,0x93,0xdc,0x13,0xd2,0x51,0x50,0xd6,0x9c,0x66,0x75,
0xfb,0x3e,0xfd,0x41,0xae,0x62,0x55,0xbf,0x54,0x20,0x2b,0x96,
0x0f,0x9a,0xac,0xd9,0x7f,0xd6,0xd2,0x84,0x1b,0x46,0x1f,0x18,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0x04,0x8b,0x65,0xb9,0xef,0x4c,0x57,0x26,0x66,0x43,0x91,0xce,
0xea,0xe2,0x41,0x83,0x4b,0x27,0x59,0x60,0xa6,0xf9,0x31,0x67,
0x99,0xf5,0xc9,0x74,0xec,0xeb,0x71,0xdf,0xb6,0xd3,0x6e,0x98,
0x9a,0xdd,0xf2,0xae,0x8c,0x4e,0x33,0x8f,0x20,0x4b,0x2c,0xd7,
0x54,0xe1,0xc4,0x3b,0x43,0xa1,0x26,0x92,0x8d,0x8d,0x81,0xce,
0x2e,0x6e,0xdb,0xc2,0x2a,0x99,0xed,0x47,0x8a,0xd3,0x48,0x7b,
0x87,0xe1,0x05,0x2b,0xce,0x2d,0x94,0xb6,0x46,0x4a,0x22,0x28,
0xea,0xb7,0x3c,0x01,0xf7,0x9d,0x6b,0x29,0x0a,0xf6,0xb2,0x18,
0xcf,
};
const uint8_t tc_yb[] = {
0x5c,0xc9,0x46,0x5b,0xdb,0x3a,0xe6,0x26,0xb7,0x75,0x21,0xea,
0x36,0x21,0x8f,0xc9,0x3a,0x96,0x93,0xc3,0x6f,0xf1,0x26,0x89,
0x9e,0x3d,0x87,0x77,0xc1,0x26,0xef,0x05,0x48,0x3e,0x34,0xc0,
0x55,0x76,0xc9,0xe8,0xc6,0x4b,0x1a,0x0b,0x6f,0x5b,0x53,0xd1,
};
const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0x04,0xcb,0x68,0x45,0x18,0x13,0x69,0x9a,0xbd,0xa3,0xdc,0x0e,
0xd9,0xd5,0x21,0xba,0xf9,0x10,0x8b,0xc2,0xc4,0xb2,0xa1,0xdb,
0xcd,0x90,0xa0,0x83,0x63,0xf5,0xe4,0x58,0x93,0x8d,0x6f,0xe6,
0x34,0xed,0x63,0x93,0xbc,0x84,0x40,0xec,0x9b,0x9f,0x8a,0x30,
0x84,0x1f,0xfc,0xc7,0xcd,0x65,0xd9,0xcf,0x96,0x17,0x15,0x5e,
0x12,0x9b,0xcc,0xde,0xd9,0x88,0x8a,0xc7,0x38,0xf7,0x8e,0x94,
0x0f,0x98,0x87,0xf9,0x08,0x9a,0xb6,0x27,0x5d,0x36,0xc3,0xab,
0x1b,0xcc,0xbd,0x04,0x8b,0x08,0x8b,0x1b,0x80,0xa0,0xf5,0x6f,
0x27,
};
const uint8_t tc_K[] = {
0xc8,0x62,0x70,0x9d,0x6b,0xfe,0x7c,0xc0,0x2f,0x0c,0x11,0xda,
0xfd,0xbf,0x4e,0xf8,0xdb,0x1c,0x5e,0x4c,0xb1,0x3a,0x22,0x98,
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0x5a,0x83,0xbe,0xf1,0x86,0x31,0x36,0x1e,0xd7,0xd8,0xcd,0x97,
0xb1,0x29,0x31,0x84,0x4b,0x7a,0xc6,0x1b,0x2f,0x31,0xd3,0x32,
};
const uint8_t tc_ISK_IR[] = {
0xdb,0x1e,0x81,0x33,0xbe,0x83,0x59,0xb9,0xaa,0x8c,0xd5,0x63,
0x04,0x3e,0xe7,0x84,0x34,0x4f,0x26,0x58,0x08,0x76,0x86,0x2e,
0x28,0xb3,0xf9,0x8b,0x51,0xb2,0xf6,0x11,0xa6,0x53,0x62,0xc1,
0xd7,0x7d,0xb6,0x6c,0x87,0x9d,0xe4,0x66,0xf5,0xb6,0x14,0x8a,
};
const uint8_t tc_ISK_SY[] = {
0x51,0x9b,0xfb,0xb1,0x47,0x76,0x52,0xe8,0xed,0x1b,0x4e,0xc5,
0x77,0x4e,0x31,0x0c,0x4f,0x44,0xda,0x46,0xf3,0xc3,0x6b,0xe9,
0x1b,0x0d,0xd6,0xb4,0xe3,0xa3,0x24,0x59,0x42,0xcf,0x4d,0x9d,
0xb8,0xf7,0x90,0x23,0xda,0xd6,0xe1,0xb5,0x7a,0xed,0x48,0x91,
};
B.6.8. Test case for scalar_mult_vfy with correct inputs
s: (length: 48 bytes)
6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9
2ae0d90bd57a564fd3500fbcce3425dc94ba0ade
X: (length: 97 bytes)
045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165
7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe
f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f
8ac1f301538badcce9d9f4036e
G.scalar_mult(s,X) (full coordinates): (length: 97 bytes)
0465c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f6105873
3173a033cda76c3e3ea5352b804e67fdbe2e334be8245dad5c8c993e
63bacf0456478f29b71b6c859f13676f84ff150d2741f028f560584a
0bdba19a63df62c08949c2fd6d
G.scalar_mult_vfy(s,X) (only X-coordinate):
(length: 48 bytes)
65c28db05fd9f9a93651c5cc31eae49c4e5246b46489b8f610587331
73a033cda76c3e3ea5352b804e67fdbe2e334be8
B.6.9. Invalid inputs for scalar_mult_vfy
For these test cases scalar_mult_vfy(y,.) MUST return the
representation of the neutral element G.I. When including Y_i1 or
Y_i2 in MSGa or MSGb the protocol MUST abort.
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s: (length: 48 bytes)
6e8a99a5cdd408eae98e1b8aed286e7b12adbbdac7f2c628d9060ce9
2ae0d90bd57a564fd3500fbcce3425dc94ba0ade
Y_i1: (length: 97 bytes)
045b4cd53c4506cc04ba4c44f2762d5d32c3e55df25b8baa5571b165
7ad9576efea8259f0684de065a470585b4be876748c7797054f3defe
f21b77f83d53bac57c89d52aa4d6dd5872bd281989b138359698009f
8ac1f301538badcce9d9f40302
Y_i2: (length: 1 bytes)
00
G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
B.7. Test vector for CPace using group NIST P-521 and hash SHA-512
B.7.1. Test vectors for calculate_generator with group NIST P-521
Inputs
H = SHA-512 with input block size 128 bytes.
PRS = b'Password' ; ZPAD length: 87 ;
DSI = b'CPaceP521_XMD:SHA-512_SSWU_NU_'
DST = b'CPaceP521_XMD:SHA-512_SSWU_NU__DST'
CI = b'\nAinitiator\nBresponder'
CI = 0a41696e69746961746f720a42726573706f6e646572
sid = 7e4b4791d6a8ef019b936c79fb7f2c57
Outputs
generator_string(PRS,G.DSI,CI,sid,H.s_in_bytes):
(length: 168 bytes)
1e4350616365503532315f584d443a5348412d3531325f535357555f
4e555f0850617373776f726457000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000160a41696e69746961746f72
0a42726573706f6e646572107e4b4791d6a8ef019b936c79fb7f2c57
generator g: (length: 133 bytes)
0400dc927958f0b69ccad8fb67ef008905354b58c7c9c92ad50060a9
e6afb10437d6ca8a26164e8573702b897275a25d05ed4407af2a3849
86dca7e243b92c5dd500d40057012121a9c8e34373fa619f918f7d47
9c23f85f0485379ef0f05284398de26653b49a155324c9d7b138be84
d0b49bb58e232b7bf697798de6ee8afd6b92b6fa2f
B.7.2. Test vector for MSGa
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Inputs
ADa = b'ADa'
ya (big endian): (length: 66 bytes)
006367e9c2aeff9f1db19af600cca73343d47cbe446cebbd1ccd783f
82755a872da86fd0707eb3767c6114f1803deb62d63bdd1e613f67e6
3e8c141ee5310e3ee819
Outputs
Ya: (length: 133 bytes)
04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d
ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5
286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178d7bf
d8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd271740469
bb322b07c179c7c225499b31727c0ea3ee65578634
Alternative correct value for Ya: g^(-ya):
(length: 133 bytes)
04003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065706d
ca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594ad5
286c068792ab7ca60ff6ea016e63be3ff18762543d0bd026be872840
27041e500e3e7ab4c2504c5f15ec0a5a03e8fc79f0fdd42d8e8bfb96
44cdd4f83e86383ddab664ce8d83f15c119aa879cb
MSGa = lv_cat(Ya,ADa): (length: 139 bytes)
850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065
706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59
4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178
d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174
0469bb322b07c179c7c225499b31727c0ea3ee6557863403414461
B.7.3. Test vector for MSGb
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Inputs
ADb = b'ADb'
yb (big endian): (length: 66 bytes)
009227bf8dc741dacc9422f8bf3c0e96fce9587bc562eaafe0dc5f6f
82f28594e4a6f98553560c62b75fa4abb198cecbbb86ebd41b0ea025
4cde78ac68d39a240ae7
Outputs
Yb: (length: 133 bytes)
0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3
bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2
82cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc4e
c691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32daf
bfd2d340d6e382d71b1fc763d7cec502fbcbcf93b4
Alternative correct value for Yb: g^(-yb):
(length: 133 bytes)
0400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6d3
bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399fa2
82cc1a78de91f3a4e30b5d005f7a4bac0dd40c236b8c794fbd1a03b1
396e011b801c3c139bf73dd5e83d943f548c6bf6ef9eb290311cd250
402d2cbf291c7d28e4e0389c28313afd0434306c4b
MSGb = lv_cat(Yb,ADb): (length: 139 bytes)
85010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1e
a6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa339
9fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5
fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee3
2dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462
B.7.4. Test vector for secret points K
scalar_mult_vfy(ya,Yb): (length: 66 bytes)
00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3
2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0
99bf8720dd8f6a07e8ed
scalar_mult_vfy(yb,Ya): (length: 66 bytes)
00503e75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c3
2cc45764d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa0
99bf8720dd8f6a07e8ed
B.7.5. Test vector for ISK calculation initiator/responder
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unordered cat of transcript : (length: 278 bytes)
850104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e1065
706dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc59
4ad5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178
d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd27174
0469bb322b07c179c7c225499b31727c0ea3ee655786340341446185
010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1ea6
d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa3399f
a282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5fc
4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee32d
afbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b403414462
DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK':
(length: 34 bytes)
4350616365503532315f584d443a5348412d3531325f535357555f4e
555f5f49534b
lv_cat(DSI,sid,K)||MSGa||MSGb: (length: 397 bytes)
224350616365503532315f584d443a5348412d3531325f535357555f
4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e
75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457
64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87
20dd8f6a07e8ed850104003701ec35caafa3dd416cad29ba1774551f
9d2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a05983
167d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e789d
abc2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703
860f022bd271740469bb322b07c179c7c225499b31727c0ea3ee6557
86340341446185010400f5cb68bf0117bd1a65412a2bc800af92013f
9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600be5
1295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc
947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab739409
10614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93
b403414462
ISK result: (length: 64 bytes)
ed208a15af3ef8a67a5cac4acb360d03154570e3b1b1c54867f53a72
53cb919d13aa47efc647375be2250cb39ad965afa4ddfcb6be47d586
d28c7eef6d654525
B.7.6. Test vector for ISK calculation parallel execution
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ordered cat of transcript : (length: 278 bytes)
85010400f5cb68bf0117bd1a65412a2bc800af92013f9969cf546e1e
a6d3bcf08643fdc482130aec1eecc33a2b5f33600be51295047fa339
9fa282cc1a78de91f3a4e30b5d01a085b453f22bf3dc947386b042e5
fc4ec691fee47fe3c3ec6408c22a17c26bc0ab73940910614d6fcee3
2dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf93b40341446285
0104003701ec35caafa3dd416cad29ba1774551f9d2ed89f7e106570
6dca230b86a11d02e4cee8b3fde64380d4a05983167d8a2414bc594a
d5286c068792ab7ca60ff6ea00919c41c00e789dabc2f42fd94178d7
bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc1703860f022bd2717404
69bb322b07c179c7c225499b31727c0ea3ee6557863403414461
DSI = G.DSI_ISK, b'CPaceP521_XMD:SHA-512_SSWU_NU__ISK':
(length: 34 bytes)
4350616365503532315f584d443a5348412d3531325f535357555f4e
555f5f49534b
lv_cat(DSI,sid,K)||oCAT(MSGa,MSGb): (length: 397 bytes)
224350616365503532315f584d443a5348412d3531325f535357555f
4e555f5f49534b107e4b4791d6a8ef019b936c79fb7f2c574200503e
75e38e012a6dc6f3561980e4cf540dbcff3de3a4a6f09d79c32cc457
64d3a6605eb45df1dc63fb7937b7879f2820da1b3266b69fa099bf87
20dd8f6a07e8ed85010400f5cb68bf0117bd1a65412a2bc800af9201
3f9969cf546e1ea6d3bcf08643fdc482130aec1eecc33a2b5f33600b
e51295047fa3399fa282cc1a78de91f3a4e30b5d01a085b453f22bf3
dc947386b042e5fc4ec691fee47fe3c3ec6408c22a17c26bc0ab7394
0910614d6fcee32dafbfd2d340d6e382d71b1fc763d7cec502fbcbcf
93b403414462850104003701ec35caafa3dd416cad29ba1774551f9d
2ed89f7e1065706dca230b86a11d02e4cee8b3fde64380d4a0598316
7d8a2414bc594ad5286c068792ab7ca60ff6ea00919c41c00e789dab
c2f42fd94178d7bfd8fbe1aff1c1854b3dafb3a0ea13f5a5fc170386
0f022bd271740469bb322b07c179c7c225499b31727c0ea3ee655786
3403414461
ISK result: (length: 64 bytes)
aae7320b73ba2516f289f71088662d41c4314d00521c48ea3c9c85ea
ca57112e55eb2b4094d4a0c7813ddd95c5d80c5596ad686d2eba876b
a1cd92f90407aa3d
B.7.7. Corresponding ANSI-C initializers
const uint8_t tc_PRS[] = {
0x50,0x61,0x73,0x73,0x77,0x6f,0x72,0x64,
};
const uint8_t tc_CI[] = {
0x0a,0x41,0x69,0x6e,0x69,0x74,0x69,0x61,0x74,0x6f,0x72,0x0a,
0x42,0x72,0x65,0x73,0x70,0x6f,0x6e,0x64,0x65,0x72,
};
const uint8_t tc_sid[] = {
0x7e,0x4b,0x47,0x91,0xd6,0xa8,0xef,0x01,0x9b,0x93,0x6c,0x79,
0xfb,0x7f,0x2c,0x57,
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};
const uint8_t tc_g[] = {
0x04,0x00,0xdc,0x92,0x79,0x58,0xf0,0xb6,0x9c,0xca,0xd8,0xfb,
0x67,0xef,0x00,0x89,0x05,0x35,0x4b,0x58,0xc7,0xc9,0xc9,0x2a,
0xd5,0x00,0x60,0xa9,0xe6,0xaf,0xb1,0x04,0x37,0xd6,0xca,0x8a,
0x26,0x16,0x4e,0x85,0x73,0x70,0x2b,0x89,0x72,0x75,0xa2,0x5d,
0x05,0xed,0x44,0x07,0xaf,0x2a,0x38,0x49,0x86,0xdc,0xa7,0xe2,
0x43,0xb9,0x2c,0x5d,0xd5,0x00,0xd4,0x00,0x57,0x01,0x21,0x21,
0xa9,0xc8,0xe3,0x43,0x73,0xfa,0x61,0x9f,0x91,0x8f,0x7d,0x47,
0x9c,0x23,0xf8,0x5f,0x04,0x85,0x37,0x9e,0xf0,0xf0,0x52,0x84,
0x39,0x8d,0xe2,0x66,0x53,0xb4,0x9a,0x15,0x53,0x24,0xc9,0xd7,
0xb1,0x38,0xbe,0x84,0xd0,0xb4,0x9b,0xb5,0x8e,0x23,0x2b,0x7b,
0xf6,0x97,0x79,0x8d,0xe6,0xee,0x8a,0xfd,0x6b,0x92,0xb6,0xfa,
0x2f,
};
const uint8_t tc_ya[] = {
0x00,0x63,0x67,0xe9,0xc2,0xae,0xff,0x9f,0x1d,0xb1,0x9a,0xf6,
0x00,0xcc,0xa7,0x33,0x43,0xd4,0x7c,0xbe,0x44,0x6c,0xeb,0xbd,
0x1c,0xcd,0x78,0x3f,0x82,0x75,0x5a,0x87,0x2d,0xa8,0x6f,0xd0,
0x70,0x7e,0xb3,0x76,0x7c,0x61,0x14,0xf1,0x80,0x3d,0xeb,0x62,
0xd6,0x3b,0xdd,0x1e,0x61,0x3f,0x67,0xe6,0x3e,0x8c,0x14,0x1e,
0xe5,0x31,0x0e,0x3e,0xe8,0x19,
};
const uint8_t tc_ADa[] = {
0x41,0x44,0x61,
};
const uint8_t tc_Ya[] = {
0x04,0x00,0x37,0x01,0xec,0x35,0xca,0xaf,0xa3,0xdd,0x41,0x6c,
0xad,0x29,0xba,0x17,0x74,0x55,0x1f,0x9d,0x2e,0xd8,0x9f,0x7e,
0x10,0x65,0x70,0x6d,0xca,0x23,0x0b,0x86,0xa1,0x1d,0x02,0xe4,
0xce,0xe8,0xb3,0xfd,0xe6,0x43,0x80,0xd4,0xa0,0x59,0x83,0x16,
0x7d,0x8a,0x24,0x14,0xbc,0x59,0x4a,0xd5,0x28,0x6c,0x06,0x87,
0x92,0xab,0x7c,0xa6,0x0f,0xf6,0xea,0x00,0x91,0x9c,0x41,0xc0,
0x0e,0x78,0x9d,0xab,0xc2,0xf4,0x2f,0xd9,0x41,0x78,0xd7,0xbf,
0xd8,0xfb,0xe1,0xaf,0xf1,0xc1,0x85,0x4b,0x3d,0xaf,0xb3,0xa0,
0xea,0x13,0xf5,0xa5,0xfc,0x17,0x03,0x86,0x0f,0x02,0x2b,0xd2,
0x71,0x74,0x04,0x69,0xbb,0x32,0x2b,0x07,0xc1,0x79,0xc7,0xc2,
0x25,0x49,0x9b,0x31,0x72,0x7c,0x0e,0xa3,0xee,0x65,0x57,0x86,
0x34,
};
const uint8_t tc_yb[] = {
0x00,0x92,0x27,0xbf,0x8d,0xc7,0x41,0xda,0xcc,0x94,0x22,0xf8,
0xbf,0x3c,0x0e,0x96,0xfc,0xe9,0x58,0x7b,0xc5,0x62,0xea,0xaf,
0xe0,0xdc,0x5f,0x6f,0x82,0xf2,0x85,0x94,0xe4,0xa6,0xf9,0x85,
0x53,0x56,0x0c,0x62,0xb7,0x5f,0xa4,0xab,0xb1,0x98,0xce,0xcb,
0xbb,0x86,0xeb,0xd4,0x1b,0x0e,0xa0,0x25,0x4c,0xde,0x78,0xac,
0x68,0xd3,0x9a,0x24,0x0a,0xe7,
};
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const uint8_t tc_ADb[] = {
0x41,0x44,0x62,
};
const uint8_t tc_Yb[] = {
0x04,0x00,0xf5,0xcb,0x68,0xbf,0x01,0x17,0xbd,0x1a,0x65,0x41,
0x2a,0x2b,0xc8,0x00,0xaf,0x92,0x01,0x3f,0x99,0x69,0xcf,0x54,
0x6e,0x1e,0xa6,0xd3,0xbc,0xf0,0x86,0x43,0xfd,0xc4,0x82,0x13,
0x0a,0xec,0x1e,0xec,0xc3,0x3a,0x2b,0x5f,0x33,0x60,0x0b,0xe5,
0x12,0x95,0x04,0x7f,0xa3,0x39,0x9f,0xa2,0x82,0xcc,0x1a,0x78,
0xde,0x91,0xf3,0xa4,0xe3,0x0b,0x5d,0x01,0xa0,0x85,0xb4,0x53,
0xf2,0x2b,0xf3,0xdc,0x94,0x73,0x86,0xb0,0x42,0xe5,0xfc,0x4e,
0xc6,0x91,0xfe,0xe4,0x7f,0xe3,0xc3,0xec,0x64,0x08,0xc2,0x2a,
0x17,0xc2,0x6b,0xc0,0xab,0x73,0x94,0x09,0x10,0x61,0x4d,0x6f,
0xce,0xe3,0x2d,0xaf,0xbf,0xd2,0xd3,0x40,0xd6,0xe3,0x82,0xd7,
0x1b,0x1f,0xc7,0x63,0xd7,0xce,0xc5,0x02,0xfb,0xcb,0xcf,0x93,
0xb4,
};
const uint8_t tc_K[] = {
0x00,0x50,0x3e,0x75,0xe3,0x8e,0x01,0x2a,0x6d,0xc6,0xf3,0x56,
0x19,0x80,0xe4,0xcf,0x54,0x0d,0xbc,0xff,0x3d,0xe3,0xa4,0xa6,
0xf0,0x9d,0x79,0xc3,0x2c,0xc4,0x57,0x64,0xd3,0xa6,0x60,0x5e,
0xb4,0x5d,0xf1,0xdc,0x63,0xfb,0x79,0x37,0xb7,0x87,0x9f,0x28,
0x20,0xda,0x1b,0x32,0x66,0xb6,0x9f,0xa0,0x99,0xbf,0x87,0x20,
0xdd,0x8f,0x6a,0x07,0xe8,0xed,
};
const uint8_t tc_ISK_IR[] = {
0xed,0x20,0x8a,0x15,0xaf,0x3e,0xf8,0xa6,0x7a,0x5c,0xac,0x4a,
0xcb,0x36,0x0d,0x03,0x15,0x45,0x70,0xe3,0xb1,0xb1,0xc5,0x48,
0x67,0xf5,0x3a,0x72,0x53,0xcb,0x91,0x9d,0x13,0xaa,0x47,0xef,
0xc6,0x47,0x37,0x5b,0xe2,0x25,0x0c,0xb3,0x9a,0xd9,0x65,0xaf,
0xa4,0xdd,0xfc,0xb6,0xbe,0x47,0xd5,0x86,0xd2,0x8c,0x7e,0xef,
0x6d,0x65,0x45,0x25,
};
const uint8_t tc_ISK_SY[] = {
0xaa,0xe7,0x32,0x0b,0x73,0xba,0x25,0x16,0xf2,0x89,0xf7,0x10,
0x88,0x66,0x2d,0x41,0xc4,0x31,0x4d,0x00,0x52,0x1c,0x48,0xea,
0x3c,0x9c,0x85,0xea,0xca,0x57,0x11,0x2e,0x55,0xeb,0x2b,0x40,
0x94,0xd4,0xa0,0xc7,0x81,0x3d,0xdd,0x95,0xc5,0xd8,0x0c,0x55,
0x96,0xad,0x68,0x6d,0x2e,0xba,0x87,0x6b,0xa1,0xcd,0x92,0xf9,
0x04,0x07,0xaa,0x3d,
};
B.7.8. Test case for scalar_mult_vfy with correct inputs
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s: (length: 66 bytes)
0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15
87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c
480965ef65ff2048b856
X: (length: 133 bytes)
0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2
26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda
c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008
860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74
c00696cb1b13e40bda15b27964d7670576647bfab9
G.scalar_mult(s,X) (full coordinates): (length: 133 bytes)
040122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7
c73cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517c
df6d99a8a2d6db19ef27bd0055af9e8ddcf337ce0a7c22a9c8099bc4
a44faeded1eb72effd26e4f322217b67d60b944b267b3df5046078fd
577f1785728f49b241fd5e8c83223a994a2d219281
G.scalar_mult_vfy(s,X) (only X-coordinate):
(length: 66 bytes)
0122f88ce73ec5aa2d1c8c5d04148760c3d97ba87daa10d8cb8bb7c7
3cf6e951fc922721bf1437995cfb13e132a78beb86389e60d3517cdf
6d99a8a2d6db19ef27bd
B.7.9. Invalid inputs for scalar_mult_vfy
For these test cases scalar_mult_vfy(y,.) MUST return the
representation of the neutral element G.I. When including Y_i1 or
Y_i2 in MSGa or MSGb the protocol MUST abort.
s: (length: 66 bytes)
0182dd7925f1753419e4bf83429763acd37d64000cd5a175edf53a15
87dd986bc95acc1506991702b6ba1a9ee2458fee8efc00198cf0088c
480965ef65ff2048b856
Y_i1: (length: 133 bytes)
0400dc5078b24c4af1620cc10fbecc6cd8cf1cab0b011efb73c782f2
26dc21c7ca7eb406be74a69ecba5b4a87c07cfc6e687b4beca9a6eda
c95940a3b4120573b26a80005e697833b0ba285fce7b3f1f25243008
860b8f1de710a0dcc05b0d20341efe90eb2bcca26797c2d85ae6ca74
c00696cb1b13e40bda15b27964d7670576647bfaf9
Y_i2: (length: 1 bytes)
00
G.scalar_mult_vfy(s,Y_i1) = G.scalar_mult_vfy(s,Y_i2) = G.I
Authors' Addresses
Michel Abdalla
DFINITY - Zurich
Email: michel.abdalla@gmail.com
Abdalla, et al. Expires 28 March 2024 [Page 73]
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Bjoern Haase
Endress + Hauser Liquid Analysis - Gerlingen
Email: bjoern.m.haase@web.de
Julia Hesse
IBM Research Europe - Zurich
Email: JHS@zurich.ibm.com
Abdalla, et al. Expires 28 March 2024 [Page 74]
