| Network Working Group | F. Kiefer |
| Internet-Draft | Mozilla |
| Intended status: Experimental | K. Kwiatkowski |
| Expires: May 10, 2019 | Cloudflare |
| November 06, 2018 |
Hybrid ECDHE-SIDH Key Exchange for TLS
draft-kiefer-tls-ecdhe-sidh-00
This draft specifies a TLS key exchange that combines the post-quantum key exchange, Supersingular elliptic curve isogenie diffie-hellman (SIDH), with elliptic curve Diffie-Hellman (ECDHE) key exchange.
This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.
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This Internet-Draft will expire on May 10, 2019.
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Supersingular elliptic curve isogenie diffie-hellman (SIDH) has been proposed [SIDH] as a diffie-hellman like key-exchange protocol secure against quantum computers. Because there’s not enough confidence in the security of SIDH yet it should only be used in combination with a classical key-exchange scheme.
This document defines a way to combine [eSIDH] with the ECDHE key exchanges defined in [RFC7748] for the TLS 1.3 [RFC8446] key-exchange.
x25519 is combined with sidh503 and x448 is combined with sidh751.
Both handshake partners have to compute the SIDH values in addition to the ECDHE values, which requires additional time for computation. The handshake messages also get larger because the SIDH values are added (see Section 4 for details).
x25519 and x448 denote the ECDHE algorithms defined over the respective curve from [RFC7748]. sidh503 and sidh751 denote the SIDH algorithms defined using a prime of bit-length 503 and 751 respectively.
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.
A hybrid key exchange takes the output of two separate key exchanges and mixes the results in a secure way.
The ECDHE and SIDH shared secrets are calculated independently. The shared secret for ECDHE-SIDH is then the concatenation of the ECDHE and the SIDH shared secrets. For x25519sidh503 for example this is
secret = x25519_secret || sidh_secret
The HKDF-Extract step used by TLS is relied on to combine entropy from both secrets.
The ECDHE shared secret calculation is performed as described in Section 7.4.2 of [RFC8446].
This document uses primes p503 and p751 defined in [eSIDH] and [SIKE] for sidh503 and sidh751. See [SIKE] for details on how to compute key-exchange messages and the shared secret. Optimised versions of the algorithms mentioned here might be used.
Each element (c=a+b*i) of the underlying quadratic field GF(p^2) is encoded as an array of bytes in little-endian order, i.e., the least significant octet appears first, where each element a,b from GF(p) is encoded using itoos from [SIKE] Section 1.2.6. In particular, an element of GF(p) is converted to
e_(n-1) * 256^(n-1) + ... + e_1 * 256 + e_0
with n 63 for p503 and 94 for p751. The octet representation of each element is then the concatenation of e_i in little endian, i.e. e_0||...||e_(n-1), and the octet representation of element c is the concatenation of a and b, a||b.
See fp2toos [SIKE] Section 1.2.6 to 1.2.8 for details.
After choosing a private key each party computes its public key (P, Q, R) using isogen_l from [SIKE] Section 1.3.5 and converts each element into octets (cf. Section 2.2.1).
See pktoos from [SIKE] Section 1.2.9 for details on converting the public key to octets.
The SIDH shared secret is calculated as described in Section 1.3.6 of [SIKE] using isoex_l.
Calculating SIDH shared secret requires each side to use isogenies of different degree. This document assumes parameterizations as described in [SIKE], which is based on 4- and 3-power degree isogenies. In order to calculate the shared secret, the client always generates an ephemeral key pair based on 4-power degree isogenies. Accordingly, the server always generates an ephemeral key pair based on 3-power degree isogenies.
The shared secret is a j-invariant and therefore an element of GF(p^2). It is converted to octets as described in Section 2.2.1.
See fp2toos [SIKE] Section 1.2.6 to 1.2.8 for details. All values are encoded without length prefixes or separators.
This document extends the enum of NamedGroups to use in the supported_groups extension from TLS 1.3 [RFC8446] Section 4.2.7. The new codepoint for the “Supported Groups Registry” is:
enum {
...,
x25519sidh503(0x0105), x448sidh751(0x0106),
} NamedGroup;
This document defines ECDHE-SIDH parameters to use in the key_share extension from TLS 1.3 (see Section 4.2.8 of [RFC8446]).
ECDHE parameters for both clients and servers are encoded in the key_exchange field of a KeyShareEntry as described in Section 4.2.8 of [RFC8446] and [RFC7748]. SIDH parameters are appended to this value.
In particular, the contents are the serialised value of the following struct:
struct {
opaque X[coordinate_length];
opaque P[sidh_coordinate_length];
opaque Q[sidh_coordinate_length];
opaque R[sidh_coordinate_length];
} UncompressedPointRepresentation;
X is the public point from x25519 or x448 as described in [RFC7748].
P, Q, and R are the binary representations of three field elements over GF(p503^2) and GF(p751^2) respectively from the public SIDH key values as described in Section 2.2.2. All values in the struct are encoded without length prefixes or separators.
Implementers MUST perform the checks to verify the SIDH public key as specified in Section 9 of [eSIDH].
Security of SIDH is based on the isogeny walk problem, assuming elliptic curves between isogenies are supersingular (see [SIKE] chapter 4.1). Algorithms solving this problem as well as usage of isogenies as drop-in replacement for Diffie-Hellman are relatively young area of research. Therefore the security behind the ECDHE-SIDH handshake does not rely on the security of SIDH exclusively.
Idea behind ECDHE-SIDH hybrid scheme is to combine an existing key-agreement algorithm with what’s believed to be a quantum-resistant one. When large quantum computers are available they will be able to break both x25519 and x448. In this case the ECDHE-SIDH scheme is still safe assuming SIDH is secure. On the other hand, if SIDH is found to be flawed, the hybrid scheme is still secure against classical attacks assuming security of x25519/x448. Security estimates for classical and quantum computers are provided in table below based on [SIKE] and [RFC7748]. [RNSL] Chapter 1 provides introduction to quantum resource estimates.
| Scheme | Classical | Quantum | NIST PQ category |
|---|---|---|---|
| x25519sidh503 | 128-bit | 64-qubit | 1 |
| x448sidh751 | 224-bit | 96-qubit | 3 |
As described in [ISOSEC] it is possible to perform active attacks on static-static or non-interactive variants of the SIDH scheme. The countermeasure for this attack was described in [KLM15]. Research proposes so-called “indirect key validation”, using Fujisaki-Okamoto type transform. However, using this transform is impractical and thus SIDH can be considered secure only if used for ephemeral keys. A more detailed discussion can be found in [URBJAO].
Security against side-channel attacks is described in [SIKE]. Implementers are encouraged to use a constant-time implementation.
The security of the described key exchange relies on the security, in particular the collision resistance, of the used key-derivation function. TLS 1.3 uses HKDF [RFC5869] as its key-derivation function. It is therefore important that the hash function used in HKDF is collision-resistant.
This section gives a test vectors for x25519sidh503.
Client SIDH-ECDHE public key:
Public Key (X25519||SIDH) [Len: 410 (32+378)]
2b 5b 10 41 56 2f c7 04 d8 56 ce 41 9c e3 7d e6
ae 0e 32 a9 98 5e a5 95 47 5a 9d a2 98 59 67 16
6f ed 78 ba b1 01 e7 f4 c4 f4 9a d6 4f ac 8d ee
ee 46 57 10 f7 12 40 41 7a 45 53 c2 35 b5 f7 a0
42 9e c3 38 d3 7c 47 11 78 f7 8d d3 c9 18 9c 79
7c dc d3 7f e4 93 ac 63 c4 77 5e 36 43 d5 2a 43
ee a9 37 b6 88 41 86 98 c9 dc d6 b7 20 66 ab 3d
d1 e4 f7 90 80 8d 8e fb 0b bf 79 bb b6 e2 13 c0
38 4e 86 20 13 49 81 be 31 f9 2c 73 a3 2a e9 3c
e1 7e b5 1b 75 2d 3f 26 79 7c c2 e5 e5 16 57 1d
6f b4 06 4b 5d b1 9b bd d2 cf 4b f1 2b cd f9 b2
5f 2c 9d 1b d1 78 55 4c b7 ec fa 7a 3a 64 dd db
6b 43 0f 67 e9 61 1d 57 fe 63 c8 d4 b3 0f 7a 2f
60 1b 0f 6a 3a e6 80 0c 14 b7 05 ae 06 bb 5c 46
71 1b 0e d7 a0 e7 bb 5d 87 37 c4 56 d8 c4 b2 e1
01 a6 39 70 14 13 50 22 4f cf d9 20 77 51 f7 c4
37 27 c0 57 5c f2 be 36 3b f7 38 1f 95 5f 54 fc
f4 ce 96 24 fa 04 d8 62 03 aa 9b 24 28 56 47 e9
c3 04 24 5e ee b5 3e 16 25 c9 b3 0d 70 6e e7 a1
a8 76 bf 8c 53 78 7f d0 a3 13 26 fd 3a b5 f6 11
05 60 af 4a ad 7e 45 0c 41 de 52 e5 29 4d a1 42
c3 7b 88 7b 6a ed 66 03 04 25 12 78 31 36 94 58
86 e6 00 59 13 99 0c c1 5d 1c d0 f7 aa c3 73 a9
dd 25 ac cd 4d 04 2a dd 77 f0 b0 96 6e 3a 0b 76
df 59 92 de 38 fe e5 10 5d 8b 6d e5 b9 1e 1e 8c
8b 9c a5 9c 52 2f 26 d6 73 0d
Client SIDH-ECDHE private key:
SIDH Private Key [Len: 32]
37 83 09 b4 a8 c4 b8 6a 83 84 36 8a 18 55 d8 48
69 f2 31 60 2e f0 a6 70 d3 24 fe 92 e5 25 82 01
X25519 Private Key [Len: 32]
a0 31 67 54 87 02 0f cb ef 07 40 af d3 ec 90 19
88 02 fa d5 83 46 46 c9 8e 0e 49 c0 e1 3e 86 1a
Client Hello:
CH [Len: 522]
01 00 02 06 03 03 bf d0 ca 2e 81 e2 6e c4 d8 01
6b 2e 86 f3 2e d9 d0 f9 83 93 85 03 50 3f 67 05
71 cf 82 1d 5c e4 00 00 02 13 01 01 00 01 db 00
00 00 0b 00 09 00 00 06 73 65 72 76 65 72 ff 01
00 01 00 00 0a 00 04 00 02 01 05 00 33 01 a0 01
9e 01 05 01 9a 2b 5b 10 41 56 2f c7 04 d8 56 ce
41 9c e3 7d e6 ae 0e 32 a9 98 5e a5 95 47 5a 9d
a2 98 59 67 16 6f ed 78 ba b1 01 e7 f4 c4 f4 9a
d6 4f ac 8d ee ee 46 57 10 f7 12 40 41 7a 45 53
c2 35 b5 f7 a0 42 9e c3 38 d3 7c 47 11 78 f7 8d
d3 c9 18 9c 79 7c dc d3 7f e4 93 ac 63 c4 77 5e
36 43 d5 2a 43 ee a9 37 b6 88 41 86 98 c9 dc d6
b7 20 66 ab 3d d1 e4 f7 90 80 8d 8e fb 0b bf 79
bb b6 e2 13 c0 38 4e 86 20 13 49 81 be 31 f9 2c
73 a3 2a e9 3c e1 7e b5 1b 75 2d 3f 26 79 7c c2
e5 e5 16 57 1d 6f b4 06 4b 5d b1 9b bd d2 cf 4b
f1 2b cd f9 b2 5f 2c 9d 1b d1 78 55 4c b7 ec fa
7a 3a 64 dd db 6b 43 0f 67 e9 61 1d 57 fe 63 c8
d4 b3 0f 7a 2f 60 1b 0f 6a 3a e6 80 0c 14 b7 05
ae 06 bb 5c 46 71 1b 0e d7 a0 e7 bb 5d 87 37 c4
56 d8 c4 b2 e1 01 a6 39 70 14 13 50 22 4f cf d9
20 77 51 f7 c4 37 27 c0 57 5c f2 be 36 3b f7 38
1f 95 5f 54 fc f4 ce 96 24 fa 04 d8 62 03 aa 9b
24 28 56 47 e9 c3 04 24 5e ee b5 3e 16 25 c9 b3
0d 70 6e e7 a1 a8 76 bf 8c 53 78 7f d0 a3 13 26
fd 3a b5 f6 11 05 60 af 4a ad 7e 45 0c 41 de 52
e5 29 4d a1 42 c3 7b 88 7b 6a ed 66 03 04 25 12
78 31 36 94 58 86 e6 00 59 13 99 0c c1 5d 1c d0
f7 aa c3 73 a9 dd 25 ac cd 4d 04 2a dd 77 f0 b0
96 6e 3a 0b 76 df 59 92 de 38 fe e5 10 5d 8b 6d
e5 b9 1e 1e 8c 8b 9c a5 9c 52 2f 26 d6 73 0d 00
2b 00 03 02 7f 1c 00 0d 00 04 00 02 04 01 00 2d
00 02 01 01 00 1c 00 02 40 01
Server selected KE = (EC)DHE. Group = 261.
Server SIDH-ECDHE public key:
Public Key (X25519||SIDH) [Len: 410 (32+378)]
f8 c6 8f 4e 57 6b fb ec b3 de 23 d9 db 89 fc 1b
f4 6f 01 a5 c0 91 61 fd c4 e7 bc 58 b4 eb 5f 76
44 c4 c7 7b a3 09 1a 60 c7 15 8f 1e d6 83 f2 1c
f8 36 13 a4 b3 c5 bc 4e 73 41 96 36 34 9a 9e 5a
bc fc 9d fa 2b c3 2c 85 17 44 9b 21 8f bf ba f7
7b 6c 19 c3 07 19 45 34 1e 88 cd 86 41 f8 32 38
41 3d 75 20 e1 c9 4a 94 03 e4 2f 4b 38 2d 93 39
b7 71 e9 84 80 b9 aa ca 97 39 5a c6 68 a7 b2 6f
b0 3e 10 f0 02 e3 e3 62 78 23 b4 f7 f1 a8 ce cd
71 a8 3a 23 81 63 ba 70 92 ea c6 9b 2c 35 93 6d
b5 58 61 6d 2c 06 a5 4d 0d 27 35 20 0b 77 d0 0d
65 f0 24 11 71 0b 71 45 2b 73 9c 42 fd d4 09 ba
8a ed d2 9e 78 9c 2f 43 91 5d e7 3a 19 0b f8 2b
71 6d 47 ae 86 e4 7a 9e e1 a0 de b5 08 bd a4 30
bb c1 3e ad db 75 79 36 a0 0a ea 70 a0 9c 64 f7
ba 92 a4 02 05 4d d4 9b ba a8 b3 9e 92 cd 28 13
0e 84 81 90 84 cd ae 09 b2 0b 12 23 1c b4 3a 18
cb 66 a1 8a 81 63 d4 e7 06 1c 16 04 29 20 2b cf
da a3 90 55 15 4a 15 ab 30 95 f1 20 b0 84 f5 7e
0f 92 f6 7f 4d 8c 22 2a a8 80 41 7b ee fa 85 f8
e2 4d 45 38 28 eb e2 fd a5 c6 1e 37 98 9f a2 ed
13 b9 dd f5 21 bc 78 10 2f 99 21 dc 30 55 57 58
c6 59 89 13 f9 76 aa e1 ec 0d 82 27 74 a1 86 b5
d1 74 12 49 5f ac a0 25 d2 91 5a 26 11 5e 0e f8
d2 7f 00 7f 8e 8b 7d 89 93 ba 69 4c 5f c7 7c df
d0 45 f4 17 3c 0c 03 df bf 1e
Server SIDH-ECDHE private key:
SIDH Private Key [Len: 32]
ca a5 1b 8d cc 2e df b0 b9 f5 ed 9d b0 1c f4 7c
b6 61 07 4d 5f e3 9d 6a 24 48 71 48 f3 11 4d 0a
X25519 Private Key [Len: 32]
a1 27 74 2c 0e ea 56 25 41 68 f4 7c d0 94 30 03
5e 7e cb 3d e0 4f 84 36 41 e8 b4 39 1e 45 99 91
Handshake secrets tls13 s hs traffic:
PRK [Len: 32]
35 7b 46 ed 6d f0 40 77 ae 2a a0 f4 47 cc df c1
78 54 74 48 d4 ff 69 05 f9 d5 2f 9a 00 1c e8 86
Hash [Len: 32]
52 a5 04 4f 78 da 41 12 b8 ac 35 f3 37 54 0c 51
18 ba c9 be c7 de 06 21 b2 f8 22 b6 e1 fa b5 96
Info [Len: 54]
00 20 12 74 6c 73 31 33 20 73 20 68 73 20 74 72
61 66 66 69 63 20 52 a5 04 4f 78 da 41 12 b8 ac
35 f3 37 54 0c 51 18 ba c9 be c7 de 06 21 b2 f8
22 b6 e1 fa b5 96
Derived key [Len: 32]
e4 31 ba e7 1e 38 f1 d6 81 17 83 56 d3 8d 0e 35
cf 42 6a 05 a2 2b 03 df d6 bb 4f 72 94 d2 9f c3
Handshake secrets tls13 c hs traffic:
Info [Len: 54]
00 20 12 74 6c 73 31 33 20 63 20 68 73 20 74 72
61 66 66 69 63 20 52 a5 04 4f 78 da 41 12 b8 ac
35 f3 37 54 0c 51 18 ba c9 be c7 de 06 21 b2 f8
22 b6 e1 fa b5 96
Derived key [Len: 32]
d2 99 cb cb 70 91 05 b6 3f 62 a5 e7 a2 5c 9c 07
2b 98 d9 0c d0 92 1c f2 0f c3 6a fa b3 57 4d 2a
Server Handshake:
SH [Len: 468]
02 00 01 d0 03 03 8b 94 f0 f0 25 bd 87 30 3f 1c
7c 86 e0 bc 25 e3 7f d7 ca 77 88 c5 a3 3c 69 34
c8 ec a9 64 15 85 00 13 01 00 01 a8 00 33 01 9e
01 05 01 9a f8 c6 8f 4e 57 6b fb ec b3 de 23 d9
db 89 fc 1b f4 6f 01 a5 c0 91 61 fd c4 e7 bc 58
b4 eb 5f 76 44 c4 c7 7b a3 09 1a 60 c7 15 8f 1e
d6 83 f2 1c f8 36 13 a4 b3 c5 bc 4e 73 41 96 36
34 9a 9e 5a bc fc 9d fa 2b c3 2c 85 17 44 9b 21
8f bf ba f7 7b 6c 19 c3 07 19 45 34 1e 88 cd 86
41 f8 32 38 41 3d 75 20 e1 c9 4a 94 03 e4 2f 4b
38 2d 93 39 b7 71 e9 84 80 b9 aa ca 97 39 5a c6
68 a7 b2 6f b0 3e 10 f0 02 e3 e3 62 78 23 b4 f7
f1 a8 ce cd 71 a8 3a 23 81 63 ba 70 92 ea c6 9b
2c 35 93 6d b5 58 61 6d 2c 06 a5 4d 0d 27 35 20
0b 77 d0 0d 65 f0 24 11 71 0b 71 45 2b 73 9c 42
fd d4 09 ba 8a ed d2 9e 78 9c 2f 43 91 5d e7 3a
19 0b f8 2b 71 6d 47 ae 86 e4 7a 9e e1 a0 de b5
08 bd a4 30 bb c1 3e ad db 75 79 36 a0 0a ea 70
a0 9c 64 f7 ba 92 a4 02 05 4d d4 9b ba a8 b3 9e
92 cd 28 13 0e 84 81 90 84 cd ae 09 b2 0b 12 23
1c b4 3a 18 cb 66 a1 8a 81 63 d4 e7 06 1c 16 04
29 20 2b cf da a3 90 55 15 4a 15 ab 30 95 f1 20
b0 84 f5 7e 0f 92 f6 7f 4d 8c 22 2a a8 80 41 7b
ee fa 85 f8 e2 4d 45 38 28 eb e2 fd a5 c6 1e 37
98 9f a2 ed 13 b9 dd f5 21 bc 78 10 2f 99 21 dc
30 55 57 58 c6 59 89 13 f9 76 aa e1 ec 0d 82 27
74 a1 86 b5 d1 74 12 49 5f ac a0 25 d2 91 5a 26
11 5e 0e f8 d2 7f 00 7f 8e 8b 7d 89 93 ba 69 4c
5f c7 7c df d0 45 f4 17 3c 0c 03 df bf 1e 00 2b
00 02 7f 1c
SH [Len: 651]
08 00 00 14 00 12 00 0a 00 04 00 02 01 05 00 1c
00 02 40 01 00 00 00 00 0b 00 01 c3 00 00 01 bf
00 01 ba 30 82 01 b6 30 82 01 1f a0 03 02 01 02
02 01 05 30 0d 06 09 2a 86 48 86 f7 0d 01 01 0b
05 00 30 13 31 11 30 0f 06 03 55 04 03 0c 08 72
73 61 5f 73 69 67 6e 30 1e 17 0d 31 38 30 36 31
30 31 30 32 34 31 32 5a 17 0d 32 38 30 36 31 30
31 30 32 34 31 32 5a 30 13 31 11 30 0f 06 03 55
04 03 0c 08 72 73 61 5f 73 69 67 6e 30 81 9f 30
0d 06 09 2a 86 48 86 f7 0d 01 01 01 05 00 03 81
8d 00 30 81 89 02 81 81 00 c6 96 5e 96 71 5a ea
40 c2 c0 60 c7 c7 4e 98 b3 40 0c 02 a3 1c 9c 8e
e7 c6 57 6b 48 8c 23 04 d4 e8 54 09 37 c2 b8 b1
ac b4 49 b7 76 ef 59 f8 3f 7c 4e e3 6a fa 32 04
53 74 85 2d 0d 8e 91 ad 2d 65 52 ec f2 54 1c 82
f1 b5 46 c8 5e ec e1 4e 6a f1 a1 c8 9f 9c 2b e1
79 3b 85 58 80 19 d5 f2 87 cb c0 13 5f 56 56 d3
75 78 bb 71 ef fa df e4 98 76 31 47 72 9b 5d 6a
fe d7 c9 58 e6 d2 c6 2c 5f 02 03 01 00 01 a3 1a
30 18 30 09 06 03 55 1d 13 04 02 30 00 30 0b 06
03 55 1d 0f 04 04 03 02 07 80 30 0d 06 09 2a 86
48 86 f7 0d 01 01 0b 05 00 03 81 81 00 07 58 be
81 c3 60 a0 cb 94 bc 79 81 0c b5 c6 ec 84 c6 c0
f8 d9 63 50 0a 7e b2 9d 80 95 5d b2 ba c9 31 72
73 d5 78 97 d1 5f e3 d9 f8 54 25 e0 1a 0e 7f 2b
ec 20 27 b6 ba ff 9c 38 42 23 ed 10 c4 51 54 f2
a3 45 54 df 59 be 83 d4 b8 00 7f 13 a3 27 40 ca
af 66 72 f5 f7 cf 1e 4d 6c 94 e4 02 46 cf f7 9d
13 7e 72 6b 77 20 15 9c c9 f7 c2 f1 5f 00 91 d3
da c6 2e 71 f0 51 82 db 13 b2 ee c7 0c 00 00 0f
00 00 84 04 01 00 80 7e c8 cb 1d d8 2e 83 d4 7e
69 8a db 8d 39 79 13 49 9f 03 21 7a 2f 5f ef df
2e 07 58 a8 0f 4b 61 85 10 25 01 3c cc 14 ef ac
35 17 c8 ed 27 17 0f 6e e5 78 7e 19 b5 0a 99 2d
bd 68 f4 47 0e 0a 11 cb 57 12 d5 73 cd 20 05 a4
b5 04 6c 13 6c 1b 9a f9 15 aa cc ca c2 22 83 fe
37 5f c3 f2 24 09 e3 d5 df 26 9f ab 84 e9 92 68
38 73 09 b1 58 55 43 79 02 59 0c 13 cc 68 4f 53
62 4b 72 d7 3b 17 86 14 00 00 20 d1 6e ba a6 9a
12 01 ec 46 fb e4 2c 8b 0e dd b5 73 a9 d7 e1 da
ba c5 c3 0d 53 5d 90 24 fd 53 60
Client finished handshake:
Client finished [Len: 36]
14 00 00 20 8e 34 2d 9d 69 fb 95 76 65 05 03 4d
cf 27 21 59 7c 45 f7 0e 3f 1e d9 29 18 4b 29 87
6a a5 c1 4d
Shared secret (server & client):
Shared secret (X25519||SIDH) [Len: 158 (32+126)]
f0 93 ac 03 ca 0b 5c 05 e6 c3 d3 7f ae 71 10 57
a6 a6 3e c7 7c 12 8d 21 8b 39 fc a5 8a 19 69 02
31 c8 0b 85 96 07 d4 f2 9b 5d ca a1 2d 69 78 2a
4f d8 1e c5 ea 87 ff 24 a2 7e b3 96 db 63 d5 66
cd f8 13 d3 34 70 e8 03 10 34 44 68 2d 6b 11 1a
a9 a0 58 cd 54 ed ce 8b 27 bc 3d ef 23 4b 2e f7
0b 28 de 95 d9 de 45 4a 73 48 d1 ad 51 21 f6 fe
fa ae 22 64 b5 2c db f7 99 7e 5b 3c 09 06 d9 eb
e1 a3 a7 8f 34 74 a2 77 a0 85 ca 11 d4 1b 44 53
ed eb 8c 67 b4 f2 62 6e 54 4c 97 a9 1a 27
TODO: register the codepoints
| [eSIDH] | Costello, C., Longa, P. and M. Naehrig, "Efficient algorithms for supersingular isogeny Diffie-Hellman", IACR-CRYPTO-2016 , 2016. |
| [ISOSEC] | Galbraith, S., Petit, C., Shani, B. and Y. Bo Ti, "On the security of supersingular isogeny cryptosystems", IACR-CRYPTO-2016 , 2016. |
| [KLM15] | Kirkwood, D., Lackey, B., McVey, J., Motley, M., Solinas, J. and D. Tuller, "Failure is not an Option: Standardization Issues for Post-Quantum Key Agreement", Workshop on Cybersecurity in a Post Quantum World, 2015 , 2015. |
| [RNSL] | Roetteler, M., Naehrig, M., Svore, K. and K. Lauter, "Quantum Resource Estimates for Computing Elliptic Curve Discrete Logarithms", arXiv , 2017. |
| [URBJAO] | Urbanik, D. and D. Jao, "SoK: The Problem Landscape of SIDH", IACR-CRYPTO-2018 , 2018. |