Internet DRAFT - draft-summermatter-set-union
draft-summermatter-set-union
Independent Stream E. Summermatter
Internet-Draft Seccom GmbH
Intended status: Informational C. Grothoff
Expires: 18 December 2021 Berner Fachhochschule
16 June 2021
Byzantine Fault Tolerant Set Reconciliation
draft-summermatter-set-union-01
Abstract
This document contains a protocol specification for Byzantine fault-
tolerant Set Reconciliation.
Status of This Memo
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This Internet-Draft will expire on 18 December 2021.
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Copyright (c) 2021 IETF Trust and the persons identified as the
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Background . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1. Bloom Filter . . . . . . . . . . . . . . . . . . . . . . 6
2.2. Counting Bloom Filter . . . . . . . . . . . . . . . . . . 7
3. Invertible Bloom Filter . . . . . . . . . . . . . . . . . . . 8
3.1. Structure . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2. Salted Element ID Calculation . . . . . . . . . . . . . . 9
3.3. HASH calculation . . . . . . . . . . . . . . . . . . . . 10
3.4. Mapping Function . . . . . . . . . . . . . . . . . . . . 10
3.5. Operations . . . . . . . . . . . . . . . . . . . . . . . 11
3.5.1. Insert Element . . . . . . . . . . . . . . . . . . . 11
3.5.2. Remove Element . . . . . . . . . . . . . . . . . . . 13
3.5.3. Extracting elements . . . . . . . . . . . . . . . . . 14
3.5.4. Set Difference . . . . . . . . . . . . . . . . . . . 16
3.6. Wire format . . . . . . . . . . . . . . . . . . . . . . . 18
4. Strata Estimator . . . . . . . . . . . . . . . . . . . . . . 18
5. Mode of Operation . . . . . . . . . . . . . . . . . . . . . . 19
5.1. Full Synchronisation Mode . . . . . . . . . . . . . . . . 20
5.2. Differential Synchronisation Mode . . . . . . . . . . . . 21
5.3. Combined Mode . . . . . . . . . . . . . . . . . . . . . . 25
6. Messages . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.1. Operation Request . . . . . . . . . . . . . . . . . . . . 25
6.1.1. Description . . . . . . . . . . . . . . . . . . . . . 25
6.1.2. Structure . . . . . . . . . . . . . . . . . . . . . . 26
6.2. IBF . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2.1. Description . . . . . . . . . . . . . . . . . . . . . 26
6.2.2. Structure . . . . . . . . . . . . . . . . . . . . . . 26
6.3. IBF Last . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3.1. Description . . . . . . . . . . . . . . . . . . . . . 28
6.3.2. Structure . . . . . . . . . . . . . . . . . . . . . . 29
6.4. Element . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.4.1. Description . . . . . . . . . . . . . . . . . . . . . 29
6.4.2. Structure . . . . . . . . . . . . . . . . . . . . . . 29
6.5. Offer . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.5.1. Description . . . . . . . . . . . . . . . . . . . . . 30
6.5.2. Structure . . . . . . . . . . . . . . . . . . . . . . 30
6.6. Inquiry . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.6.1. Description . . . . . . . . . . . . . . . . . . . . . 31
6.6.2. Structure . . . . . . . . . . . . . . . . . . . . . . 31
6.7. Demand . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.7.1. Description . . . . . . . . . . . . . . . . . . . . . 32
6.7.2. Structure . . . . . . . . . . . . . . . . . . . . . . 32
6.8. Done . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.8.1. Description . . . . . . . . . . . . . . . . . . . . . 33
6.8.2. Structure . . . . . . . . . . . . . . . . . . . . . . 33
6.9. Full Done . . . . . . . . . . . . . . . . . . . . . . . . 33
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6.9.1. Description . . . . . . . . . . . . . . . . . . . . . 33
6.9.2. Structure . . . . . . . . . . . . . . . . . . . . . . 34
6.10. Request Full . . . . . . . . . . . . . . . . . . . . . . 34
6.10.1. Description . . . . . . . . . . . . . . . . . . . . 34
6.10.2. Structure . . . . . . . . . . . . . . . . . . . . . 34
6.11. Send Full . . . . . . . . . . . . . . . . . . . . . . . . 35
6.11.1. Description . . . . . . . . . . . . . . . . . . . . 35
6.11.2. Structure . . . . . . . . . . . . . . . . . . . . . 35
6.12. Strata Estimator . . . . . . . . . . . . . . . . . . . . 36
6.12.1. Description . . . . . . . . . . . . . . . . . . . . 36
6.12.2. Structure . . . . . . . . . . . . . . . . . . . . . 37
6.13. Strata Estimator Compressed . . . . . . . . . . . . . . . 38
6.13.1. Description . . . . . . . . . . . . . . . . . . . . 38
6.14. Full Element . . . . . . . . . . . . . . . . . . . . . . 38
6.14.1. Description . . . . . . . . . . . . . . . . . . . . 38
6.14.2. Structure . . . . . . . . . . . . . . . . . . . . . 39
7. Performance Considerations . . . . . . . . . . . . . . . . . 39
7.1. Formulas . . . . . . . . . . . . . . . . . . . . . . . . 39
7.1.1. Operation Mode . . . . . . . . . . . . . . . . . . . 40
7.1.2. IBF Size . . . . . . . . . . . . . . . . . . . . . . 42
7.1.3. Number of Buckets an Element is Hashed into . . . . . 43
7.2. Variable Counter Size . . . . . . . . . . . . . . . . . . 44
7.3. Multi Strata Estimators . . . . . . . . . . . . . . . . . 47
8. Security Considerations . . . . . . . . . . . . . . . . . . . 48
8.1. General Security Checks . . . . . . . . . . . . . . . . . 48
8.1.1. Input validation . . . . . . . . . . . . . . . . . . 49
8.1.2. Byzantine Boundaries . . . . . . . . . . . . . . . . 49
8.1.3. Valid State . . . . . . . . . . . . . . . . . . . . . 50
8.1.4. Message Flow Control . . . . . . . . . . . . . . . . 50
8.1.5. Limit Active/Passive Decoding changes . . . . . . . . 51
8.1.6. Full Synchronisation Plausibility Check . . . . . . . 52
8.2. States . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2.1. Expecting IBF . . . . . . . . . . . . . . . . . . . . 54
8.2.2. Full Sending . . . . . . . . . . . . . . . . . . . . 54
8.2.3. Expecting IBF Last . . . . . . . . . . . . . . . . . 54
8.2.4. Active Decoding . . . . . . . . . . . . . . . . . . . 55
8.2.5. Finish Closing . . . . . . . . . . . . . . . . . . . 57
8.2.6. Finished . . . . . . . . . . . . . . . . . . . . . . 57
8.2.7. Expect SE . . . . . . . . . . . . . . . . . . . . . . 57
8.2.8. Full Receiving . . . . . . . . . . . . . . . . . . . 57
8.2.9. Passive Decoding . . . . . . . . . . . . . . . . . . 58
8.2.10. Finish Waiting . . . . . . . . . . . . . . . . . . . 59
9. Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10. GANA Considerations . . . . . . . . . . . . . . . . . . . . . 60
11. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 61
12. Normative References . . . . . . . . . . . . . . . . . . . . 62
Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . 63
A.1. Map Function . . . . . . . . . . . . . . . . . . . . . . 63
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A.2. ID Calculation Function . . . . . . . . . . . . . . . . . 63
A.3. Counter Compression Function . . . . . . . . . . . . . . 64
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 64
1. Introduction
This document describes a byzantine fault tolerant set reconciliation
protocol used to efficient and securely compute the union of two sets
across a network.
This byzantine fault tolerant set reconciliation protocol can be used
in a variety of applications. Our primary envisioned application
domain is the distribution of revocation messages in the GNU Name
System (GNS) [GNS]. In GNS, key revocation messages are usually
flooded across the peer-to-peer overlay network to all connected
peers whenever a key is revoked. However, as peers may be offline or
the network might have been partitioned, there is a need to reconcile
revocation lists whenever network partitions are healed or peers go
online. The GNU Name System uses the protocol described in this
specification to efficiently distribute revocation messages whenever
network partitions are healed. Another application domain for the
protocol described in this specification are Byzantine fault-tolerant
bulletin boards, like those required in some secure multiparty
computations. A well-known example for secure multiparty
computations are various E-voting protocols
[CryptographicallySecureVoting] which use a bulletin board to share
the votes and intermediate computational results. We note that for
such systems, the set reconciliation protocol is merely a component
of a multiparty consensus protocol, such as the one described in
Dold's "Byzantine set-union consensus using efficient set
reconciliation"
[ByzantineSetUnionConsensusUsingEfficientSetReconciliation].
The protocol described in this report is generic and suitable for a
wide range of applications. As a result, the internal structure of
the elements in the sets MUST be defined and verified by the
application using the protocol. This document thus does not cover
the element structure, except for imposing a limit on the maximum
size of an element.
The protocol faces an inherent trade-off between minimizing the
number of network round-trips and the number of bytes sent over the
network. Thus, for the protocol to choose the right parameters for a
given situation, applications using an implementation of the protocol
SHOULD provide a parameter that specifies the cost-ratio of round-
trips vs. bandwidth usage. Given this trade-off factor, an
implementation CAN then choose parameters that minimize total
execution cost. In particular, there is one major choice to be made,
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namely between sending the complete set of elements, or computing the
set differences and transmitting only the elements in the set
differences. In the latter case, our design is basically a concrete
implementation of a proposal by Eppstein.[Eppstein]
We say that our set reconciliation protocol is Byzantine fault-
tolerant because it provides cryptographic and probabilistic methods
to discover if the other peer is dishonest or misbehaving. Here, the
security objective is to limit resources wasted on malicious actors.
Malicious actors could send malformed messages, including malformed
set elements, claim to have much larger numbers of valid set elements
than they actually hold, or request the retransmission of elements
that they have already received in previous interactions. Bounding
resources consumed by malicous actors is important to ensure that
higher-level protocols can use set reconciliation and still meet
their resource targets. This can be particularly critical in multi-
round synchronous consensus protocols where peers that cannot answer
in a timely fashion would have to be treated as failed or malicious.
To defend against some of these attacks, applications SHOULD remember
the number of elements previously shared with a peer, and SHOULD
provide a way to check that elements are well-formed. Applications
MAY also provide an upper bound on the total number of valid elements
that exist. For example, in E-voting, the number of eligible voters
MAY be used to provide such an upper bound.
A first draft of this RFC is part of Elias Summermatter's bachelor
thesis. Many of the algorithms and parameters documented in this RFC
are derived from experiments detailed in this thesis.
[byzantine_fault_tolerant_set_reconciliation]
This document defines the normative wire format of resource records,
resolution processes, cryptographic routines and security
considerations for use by implementors. SETU requires a
bidirectional secure communication channel between the two parties.
Specification of the communication channel is out of scope of this
document.
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
2. Background
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2.1. Bloom Filter
A Bloom filter (BF) is a space-efficient probabilistic datastructure
to test if an element is part of a set of elements. Elements are
identified by an element ID. Since a BF is a probabilistic
datastructure, it is possible to have false-positives: when asked if
an element is in the set, the answer from a BF is either "no" or
"maybe".
A BF consists of L buckets. Every bucket is a binary value that can
be either 0 or 1. All buckets are initialized to 0. A mapping
function M is used to map each ID of each element from the set to a
subset of k buckets. In the original proposal by Bloom, M is non-
injective and can thus map the same element multiple times to the
same bucket. The type of the mapping function can thus be described
by the following mathematical notation:
------------------------------------
# M: E->B^k
------------------------------------
# L = Number of buckets
# B = 0,1,2,3,4,...L-1 (the buckets)
# k = Number of buckets per element
# E = Set of elements
------------------------------------
Example: L=256, k=3
M('element-data') = {4,6,255}
Figure 1
A typical mapping function is constructed by hashing the element, for
example using the well-known Section 2 of HKDF construction
[RFC5869].
To add an element to the BF, the corresponding buckets under the map
M are set to 1. To check if an element may be in the set, one tests
if all buckets under the map M are set to 1.
In the BF the buckets are set to 1 if the corresponding bit in the
bitstream is 1. If there is a collision and a bucket is already set
to 1, the bucket stays at 1.
In the following example the element e0 with M(e0) = {1,3} has been
added:
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bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
| 0 | 1 | 0 | 1 |
+-------------+-------------+-------------+-------------+
Figure 2
It is easy to see that an element e1 with M(e1) = {0,3} could have
been added to the BF below, while an element e2 with M(e2) = {0,2}
cannot be in the set represented by the BF below:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
| 1 | 0 | 0 | 1 |
+-------------+-------------+-------------+-------------+
Figure 3
The parameters L and k depend on the set size and MUST be chosen
carefully to ensure that the BF does not return too many false-
positives.
It is not possible to remove an element from the BF because buckets
can only be set to 1 or 0. Hence it is impossible to differentiate
between buckets containing one or more elements. To remove elements
from the BF a Counting Bloom Filter is required.
2.2. Counting Bloom Filter
A Counting Bloom Filter (CBF) is a variation on the idea of a Bloom
Filter. With a CBF, buckets are unsigned numbers instead of binary
values. This allows the removal of an element from the CBF.
Adding an element to the CBF is similar to the adding operation of
the BF. However, instead of setting the buckets to 1 the numeric
value stored in the bucket is increased by 1. For example, if two
colliding elements M(e1) = {1,3} and M(e2) = {0,3} are added to the
CBF, bucket 0 and 1 are set to 1 and bucket 3 (the colliding bucket)
is set to 2:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
| 1 | 1 | 0 | 2 |
+-------------+-------------+-------------+-------------+
Figure 4
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The counter stored in the bucket is also called the order of the
bucket.
To remove an element form the CBF the counters of all buckets the
element is mapped to are decreased by 1.
For example, removing M(e2) = {1,3} from the CBF above results in:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
| 1 | 0 | 0 | 1 |
+-------------+-------------+-------------+-------------+
Figure 5
In practice, the number of bits available for the counters is often
finite. For example, given a 4-bit counter, a CBF bucket would
overflow 16 elements are mapped to the same bucket. To handle this
case, the maximum value (15 in our example) is considered to
represent "infinity". Once the order of a bucket reaches "infinity",
it is no longer incremented or decremented.
The parameters L and k and the number of bits allocated to the
counters SHOULD depend on the set size. A CBF will degenerate when
subjected to insert and remove iterations of different elements, and
eventually all buckets will reach "infinity". The speed of the
degradation will depend on the choice of L and k in relation to the
number of elements stored in the IBF.
3. Invertible Bloom Filter
An Invertible Bloom Filter (IBF) is a further extension of the
Counting Bloom Filter. An IBF extends the Counting Bloom Filter with
two more operations: decode and set difference. This two extra
operations are key to efficiently obtain small differences between
large sets.
3.1. Structure
An IBF consists of an injective mapping function M mapping elements
to k out of L buckets. Each of the L buckets stores a signed
COUNTER, an IDSUM and an XHASH. An IDSUM is the XOR of various
element IDs. An XHASH is the XOR of various hash values. As before,
the values used for k, L and the number of bits used for the signed
counter and the XHASH depend on the set size and various other trade-
offs.
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If the IBF size is too small or the mapping function does not spread
out the elements uniformly, the signed counter can overflow or
underflow. As with the CBF, the "maximum" value is thus used to
represent "infinite". As there is no need to distinguish between
overflow and underflow, the most canonical representation of
"infinite" would be the minimum value of the counter in the canonical
2-complement interpretation. For example, given a 4-bit counter a
value of -8 would be used to represent "infinity".
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+-------
count | COUNTER | COUNTER | COUNTER | COUNTER | C...
+-------------+-------------+-------------+-------------+------
idSum | IDSUM | IDSUM | IDSUM | IDSUM | I...
+-------------+-------------+-------------+-------------+------
hashSum | HASHSUM | HASHSUM | HASHSUM | HASHSUM | H..
+-------------+-------------+-------------+-------------+-------
Figure 6
3.2. Salted Element ID Calculation
IBFs are a probabilistic data structure, hence it can be necessary to
recompute the IBF in case operations fail, and then try again. The
recomputed IBF would ideally be statistically independent of the
failed IBF. This is achieved by introducing an IBF-salt. Given that
with benign peers failures should be rare, and that we need to be
able to "invert" the application of the IBF-salt to the element IDs,
we use an unsigned 32 bit non-random IBF-salt value of which the
lowest 6 bits will be used to rotate bits in the element ID
computation.
64-bit element IDs are generated from a Section 2 of HKDF
construction [RFC5869] with HMAC-SHA512 as XTR and HMAC-SHA256 as PRF
with a 16-bit KDF-salt set to a unsigned 16-bit representation of
zero. The output of the KDF is then truncated to 64-bit. Finally,
salting is done by calculating the IBF-salt modulo 64 (effectively
using only the lowest 6-bits of the IBF-salt) and doing a bitwise
right rotation of the output of KDF. We note that this operation was
chosen as it is easily inverted, allowing applications to easily
derive element IDs with one IBF-salt value from element IDs generated
with a different IBF-salt value.
In case the IBF does not decode, the IBF-salt can be changed to
compute different element IDs, which will (likely) be mapped to
different buckets, likely allowing the IBF to decode in a subsequent
iteration.
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# INPUTS:
# key: Pre calculated and truncated key from id_calculation function
# ibf_salt: Salt of the IBF
# OUTPUT:
# value: salted key
FUNCTION salt_key(key,ibf_salt):
s = (ibf_salt * 7) modulo 64;
/* rotate key */
return (key >> s) | (key << (64 - s))
END FUNCTION
# INPUTS:
# element: element for which we are to calculate the element ID
# ibf_salt: Salt of the IBF
# OUTPUT:
# value: the ID of the element
FUNCTION id_calculation (element,ibf_salt):
kdf_salt = 0 // 16 bits
XTR=HMAC-SHA256
PRF=HMAC-SHA256
key = HKDF(XTR, PRF, kdf_salt, element) modulo 2^64
return salt_key(key, ibf_salt)
END FUNCTION
Figure 7
3.3. HASH calculation
The HASH of an element ID is computed by calculating the CRC32
checksum of the 64-bit ID value, which returns a 32-bit value.CRC32
is well-known and described in Section 4.1 of the RFC [RFC3385].
3.4. Mapping Function
The mapping function M decides which buckets a given ID is mapped to.
For an IBF, it is beneficial to use an injective mapping function M.
The first index is simply the CRC32 of the ID modulo the IBF size.
The second index is calculated by creating a new 64-bit value by
shifting the previous 32-bit value left and setting the lower 32 bits
to the number of indices already processed. From the resulting
64-bit value, another CRC32 checksum is computed. The subsequent
index is the modulo of this CRC32 output. The process is repeated
until the desired number of indices is generated. In the case the
process computes the same index twice, which would mean this bucket
could not get pure again, the second hit is just skipped and the next
iteration is used instead, creating an injective mapping function.
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# INPUTS:
# key: the ID of the element calculated
# k: numbers of buckets per element
# L: total number of buckets in the IBF
# OUTPUT:
# dst: Array with k bucket IDs
FUNCTION get_bucket_id (key, k, L)
bucket = CRC32(key)
i = 0 // unsigned 32-bit index
filled = 0
WHILE filled < k DO
element_already_in_bucket = false
j = 0
WHILE j < filled DO
IF dst[j] == bucket modulo L THEN
element_already_in_bucket = true
END IF
j++
END WHILE
IF !element_already_in_bucket THEN
dst[filled] = bucket modulo L
filled = filled + 1
END IF
x = (bucket << 32) | i // 64 bit result
bucket = CRC32(x)
i = i + 1
END WHILE
return dst
END FUNCTION
Figure 8
3.5. Operations
When an IBF is created, all counters and IDSUM and HASHSUM values of
all buckets are initialized to zero.
3.5.1. Insert Element
To add an element to an IBF, the element is mapped to a subset of k
buckets using the injective mapping function M as described in
section Mapping Function. For the buckets selected by the mapping
function, the counter is increased by one and the IDSUM field is set
to the XOR of the element ID computed as described in section Salted
Element ID Calculation and the previously stored IDSUM. Furthermore,
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the HASHSUM is set to the XOR of the previously stored HASHSUM and
the hash of the element ID computed as described in section HASH
calculation.
In the following example, the insert operation is illustrated using
an element with the ID 0x0102 mapped to {1,3} with a hash of 0x4242,
and a second element with the ID 0x0304 mapped to {0,1} and a hash of
0x0101.
Empty IBF:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 0 | 0 | 0 | 0 |
+-------------+-------------+-------------+-------------+
idSum | 0x0000 | 0x0000 | 0x0000 | 0x0000 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0000 | 0x0000 | 0x0000 | 0x0000 |
+-------------+-------------+-------------+-------------+
Figure 9
Insert first element with ID 0x0102 and hash 0x4242 into {1,3}:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 0 | 1 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0000 | 0x0102 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0000 | 0x4242 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 10
Insert second element with ID 0x0304 and hash 0101 into {0,1}:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 1 | 2 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0304 | 0x0206 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0101 | 0x4343 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 11
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3.5.2. Remove Element
To remove an element from the IBF the element is again mapped to a
subset of the buckets using M. Then all the counters of the buckets
selected by M are reduced by one, the IDSUM is replaced by the XOR of
the old IDSUM and the ID of the element being removed, and the
HASHSUM is similarly replaced with the XOR of the old HASHSUM and the
hash of the ID.
In the following example the remove operation is illustrated.
IBF with two encoded elements:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 1 | 2 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0304 | 0x0206 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0101 | 0x4343 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 12
After removal of element with ID 0x0304 and hash 0x0101 mapped to
{0,1} from the IBF:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 0 | 1 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0000 | 0x0102 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0000 | 0x4242 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 13
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Note that it is possible to "remove" elements from an IBF that were
never present in the IBF in the first place. A negative counter
value is thus indicative of elements that were removed without having
been added. Note that an IBF bucket counter of zero no longer
guarantees that an element mapped to that bucket is not present in
the set: a bucket with a counter of zero can be the result of one
element being added and a different element (mapped to the same
bucket) being removed. To check that an element is not present
requires a counter of zero and an IDSUM and HASHSUM of zero --- and
some certainty that there was no collision due to the limited number
of bits in IDSUM and HASHSUM. Thus, IBFs are not suitable to replace
BFs or IBFs.
Buckets in an IBF with a counter of 1 or -1 are crucial for decoding
an IBF, as they MIGHT represent only a single element, with the IDSUM
being the ID of that element. Following Eppstein [Eppstein], we will
call buckets that only represent a single element _pure buckets_.
Note that due to the possibility of multiple insertion and removal
operations affecting the same bucket, not all buckets with a counter
of 1 or -1 are actually pure buckets. Sometimes a counter can be 1
or -1 because N elements mapped to that bucket were added while N-1
or N+1 different elements also mapped to that bucket were removed.
3.5.3. Extracting elements
Extracting elements from an IBF yields IDs of elements from the IBF.
Elements are extracted from an IBF by repeatedly performing a decode
operation on the IBF.
A decode operation requires a pure bucket, that is a bucket to which
M only mapped a single element, to succeed. Thus, if there is no
bucket with a counter of 1 or -1, decoding fails. However, as a
counter of 1 or -1 is not a guarantee that the bucket is pure, there
is also a chance that the decoder returns an IDSUM value that is
actually the XOR of several IDSUMs. This is primarily detected by
checking that the HASHSUM is the hash of the IDSUM. Only if the
HASHSUM also matches, the bucket could be pure. Additionally, one
MUST check that the IDSUM value actually would be mapped by M to the
respective bucket. If not, there was a hash collision and the bucket
is also not pure.
The very rare case that after all these checks a bucket is still
falsely identified as pure MUST be detected (say by determining that
extracted element IDs do not match any actual elements), and
addressed at a higher level in the protocol. As these failures are
probabilistic and depend on element IDs and the IBF construction,
they can typically be avoided by retrying with different parameters,
such as a different way to assign element IDs to elements (by varying
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the IBF-salt), using a larger value for L, or a different mapping
function M. A more common scenario (especially if L was too small)
is that IBF decoding fails because there is no pure bucket. In this
case, the higher-level protocol generally MUST also retry using
different parameters (except if an attack is detected).
Suppose the IBF contains a pure bucket. In this case, the IDSUM in
the bucket is the ID of an element. Furthermore, it is then possible
to remove that element from the IBF (by inserting it if the counter
was negative, and by removing it if the counter was positive). This
is likely to cause other buckets to become pure, allowing further
elements to be decoded. Eventually, decoding ought to finish with
all counters and IDSUM and HASHSUM values reach zero. However, it is
also possible that an IBF only partly decodes and then decoding fails
due to the lack of pure buckets after extracting some element IDs.
In the following example the successful decoding of an IBF containing
the two elements previously added in our running example.
We begin with an IBF with two elements added:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 1 | 2 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0304 | 0x0206 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0101 | 0x4343 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 14
In the IBF are two pure buckets to decode (buckets 0 and 3) we choose
to start with decoding bucket 0. This yields the element with the
hash ID 0x0304 and hash 1010. This element ID is mapped to buckets
{0,1}. Subtracting this element results in bucket 1 becoming pure:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 0 | 1 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0000 | 0x0102 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0000 | 0x4242 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 15
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We can now decoding bucket 2 and extract the element with the ID
0x0102 and hash 0x4242. Now the IBF is empty. Extraction completes
with the status that the IBF has been successfully decoded.
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 0 | 0 | 0 | 0 |
+-------------+-------------+-------------+-------------+
idSum | 0x0000 | 0x0000 | 0x0000 | 0x0000 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0000 | 0x0000 | 0x0000 | 0x0000 |
+-------------+-------------+-------------+-------------+
Figure 16
3.5.4. Set Difference
Given addition and removal as defined above, it is possible to define
an operation on IBFs that computes an IBF representing the set
difference. Suppose IBF1 represents set A, and IBF2 represents set
B. Then this set difference operation will compute IBF3 which
represents the set A - B. Note that this computation can be done
only on the IBFs, and does not require access to the elements from
set A or B. To calculate the IBF representing this set difference,
both IBFs MUST have the same length L, the same number of buckets per
element k and use the same map M. Naturally, all IDs must have been
computed using the same IBF-salt. Given this, one can compute the
IBF representing the set difference by taking the XOR of the IDSUM
and HASHSUM values of the respective buckets and subtracting the
respective counters. Care MUST be taken to handle overflows and
underflows by setting the counter to "infinity" as necessary. The
result is a new IBF with the same number of buckets representing the
set difference.
This new IBF can be decoded as described in section 3.5.3. The new
IBF can have two types of pure buckets with counter set to 1 or -1.
If the counter is set to 1 the element is missing in the secondary
set, and if the counter is set to -1 the element is missing in the
primary set.
To demonstrate the set difference operation we compare IBF-A with
IBF-B and generate as described IBF-AB
IBF-A contains the elements with ID 0x0304 and hash 0x0101 mapped to
{0,1}, and ID 0x0102 and hash 0x4242 mapped to {1,3}:
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bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 1 | 2 | 0 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0304 | 0x0206 | 0x0000 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0101 | 0x4343 | 0x0000 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 17
IBF-B also contains the element with ID 0x0102 and and another
element with ID 0x1345 and hash 0x5050 mapped to {1,2}.
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 0 | 1 | 1 | 1 |
+-------------+-------------+-------------+-------------+
idSum | 0x0000 | 0x1447 | 0x1345 | 0x0102 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0000 | 0x9292 | 0x5050 | 0x4242 |
+-------------+-------------+-------------+-------------+
Figure 18
IBF-A minus IBF-B is then:
bucket-0 bucket-1 bucket-2 bucket-3
+-------------+-------------+-------------+-------------+
count | 1 | 0 | -1 | 0 |
+-------------+-------------+-------------+-------------+
idSum | 0x0304 | 0x1049 | 0x1345 | 0x0000 |
+-------------+-------------+-------------+-------------+
hashSum | 0x0101 | 0x5151 | 0x5050 | 0x0000 |
+-------------+-------------+-------------+-------------+
Figure 19
After calculating and decoding the IBF-AB shows clear that in IBF-A
the element with the hash 0x5050 is missing (-1 in bucket 2) while in
IBF-B the element with the hash 0101 is missing (1 in bucket 0). The
element with hash 0x4242 is present in IBF-A and IBF-B and is removed
by the set difference operation. Bucket 2 is not empty.
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3.6. Wire format
For the counter field, we use a variable-size encoding to ensure that
even for very large sets the counter should never reach "infinity",
while also ensuring that the encoding is compact for small sets.
Hence, the counter size transmitted over the wire varies between 1
and 64 bits, depending on the maximum counter in the IBF. A range of
1 to 64 bits should cover most areas of application and can be
efficiently implemented on most contemporary CPU architectures and
programming languages. The bit length for the transmitted IBF will
be communicated in the header of the _IBF_ message in the "IMCS"
field as unsigned 8-bit integer. For implementation details see
section Variable Counter Size.
For the "IDSUM", we always use a 64-bit representation. The IDSUM
value MUST have sufficient entropy for the mapping function M to
yield reasonably random buckets even for very large values of L.
With a 32 bit value the chance that multiple elements may be mapped
to the same ID would be quite high, even for moderately large sets.
Using more than 64 bits would at best make sense for very large sets,
but then it is likely always better to simply afford additional round
trips to handle the occasional collision. 64 bits are also a
reasonable size for many CPU architectures.
For the "HASHSUM", we always use a 32-bit representation. Here, it
is most important to avoid collisions, where different elements are
mapped to the same hash, possibly resulting in a bucket being falsely
classified as pure. While with 32 bits there remains a non-
negligible chance of accidental collisions, our protocol is designed
to handle occasional collisions. Hence, at 32 bit the chance is
believed to be sufficiently small enough for the protocol to handle
those cases efficiently. Smaller hash values would safe bandwidth,
but also substantially increase the chance of collisions. 32 bits are
also again a reasonable size for many CPU architectures.
4. Strata Estimator
Strata Estimators help estimate the size of the set difference
between two sets of elements. This is necessary to efficiently
determinate the tuning parameters for an IBF, in particular a good
value for L.
Basically a Strata Estimator (SE) is a series of IBFs (with a rather
small value of L=79) in which increasingly large subsets of the full
set of elements are added to each IBF. For the n-th IBF, the
function selecting the subset of elements MUST sample to select
(probabilistically) 1/(2^n) of all elements. This can be done by
counting the number of trailing bits set to "1" in an element ID, and
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then inserting the element into the IBF identified by that counter.
As a result, all elements will be mapped to one IBF, with the n-th
IBF being statistically expected to contain 1/(2^n) elements.
Given two SEs, the set size difference can be estimated by attempting
to decode all of the IBFs. Given that L is set to a fixed and rather
small value, IBFs containing large strata will likely fail to decode.
For IBFs that failed to decode, one simply extrapolates the number of
elements by scaling the numbers obtained from the other IBFs that did
decode. If none of the IBFs of the SE decoded (which given a
reasonable number of IBFs in the SE should be highly unlikely), one
can theoretically retry using a different IBF-salt.
When decoding the IBFs in the strata estimator, it is possible to
determine on which side which part of the difference is. For this
purpose, the pure buckets with counter 1 and -1 must be distinguished
and assigned to the respective side when decoding the IBFs.
5. Mode of Operation
Depending on the state of the two sets the set union protocol uses
different modes of operation to efficiently determinate missing
elements between the two sets.
The simplest mode is the _full synchronisation mode_. If the
difference between the sets of the two peers exceeds a certain
threshold, the overhead to determine which elements are different
would outweigh the overhead of simply sending the complete set.
Hence, the protocol may determine that the most efficient method is
to exchange the full sets.
The second possibility is that the difference between the sets is
relatively small compared to the set size. In this case, the
_differential synchronisation mode_ is more efficient. Given these
two possibilities, the first steps of the protocol are used to
determine which mode MUST be used.
Thus, the set union protocol always begins with the following
operation mode independent steps:
The initiating peer begins in the *Initiating Connection* state and
the receiving peer in the *Expecting Connection* state. The first
step for the initiating peer in the protocol is to send an _Operation
Request_ to the receiving peer and transition into the *Expect SE*
state. After receiving the _Operation Request_ the receiving peer
transitions to the *Expecting IBF* state and answers with the _Strata
Estimator_ message. When the initiating peer receives the _Strata
Estimator_ message, it decides with some heuristics which operation
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mode is likely more suitable for the estimated set difference and the
application-provided latency-bandwidth tradeoff. The detailed
algorithm used to choose between the Full Synchronisation Mode and
the Differential Synchronisation Mode is explained in the section
Combined Mode below.
5.1. Full Synchronisation Mode
When the initiating peer decides to use the full synchronisation mode
and it is better that the other peer sends his set first, the
initiating peer sends a _Request Full_ message, and transitions from
*Expecting SE* to the *Full Receiving* state. If it has been
determined that it is better that the initiating peer sends his set
first, the initiating peer sends a _Send Full_ message followed by
all set elements in _Full Element_ messages to the other peer,
followed by the _Full Done_ message, and transitions into the *Full
Sending* state.
A state diagram illustrating the state machine used during full
synchronization is provided here
(https://git.gnunet.org/lsd0003.git/plain/statemachine/
state_machine_full.png).
*The behavior of the participants the different state is described
below:*
*Expecting IBF:* If a peer in the *Expecting IBF* state receives a
_Request Full_ message from the other peer, the peer sends all the
elements of his set followed by a _Full Done_ message to the other
peer, and transitions to the *Full Sending* state. If the peer
receives an _Send Full_ message followed by _Full Element_
messages, the peer processes the element and transitions to the
*Full Receiving* state.
*Full Sending:* While a peer is in *Full Sending* state the peer
expects to continuously receive elements from the other peer. As
soon as a the _Full Done_ message is received, the peer
transitions into the *Finished* state.
*Full Receiving:* While a peer is in the *Full Receiving* state, it
expects to continuously receive elements from the other peer. As
soon as a the _Full Done_ message is received, it sends the
remaining elements (those it did not receive) from his set to the
other peer, followed by a _Full Done_. After sending the last
message, the peer transitions into the *Finished* state.
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5.2. Differential Synchronisation Mode
The message format used by the protocol limits the maximum message
size to 64 kb. Given that L can be large, an IBF will not always fit
within that size limit. To deal with this, larger IBFs are split
into multiple messages.
When the initiating peer in the *Expected SE* state decides to use
the differential synchronisation mode, it sends an IBF, which may
consist of several _IBF_ messages, to the receiving peer and
transitions into the *Passive Decoding* state.
The receiving peer in the *Expecting IBF* state receives the first
_IBF_ message from the initiating peer, and transitions into the
*Expecting IBF Last* state if the IBF was split into multiple _IBF_
messages. If there is just a single _IBF_ message, the receiving
peer transitions directly to the *Active Decoding* state.
The peer that is in the *Active Decoding*, *Finish Closing* or in the
*Expecting IBF Last* state is called the active peer, and the peer
that is in either the *Passive Decoding* or the *Finish Waiting*
state is called the passive peer.
A state diagram illustrating the state machine used during
differential synchronization is provided here
(https://git.gnunet.org/lsd0003.git/plain/statemachine/
differential_state_machine.png).
*The behavior of the participants the different states is described
below:*
*Passive Decoding:* In the *Passive Decoding* state the passive peer
reacts to requests from the active peer. The action the passive
peer executes depends on the message the passive peer receives in
the *Passive Decoding* state from the active peer and is described
below on a per message basis.
_Inquiry_ message: The _Inquiry_ message is received if the
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active peer requests the SHA-512 hash of one or more elements
(by sending the 64 bit element ID) that are missing from the
active peer's set. In this case the passive peer answers with
_Offer_ messages which contain the SHA-512 hash of the
requested element. If the passive peer does not have an
element with a matching element ID, it MUST ignore the inquiry
(in this case, a bucket was falsely classified as pure,
decoding the IBF will eventually fail, and roles will be
swapped). It should be verified that after an falsely
classified pure bucket a role change is made. If multiple
elements match the 64 bit element ID, the passive peer MUST
send offers for all of the matching elements.
_Demand_ message: The _Demand_ message is received if the active
peer requests a complete element that is missing in the active
peers set in response to an offer. If the requested element is
known and has not yet been transmitted the passive peer answers
with an _Element_ message which contains the full, application-
dependent data of the requested element. If the passive peer
receives a demand for a SHA-512 hash for which it has no
corresponding element, a protocol violation is detected and the
protocol MUST be aborted. Implementations MUST also abort when
facing demands without previous matching offers or for which
the passive peer previously transmitted the element to the
active peer.
_Offer_ message: The _Offer_ message is received if the active
peer has decoded an element that is present in the active peers
set and is likely be missing in the set of the passive peer.
If the SHA-512 hash of the offer is indeed not a hash of any of
the elements from the set of the passive peer, the passive peer
MUST answer with a _Demand_ message for that SHA-512 hash and
remember that it issued this demand. The demand thus needs to
be added to a list with unsatisfied demands.
_Element_ message: When a new _Element_ message has been received
the peer checks if a corresponding _Demand_ for the element has
been sent and the demand is still unsatisfied. If the element
has been demanded the peer checks the element for validity,
removes it from the list of pending demands and then saves the
element to the set. Otherwise the peer ignores the element.
_IBF_ message: If an _IBF_ message is received, this indicates
that decoding of the IBF on the active site has failed and
roles will be swapped. The receiving passive peer transitions
into the *Expecting IBF Last* state, and waits for more _IBF_
messages. There, once the final _IBF Last_ message has been
received, it transitions to *Active Decoding*.
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_IBF Last_ message: If an _IBF Last_ message is received this
indicates that there is just one IBF slice left and a direct
state and role transition from *Passive Decoding* to *Active
Decoding* is initiated.
_Done_ message: Receiving the _Done_ message signals the passive
peer that all demands of the active peer have been satisfied.
Alas, the active peer will continue to process demands from the
passive peer. Upon receiving this message, the passive peer
transitions into the *Finish Waiting* state.
*Active Decoding:* In the *Active Decoding* state the active peer
decodes the IBFs and evaluates the set difference between the
active and passive peer. Whenever an element ID is obtained by
decoding the IBF, the active peer sends either an offer or an
inquiry to the passive peer, depending on which site the decoded
element is missing.
If the IBF decodes a positive (1) pure bucket, the element is
missing on the passive peers site. Thus, the active peer sends an
_Offer_ to the passive peer. A negative (-1) pure bucket
indicates that an element is missing in the active peers set, so
the active peer sends a _Inquiry_ to the passive peer.
In case the IBF does not successfully decode anymore, the active
peer sends a new IBF computed with a different IBF-salt to the
passive peer and changes into *Passive Decoding* state. This
initiates a role swap. To reduce overhead and prevent double
transmission of offers and elements, the new IBF is created on the
local set after updating it with the all of the elements that have
been successfully demanded. Note that the active peer MUST NOT
wait for all active demands to be satisfied, as demands can fail
if a bucket was falsely classified as pure.
As soon as the active peer successfully finished decoding the IBF,
the active peer sends a _Done_ message to the passive peer.
All other actions taken by the active peer depend on the message
the active peer receives from the passive peer. The actions are
described below on a per message basis:
_Offer_ message: The _Offer_ message indicates that the passive
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peer received a _Inquiry_ message from the active peer. If a
inquiry has been sent and the offered element is missing in the
active peers set, the active peer sends a _Demand_ message to
the passive peer. The demand needs to be added to a list of
unsatisfied demands. In case the received offer is for an
element that is already in the set of the peer, the offer MUST
BE ignored.
_Demand_ message: The _Demand_ message indicates that the passive
peer received a _Offer_ from the active peer. The active peer
satisfies the demand of the passive peer by sending an
_Element_ message if a offer request for the element was sent
earlier. Otherwise, the protocol MUST be aborted, as peers
must never send demands for hashes that they have never been
offered.
_Element_ message: If element is received that was not demanded
or for which the application-specific validation logic fails,
the protocol MUST be aborted. Otherwise, the corresponding
demand is marked as satisfied. Note that this applies only to
the differential synchronization mode. In full
synchronization, it is perfectly normal to receive Full Element
messages for elements that were not demanded and that might
even already be known locally.
_Done_ message: Receiving the message _Done_ indicates that all
demands of the passive peer have been satisfied. The active
peer then changes into the *Finish Closing* state. If the IBF
has not finished decoding and the _Done_ is received, the other
peer is not in compliance with the protocol and the protocol
MUST be aborted.
*Expecing IBF Last* In this state the active peer continuously
receives _IBF_ messages from the passive peer. When the last _IBF
Last_ message is received, the peer changes into the *Active
Decoding* state.
*Finish Closing* / *Finish Waiting* In this states the peers are
waiting for all demands to be satisfied and for the
synchronisation to be completed. When all demands are satisfied
the peer changes into *Finished* state.
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5.3. Combined Mode
In the _combined mode_ the protocol decides between Full
Synchronisation Mode and the Differential Synchronisation Mode to
minimize resource consumption. Typically, the protocol always runs
in combined mode, but implementations MAY allow applications to force
the use of one of the modes for testing. In this case, applications
MUST ensure that the respective options to force a particular mode
are used by both participants.
The Differential Synchronisation Mode is only efficient on small set
differences or if the byte-size of the elements is large. If the set
difference is estimated to be large the Full Synchronisation Mode is
more efficient. The exact heuristics and parameters on which the
protocol decides which mode MUST be used are described in the
Performance Considerations section of this document.
There are two main cases when a Full Synchronisation Mode is always
used. The first case is when one of the peers announces having an
empty set. This is announced by setting the SETSIZE field in the
_Strata Estimator_ to 0. The second case is if the application
requests full synchronisation explicitly. This is useful for testing
and MUST NOT be used in production.
The state diagram illustrating the combined mode can be found here
(https://git.gnunet.org/lsd0003.git/plain/statemachine/
full_state_machine.png).
6. Messages
This section describes the various message formats used by the
protocol.
6.1. Operation Request
6.1.1. Description
This message is the first message of the protocol and it is sent to
signal to the receiving peer that the initiating peer wants to
initialize a new connection.
This message is sent in the transition between the *Initiating
Connection* state and the *Expect SE* state.
If a peer receives this message and is willing to run the protocol,
it answers by sending back a _Strata Estimator_ message. Otherwise
it simply closes the connection.
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6.1.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | ELEMENT COUNT |
+-----+-----+-----+-----+-----+-----+-----+-----+
| APX
+-----+-----+-----+-----+-----+-----+-----+-----+
/ APPLICATION DATA /
/ /
Figure 20
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included.
MSG TYPE is the type of SETU_P2P_OPERATION_REQUEST as registered in
GANA Considerations, in network byte order.
ELEMENT COUNT is the number of the elements the requesting party has
in its set, as a 32-bit unsigned integer in network byte order.
APX is a SHA-512 hash that identifies the application.
APPLICATION DATA is optional, variable-size application specific
data that can be used by the application to decide if it would
like to answer the request.
6.2. IBF
6.2.1. Description
The IBF message contains a slice of the IBF.
The _IBF_ message is sent at the start of the protocol from the
initiating peer in the transaction between *Expect SE* -> *Expecting
IBF Last* or when the IBF does not decode and there is a role change
in the transition between *Active Decoding* -> *Expecting IBF Last*.
This message is only sent if there is more than one IBF slice to be
sent. If there is just one slice, then only the IBF Last message is
sent.
6.2.2. Structure
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0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | IBF SIZE |
+-----+-----+-----+-----+-----+-----+-----+-----+
| OFFSET | SALT | IMCS |
+-----+-----+-----+-----+-----+-----+-----+-----+
| IBF-SLICE
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 21
where:
MSG SIZE is a 16-bit unsigned integer in network byte
orderwhichdescribes the message size in bytes with the header
included.
MSG TYPE the type of SETU_P2P_REQUEST_IBF as registered in GANA
Considerations in network byte order.
IBF SIZE is a 32-bit unsigned integer which signals the total number
of buckets in the IBF. The minimal number of buckets is 37.
OFFSET is a 32-bit unsigned integer which signals the offset of the
following IBF slices in the original.
SALT is a 16-bit unsigned integer that contains the IBF-salt which
was used to create the IBF.
IMCS is a 16-bit unsigned integer, which describes the number of
bits that are required to store a single counter. This is used
for the unpacking function as described in the Variable Counter
Size section.
IBF-SLICE are variable numbers of slices in an array. A single
slice contains multiple 64-bit IDSUMS, 32-bit HASHSUMS and (1-64)-
bit COUNTERS of variable size. All values are in the network byte
order. The array of IDSUMS is serialized first, followed by an
array of HASHSUMS. Last comes an array of unsigned COUNTERS
(details of the COUNTERS encoding are described in section
Section 7.2). The length of the array is defined by MIN( SIZE -
OFFSET, MAX_BUCKETS_PER_MESSAGE). MAX_BUCKETS_PER_MESSAGE is
defined as 32768 divided by the BUCKET_SIZE which ranges between
97-bits when counter uses bit 1 (IMCS=1) and 160-bit when counter
size uses 64 bit (IMCS=64).
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To get the IDSUM field, all IDs (computed with the IBF-salt)
hitting a bucket under the map M are added up with a binary XOR
operation. See Salted Element ID Calculation details about ID
generation.
The calculation of the HASHSUM field is done accordingly to the
calculation of the IDSUM field: all HASHes are added up with a
binary XOR operation. The HASH value is calculated as described
in detail in section HASH calculation.
The algorithm to find the correct bucket in which the ID and the
HASH have to be added is described in detail in section Mapping
Function.
Test vectors for an implementation can be found in the Test
Vectors section
IBF-SLICE
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| IDSUMS |
+-----+-----+-----+-----+-----+-----+-----+-----+
| IDSUMS |
+-----+-----+-----+-----+-----+-----+-----+-----+
| HASHSUMS | HASHSUMS |
+-----+-----+-----+-----+-----+-----+-----+-----+
| COUNTERS* | COUNTERS* |
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
* Counter size is variable. In this example the IMCS is 32 (4 bytes).
Figure 22
6.3. IBF Last
6.3.1. Description
This message indicates to the remote peer that this is the last slice
of the Bloom filter. The receiving peer MUST check that the sizes
and offsets of all received IBF slices add up to the total IBF SIZE
that was given.
Receiving this message initiates the state transmissions *Expecting
IBF Last* -> *Active Decoding*, *Expecting IBF* -> *Active Decoding*
and *Passive Decoding* -> *Active Decoding*. This message can
initiate a peer the roll change from *Active Decoding* to *Passive
Decoding*.
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6.3.2. Structure
The binary structure is exactly the same as the Structure of the
message IBF with a different "MSG TYPE" which is defined in GANA
Considerations "SETU_P2P_IBF_LAST".
6.4. Element
6.4.1. Description
The _Element_ message contains an element that is synchronized in the
Differential Synchronisation Mode and transmits a full element
between the peers.
This message is sent in the state *Active Decoding* and *Passive
Decoding* as answer to a _Demand_ message from the remote peer. The
_Element_ message can also be received in the *Finish Closing* or
*Finish Waiting* state after receiving a _Done_ message from the
remote peer. In this case the peer changes to the *Finished* state
as soon as all demands for elements have been satisfied.
This message is exclusively used in the Differential Synchronisation
Mode.
6.4.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | E TYPE | PADDING |
+-----+-----+-----+-----+-----+-----+-----+-----+
| E SIZE | DATA
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 23
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included.
MSG TYPE is SETU_P2P_ELEMENTS as registered in GANA Considerations
in network byte order.
E TYPE is a 16-bit unsigned integer which defines the element type
for the application.
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PADDING is 16-bit always set to zero.
E SIZE is a 16-bit unsigned integer that signals the size of the
elements data part.
DATA is a field with variable length that contains the data of the
element.
6.5. Offer
6.5.1. Description
The _Offer_ message is an answer to an _Inquiry_ message and
transmits the full hash of an element that has been requested by the
other peer. This full hash enables the other peer to check if the
element is really missing in his set and eventually sends a _Demand_
message for that element.
The offer is sent and received only in the *Active Decoding* and in
the *Passive Decoding* state.
This message is exclusively sent in the Differential Synchronisation
Mode.
6.5.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | HASH 1
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
+-----+-----+-----+-----+-----+-----+-----+-----+
HASH 1 | HASH 2
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
+-----+-----+-----+-----+-----+-----+-----+-----+
HASH 2 | HASH n
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 24
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes header included.
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MSG TYPE is SETU_P2P_OFFER as registered in GANA Considerations in
network byte order.
HASH n contains n (one or more) successive SHA 512-bit hashes of the
elements that are being requested with _Inquiry_ messages.
6.6. Inquiry
6.6.1. Description
The _Inquiry_ message is exclusively sent by the active peer in
*Active Decoding* state to request the full hash of an element that
is missing in the active peers set. This is normally answered by the
passive peer with _Offer_ message.
This message is exclusively sent in the Differential Synchronisation
Mode.
6.6.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | SALT |
+-----+-----+-----+-----+-----+-----+-----+-----+
| IBF KEY 1 |
+-----+-----+-----+-----+-----+-----+-----+-----+
| IBF KEY 2 |
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
+-----+-----+-----+-----+-----+-----+-----+-----+
| IBF KEY n |
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 25
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included.
MSG TYPE is SETU_P2P_INQUIRY as registered in GANA Considerations in
network byte order.
IBF KEY contains n (one or more) successive ibf keys (64-bit
unsigned integer) for which the inquiry is sent.
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6.7. Demand
6.7.1. Description
The _Demand_ message is sent in the *Active Decoding* and in the
*Passive Decoding* state. It is an answer to a received _Offer_
message and is sent if the element described in the _Offer_ message
is missing in the peers set. In the normal workflow the answer to
the _Demand_ message is an _Element_ message.
This message is exclusively sent in the Differential Synchronisation
Mode.
6.7.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | HASH 1
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
+-----+-----+-----+-----+-----+-----+-----+-----+
HASH 1 | HASH 2
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
+-----+-----+-----+-----+-----+-----+-----+-----+
HASH 2 | HASH n
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 26
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes and the header is included.
MSG TYPE the type of SETU_P2P_DEMAND as registered in GANA
Considerations in network byte order.
HASH n contains n (one or more) successive SHA 512-bit hashes of the
elements that are being demanded.
6.8. Done
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6.8.1. Description
The _Done_ message is sent when all _Demand_ messages have been
successfully satisfied and from the perspective of the sender the set
is completely synchronized.
This message is exclusively sent in the Differential Synchronisation
Mode.
6.8.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | FINAL CHECKSUM
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 27
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included. The
value is always 4 for this message type.
MSG TYPE is SETU_P2P_DONE as registered in GANA Considerations in
network byte order.
FINAL CHECKSUM a SHA-512 hash XOR sum of the full set after
synchronization. This should ensure that the sets are identical
in the end!
6.9. Full Done
6.9.1. Description
The _Full Done_ message is sent in the Full Synchronisation Mode to
signal that all remaining elements of the set have been sent. The
message is received and sent in the *Full Sending* and in the *Full
Receiving* state. When the _Full Done_ message is received in *Full
Sending* state the peer changes directly into *Finished* state. In
*Full Receiving* state receiving a _Full Done_ message initiates the
sending of the remaining elements that are missing in the set of the
other peer.
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6.9.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | FINAL CHECKSUM
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 28
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included. The
value is always 4 for this message type.
MSG TYPE the type of SETU_P2P_FULL_DONE as registered in GANA
Considerations in network byte order.
FINAL CHECKSUM a SHA-512 hash XOR sum of the full set after
synchronization. This should ensure that the sets are identical
in the end!
6.10. Request Full
6.10.1. Description
The _Request Full_ message is sent by the initiating peer in *Expect
SE* state to the receiving peer, if the operation mode "Full
Synchronisation Mode" is determined to be the superior Mode of
Operation and that it is the better choice that the other peer sends
his elements first. The initiating peer changes after sending the
_Request Full_ message into *Full Receiving* state.
The receiving peer receives the _Request Full_ message in the
*Expecting IBF*, afterwards the receiving peer starts sending his
complete set in Full Element messages to the initiating peer.
6.10.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | REMOTE SET DIFF |
+-----+-----+-----+-----+-----+-----+-----+-----+
| REMOTE SET SIZE | LOCAL SET DIFF |
+-----+-----+-----+-----+-----+-----+-----+-----+
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Figure 29
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included. The
value is always 16 for this message type.
MSG TYPE is SETU_P2P_REQUEST_FULL as registered in GANA
Considerations in network byte order.
REMOTE SET DIFF is a 32-bit unsigned integer in network byte order,
which represents the remote (from the perspective of the sending
peer) set difference calculated with strata estimator.
REMOTE SET SIZE is a 32-bit unsigned integer in network byte order,
which represents the total remote (from the perspective of the
sending peer) set size.
LOCAL SET DIFF is a 32-bit unsigned integer in network byte order,
which represents the local (from the perspective of the sending
peer) set difference calculated with strata estimator.
6.11. Send Full
6.11.1. Description
The _Send Full_ message is sent by the initiating peer in *Expect SE*
state to the receiving peer if the operation mode "Full
Synchronisation Mode" is determined as superior Mode of Operation and
that it is the better choice that the peer sends his elements first.
The initiating peer changes after sending the _Request Full_ message
into *Full Sending* state.
The receiving peer receives the _Send Full_ message in the *Expecting
IBF* state, afterwards the receiving peer changes into *Full
Receiving* state and expects to receive the set of the remote peer.
6.11.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | REMOTE SET DIFF |
+-----+-----+-----+-----+-----+-----+-----+-----+
| REMOTE SET SIZE | LOCAL SET DIFF |
+-----+-----+-----+-----+-----+-----+-----+-----+
Figure 30
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where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included. The
value is always 16 for this message type.
MSG TYPE is SETU_P2P_REQUEST_FULL as registered in GANA
Considerations in network byte order.
REMOTE SET DIFF is a 32-bit unsigned integer in network byte order,
which represents the remote (from the perspective of the sending
peer) set difference calculated with strata estimator.
REMOTE SET SIZE is a 32-bit unsigned integer in network byte order,
which represents the total remote (from the perspective of the
sending peer) set size.
LOCAL SET DIFF is a 32-bit unsigned integer in network byte order,
which represents the local (from the perspective of the sending
peer) set difference calculated with strata estimator.
6.12. Strata Estimator
6.12.1. Description
The strata estimator is sent by the receiving peer at the start of
the protocol, right after the Operation Request message has been
received.
The strata estimator is used to estimate the difference between the
two sets as described in section Strata Estimator.
When the initiating peer receives the strata estimator, the peer
decides which Mode of Operation to use for the synchronisation.
Depending on the size of the set difference and the Mode of Operation
the initiating peer changes into *Full Sending*, *Full Receiving* or
*Passive Decoding* state.
The _Strata Estimator_ message can contain one, two, four or eight
strata estimators with different salts, depending on the initial size
of the sets. More details can be found in section Multi Strata
Estimators.
The IBFs in a strata estimator always have 79 buckets. The reason
why can be found in [byzantine_fault_tolerant_set_reconciliation] in
section 3.4.2.
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6.12.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | SEC | SETSIZE
+-----+-----+-----+-----+-----+-----+-----+-----+
SETSIZE | SE-SLICES
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 31
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included.
MSG TYPE is SETU_P2P_SE as registered in GANA Considerations in
network byte order.
SEC is a 8-bit unsigned integer in network byte order, which
indicates how many strata estimators with different salts are
attached to the message. Valid values are 1,2,4 or 8, more
details can be found in the section Multi Strata Estimators.
SETSIZE is a 64-bit unsigned integer that is defined by the size of
the set the SE is
SE-SLICES are variable numbers of slices in an array. A slice can
contain one or more Strata Estimators which contain multiple IBFs
as described in IBF-SLICES in Section 6.2.2. A SE slice can
contain one to eight Strata Estimators which contain 32 (Defined
as Constant SE_STRATA_COUNT) IBFs. Every IBF in a SE contains 79
Buckets.
The different SEs are built as in detail described in Section 7.3.
Simply put, the IBFs in each SE are serialized as described in
Section 6.2.2 starting with the highest stratum. Then the created
SEs are appended one after the other starting with the SE that was
created with a salt of zero.
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SE-SLICE
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| SE_1 -> IBF_1
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
+-----+-----+-----+-----+-----+-----+-----+-----+
| SE_1 -> IBF_30
+-----+-----+-----+-----+-----+-----+-----+-----+
| SE_2 -> IBF_1
+-----+-----+-----+-----+-----+-----+-----+-----+
... ...
/ /
/ /
Figure 32
6.13. Strata Estimator Compressed
6.13.1. Description
The Strata Estimator can be compressed with gzip as described in
[RFC1951] to improve performance. This can be recognized by the
different message type number from GANA Considerations.
6.13.1.1. Structure
The key difference between the compressed and the uncompressed Strata
Estimator is that the SE slices are compressed with gzip ([RFC1951])
in the compressed SE. But the header remains uncompressed with both.
Since the content of the message is the same as the uncompressed
Strata Estimator, the details are not repeated here. For details see
section 6.12.
6.14. Full Element
6.14.1. Description
The _Full Element_ message is the equivalent of the Element message
in the Full Synchronisation Mode. It contains a complete element
that is missing in the set of the peer that receives this message.
The _Full Element_ message is exclusively sent in the transitions
*Expecting IBF* -> *Full Receiving* and *Full Receiving* ->
*Finished*. The message is only received in the *Full Sending* and
*Full Receiving* state.
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After the last _Full Element_ message has been sent, the _Full Done_
message is sent to conclude the full synchronisation of the element
sending peer.
6.14.2. Structure
0 8 16 24 32 40 48 56
+-----+-----+-----+-----+-----+-----+-----+-----+
| MSG SIZE | MSG TYPE | E TYPE | PADDING |
+-----+-----+-----+-----+-----+-----+-----+-----+
| SIZE | AE TYPE | DATA
+-----+-----+-----+-----+-----+-----+-----+-----+
/ /
/ /
Figure 33
where:
MSG SIZE is a 16-bit unsigned integer in network byte order, which
describes the message size in bytes with the header included.
MSG TYPE is SETU_P2P_REQUEST_FULL_ELEMENT as registered in GANA
Considerations in network byte order.
E TYPE is a 16-bit unsigned integer which defines the element type
for the application.
PADDING is 16-bit always set to zero
E SIZE is a 16-bit unsigned integer that signals the size of the
elements data part.
AE TYPE is a 16-bit unsigned integer that is needed to identify the
type of element that is in the data field
DATA is a field with variable length that contains the data of the
element.
7. Performance Considerations
7.1. Formulas
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7.1.1. Operation Mode
The decision which Mode of Operation is used is described by the
following code. More detailed explanations motivating the design can
be found in the accompanying thesis in section
4.5.3.[byzantine_fault_tolerant_set_reconciliation]
The function takes as input the average element size, the local set
size, the remote set size, the set differences as estimated from the
strata estimator for both the local and remote sets, and the
bandwidth/roundtrip tradeoff. The function returns the exact Mode of
Operation that is predicted to be best as output:
FULL_SYNC_REMOTE_SENDING_FIRST if it is likely cheapest that the
other peer transmits his elements first,
FULL_SYNC_LOCAL_SENDING_FIRST if it is likely cheapest that the
elements are transmitted to the other peer directly, and
DIFFERENTIAL_SYNC if the differential synchronisation is likely
cheapest.
The constant IBF_BUCKET_NUMBER_FACTOR is always 2 and IBF_MIN_SIZE is
37. The method for deriving this can be found in the IBF parameter
study in [byzantine_fault_tolerant_set_reconciliation] in section
4.5.2.
# CONSTANTS:
# IBF_BUCKET_NUMBER_FACTOR = 2: The amount the IBF gets increased if
# decoding fails
# RTT_MIN_FULL = 2: Minimal round trips used for full Synchronisation
# IBF_MIN_SIZE = 37: The minimal size of an IBF
# MAX_BUCKETS_PER_MESSAGE: Custom value depending on the underlying
# protocol
# INPUTS:
# avg_es: The average element size
# lss: The initial local set size
# rss: The remote set size
# lsd: the estimated local set difference calculated by the SE
# rsd: the estimated remote set difference calculated by the SE
# rtt: the tradeoff between round trips and bandwidth
# OUTPUT:
# FULL_SYNC_REMOTE_SENDING_FIRST, FULL_SYNC_LOCAL_SENDING_FIRST or
# DIFFERENTIAL_SYNC
FUNCTION decide_operation_mode(avg_es,
lss,
rss,
lsd
rsd,
rtt)
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# If a set size is zero always do full sync
IF 0 == rss THEN
RETURN FULL_SYNC_LOCAL_SENDING_FIRST
END IF
IF 0 == lss THEN
RETURN FULL_SYNC_REMOTE_SENDING_FIRST
END IF
# Estimate required transferred bytes when doing a full
# synchronisation and transmitting local set first.
semh = sizeof(ELEMENT_MSG_HEADER)
estimated_total_diff = rsd + lsd
total_elements_local_send = rsd + lss
cost_local_full_sync = avg_es * total_elements_local_send
+ total_elements_local_send * semh
+ sizeof(FULL_DONE_MSG_HEADER) * 2
+ RTT_MIN_FULL * rtt
# Estimate required transferred bytes when doing a full
# synchronisation and transmitting remote set first.
total_elements_remote_send = lsd + rss
cost_remote_full_sync = avg_es * total_elements_remote_send
+ total_elements_remote_send * semh
+ sizeof(FULL_DONE_MSG_HEADER) * 2
+ (RTT_MIN_FULL + 0.5) * rtt
+ sizeof(REQUEST_FULL_MSG)
# Estimate required transferred bytes when doing a differential
# synchronisation
# Estimate messages required to transfer IBF
ibf_bucket_count = estimated_total_diff * IBF_BUCKET_NUMBER_FACTOR
IF ibf_bucket_count <= IBF_MIN_SIZE THEN
ibf_bucket_count = IBF_MIN_SIZE
END IF
ibf_message_count = ceil (ibf_bucket_count / MAX_BUCKETS_PER_MESSAGE)
# Estimate average counter length with variable counter
estimated_counter_bits = MIN (2 * LOG2(lss / ibf_bucket_count),
LOG2(lss))
estimated_counter_bytes = estimated_counter_bits / 8
# Sum up all messages required to do differential synchronisation
ibf_bytes = sizeof(IBF_MESSAGE) * ibf_message_count
+ ibf_bucket_count * sizeof(IBF_KEY)
+ ibf_bucket_count * sizeof(IBF_KEYHASH)
+ ibf_bucket_count * estimated_counter_bytes
# Add 20% overhead to cover IBF retries due to decoding failures
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total_ibf_bytes = ibf_bytes * 1.2
# Estimate other message sizes to be transfered in diff sync
# Note that we simplify by adding the header each time;
# if the implementation combines multiple INQUIRY/DEMAND/OFFER
# requests in one message, the bandwidth would be lower.
done_size = sizeof(DONE_HEADER)
element_size = (avg_es + sizeof(ELEMENT_MSG_HEADER))
* estimated_total_diff
inquery_size = (sizeof(IBF_KEY) + sizeof(INQUERY_MSG_HEADER))
* estimated_total_diff
demand_size = (sizeof(HASHCODE) + sizeof(DEMAND_MSG_HEADER))
* estimated_total_diff
offer_size = (sizeof(HASHCODE) + sizeof(OFFER_MSG_HEADER))
* estimated_total_diff
# Estimate total cost
diff_cost = element_size + done_size + inquery_size
+ demand_size + offer_size + total_ibf_bytes
+ DIFFERENTIAL_RTT_MEAN * rtt
# Decide for a optimal mode of operation
full_cost_min = MIN (cost_local_full_sync,
cost_remote_full_sync)
IF full_cost_min < diff_cost THEN
IF cost_remote_full_sync > cost_local_full_sync THEN
RETURN FULL_SYNC_LOCAL_SENDING_FIRST
ELSE
RETURN FULL_SYNC_REMOTE_SENDING_FIRST
END IF
ELSE
RETURN DIFFERENTIAL_SYNC
END IF
END FUNCTION
Figure 34
7.1.2. IBF Size
The functions, described in this section, calculate a good initial
size (initial_ibf_size) and in case of decoding failure, a good next
IBF size (get_next_ibf_size).
These algorithms are described and justified in more details in
[byzantine_fault_tolerant_set_reconciliation] in the parameter study
in section 3.5.2, the max IBF counter in section 3.10 and the
Improved IBF size in section 3.11.
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# CONSTANTS:
# IBF_BUCKET_NUMBER_FACTOR = 2: The amount the IBF gets increased
if decoding fails
# Inputs:
# sd: Estimated set difference
# Output:
# next_size: Size of the initial IBF
FUNCTION initial_ibf_size(sd)
# We do not go below 37, as 37 buckets should
# basically always be below one MTU, so there is
# little to be gained, while a smaller IBF would
# increase the chance of a decoding failure.
RETURN MAX(37, IBF_BUCKET_NUMBER_FACTOR * sd);
END FUNCTION
# CONSTANTS:
# IBF_BUCKET_NUMBER_FACTOR = 2: The amount the IBF gets increased if
# decoding fails
# Inputs:
# de: Number of elements that have been successfully decoded
# lis: The number of buckets of the last IBF
# Output:
# number of buckets for the next IBF
FUNCTION get_next_ibf_size(de, lis)
next_size = IBF_BUCKET_NUMBER_FACTOR * (lis - de)
# The MAX operation here also ensures that the
# result is positive.
RETURN MAX(37, next_size);
END FUNCTION
Figure 35
7.1.3. Number of Buckets an Element is Hashed into
The number of buckets an element is hashed to is hardcoded to 3.
Reasoning and justification can be found in
[byzantine_fault_tolerant_set_reconciliation] in the IBF parameter
performance study in section 4.5.2.
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7.2. Variable Counter Size
The number of bits required to represent the counters of an IBF
varies due to different parameters as described in section 3.2 of
[byzantine_fault_tolerant_set_reconciliation]. Therefore, a packing
algorithm has been implemented. This algorithm encodes the IBF
counters in their optimal bit-width and thus minimizes the bandwidth
needed to transmit the IBF.
A simple algorithm is used for the packing. In a first step it is
determined, which is the largest counter. The the base 2 logarithm
then determines how many bits are needed to store it. In a second
step for every counter of every bucket, the counter is stored using
this many bits. The resulting bit sequence is then simply
concatenated.
Three individual functions are used for this purpose. The first one
is a function that iterates over each bucket of the IBF to get the
maximum counter in the IBF. The second function packs the counters
of the IBF, and the third function that unpacks the counters.
As a plausibly check to prevent the byzantine upper bound checks in
Section 8.1.2 to fail, implementations must ensure that the estimates
of the set size difference added together never exceed the set
byzantine upper bound. This could for example happen in case the
strata estimator overestimates the set difference.
# INPUTS:
# ibf: The IBF
# OUTPUTS:
# returns: Minimal amount of bits required to store the counter
FUNCTION ibf_get_max_counter(ibf)
max_counter=1 # convince static analysis that we never take log2(0)
FOR bucket IN ibf DO
IF bucket.counter > max_counter THEN
max_counter = bucket.counter
END IF
END FOR
# next bigger discrete number of the binary logarithm of the
# max counter
RETURN CEILING( LOG2( max_counter ) )
END FUNCTION
# INPUTS:
# ibf: The IBF
# offset: The offset which defines the starting point from which bucket
# the pack operation starts
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# count: The number of buckets in the array that will be packed
# OUTPUTS:
# returns: A byte array of packed counters to send over the network
# INPUTS:
# ibf: The IBF
# offset: The offset which defines the starting point from which bucket
# the pack operation starts
# count: The number of buckets in the array that will be packed
# OUTPUTS:
# returns: A byte array of packed counters to send over the network
FUNCTION pack_counter(ibf, offset, count)
counter_bytes = ibf_get_max_counter(ibf)
store_bits = 0
store = 0
byte_ctr = 0
buf=[]
FOR bucket IN ibf[offset] TO ibf[count] DO
counter = bucket.counter
byte_len = counter_bytes
WHILE byte_len + store_bits < 8 DO
bit_to_shift = 0
IF store_bits > 0 OR byte_len > 8 THEN
bit_free = 8 - store_bits
bit_to_shift = byte_len - bit_free
store = store << bit_free
END IF
buf[byte_ctr] = (( counter >> bit_to_shift) | store) & 0xFF
byte_ctr = byte_ctr + 1
byte_len -= 8 - store_bits
counter = counter & ((1 << byte_len) - 1)
store = 0
store_bits = 0
END WHILE
store = (store << byte_len) | counter
store_bits = store_bits + byte_len
byte_len = 0
END FOR
# Write the last partial packed byte to the buffer
IF store_bits > 0 THEN
buf[byte_ctr] = store << (8 - store_bits)
byte_ctr = byte_ctr + 1
END IF
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RETURN buf
FUNCTION END
# INPUTS:
# ibf: The IBF
# offset: The offset which defines the starting point from which bucket
the packed operation starts
# count: The number of buckets in the array that will be packed
# cbl: The bit length of the counter can be found in the
ibf message in the ibf_counter_bit_length field
# pd: A byte array which contains the data packed with the pack_counter
function
# OUTPUTS:
# returns: Nothing because the unpacked counter is saved directly
into the IBF
FUNCTION unpack_counter(ibf, offset, count, cbl, pd)
ibf_bucket_ctr = 0
store = 0
store_bits = 0
byte_ctr = 0
WHILE TRUE
byte_read = pd[byte_ctr]
bit_to_pack_left = 8
byte_ctr++
WHILE bit_to_pack_left >= 0 DO
# Prevent packet from reading more than required
IF ibf_bucket_ctr > (count - 1) THEN
RETURN
END IF
IF store_bits + bit_to_pack_left >= cbl THEN
bit_use = cbl - store_bits
IF store_bits > 0 THEN
store = store << bit_use
END IF
bytes_to_shift = bit_to_pack_left - bit_use
counter_partial = byte_read >> bytes_to_shift
store = store | counter_partial
ibf.counter[ibf_bucket_ctr + offset] = store
byte_read = byte_read & (( 1 << bytes_to_shift ) - 1)
bit_to_pack_left -= bit_use
ibf_bucket_ctr++
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store = 0
store_bits = 0
ELSE
store_bits = store_bits + bit_to_pack_left
IF 0 == store_bits THEN
store = byte_read
ELSE
store = store << bit_to_pack_left
store = store | byte_read
END IF
BREAK
END IF
END WHILE
END WHILE
END FUNCTION
Figure 36
7.3. Multi Strata Estimators
In order to improve the precision of the estimates not only one
strata estimator is transmitted for larger sets. One, two, four or
eight strata estimators can be transferred. Transmitting multiple
strata estimators has the disadvantage that additional bandwidth will
be used, so despite the higher precision, it is not always optimal to
transmit eight strata estimators. Therefore, the following rules are
used, which are based on the average element size multiplied by the
number of elements in the set. This value is denoted as "b" in the
table:
SEs Rule
1 b < 68kb
2 b > 68kb
4 b > 269kb
8 b > 1'077kb
When creating multiple strata estimators, it is important to salt the
keys for the IBFs in the strata estimators differently, using the
following bit rotation based salting method:
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# Inputs:
# value: Input value to salt (needs to be 64 bit unsigned)
# salt: Salt to salt value with; Should always be ascending and start
# at zero
i.e. SE1 = Salt 0; SE2 = Salt 1 etc.
# Output:
# Returns: Salted value
FUNCTION se_key_salting(value, salt)
s = (salt * 7) modulo 64
RETURN (value >> s) | (value << (64 - s))
END FUNCTION
Figure 37
Performance study and details about the reasoning for the used
methods can be found in [byzantine_fault_tolerant_set_reconciliation]
in section 3.4.1 under the title "Added Support for Multiple Strata
Estimators". [byzantine_fault_tolerant_set_reconciliation]
8. Security Considerations
The security considerations in this document focus mainly on the
security goal of availability. The primary goal of the protocol is
to prevent an attacker from wasting computing and network resources
of the attacked peer.
To prevent denial of service attacks, it is vital to check that peers
can only reconcile a set once in a predefined time span. This is a
predefined value and needs to be adapted per use basis. To enhance
reliability and to allow for legitimate failures, say due to network
connectivity issues, applications SHOULD define a threshold for the
maximum number of failed reconciliation attempts in a given time
period.
It is important to close and purge connections after a given timeout
to prevent draining attacks.
8.1. General Security Checks
In this section general checks are described which should be applied
to multiple states.
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8.1.1. Input validation
The format of all received messages needs to be properly validated.
This is important to prevent many attacks on the code. The
application data MUST be validated by the application using the
protocol not by the implementation of the protocol. In case the
format validation fails the set operation MUST be terminated.
8.1.2. Byzantine Boundaries
To restrict an attacker there should be an upper and lower bound
defined and checked at the beginning of the protocol, based on prior
knowledge, for the number of elements. The lower byzantine bound can
be, for example, the number of elements the other peer had in his set
at the last contact. The upper byzantine bound can be a practical
maximum e.g. the number of e-voting votes, in Switzerland.
# Input:
# rec: Number of elements in remote set
# rsd: Number of elements differ in remote set
# lec: Number of elements in local set
# lsd: Number of elements differ in local set
# UPPER_BOUND: Given byzantine upper bound
# LOWER_BOUND: Given byzantine lower bound
# Output:
# returns TRUE if parameters in byzantine bounds otherwise returns FALSE
FUNCTION check_byzantine_bounds (rec,rsd,lec,lsd)
IF rec + rsd > UPPER_BOUND THEN
RETURN FALSE
END IF
IF lec + lsd > UPPER_BOUND THEN
RETURN FALSE
END IF
IF rec < LOWER_BOUND THEN
RETURN FALSE
END IF
RETURN TRUE
END FUNCTION
Figure 38
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8.1.3. Valid State
To harden the protocol against attacks, controls were introduced in
the improved implementation that check for each message whether the
message was received in the correct state. This is central so that
an attacker finds as little attack surface as possible and makes it
more difficult for the attacker to send the protocol into an endless
loop, for example.
8.1.4. Message Flow Control
For most messages received and sent there needs to be a check in
place that checks that a message is not received multiple times.
This is solved with a global store (message) and the following code
The sequence in which messages are received and sent is arranged in a
chain. The messages are dependent on each other. There are
dependencies that are mandatory, e.g. for a sent "Demand" message, an
"Element" message must always be received. But there are also
messages for which a response is not mandatory, e.g. the _Inquiry_
message is only followed by an "Offer" message, if the corresponding
element is in the set. Due to this fact, checks can be installed to
verify compliance with the following chain.
Chain for
elements +---------+ +---------+ +---------+ +---------+
NOT in IBF | INQUIRY |--->| OFFER |===>| DEMAND |===>| ELEMENT |
decoding +---------+ +---------+ +---------+ +---------+
peers set
Chain for
elements +---------+ +---------+ +---------+
in IBF | OFFER |--->| DEMAND |===>| ELEMENT |
decoding +---------+ +---------+ +---------+
peers set
--->: Answer not mandatory
===>: Always answer needed.
Figure 39
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In the message control flow its important to ensure that no
duplicated messages are received (Except inquiries where collisions
are possible) and only messages are received which are compliant with
the flow in Figure 39. To link messages the SHA-512 element hashes,
that are part of all messages, except in the _Inquiry_ messages, can
be used. To link an _Inquiry_ message to an _Offer_ message the
SHA-512 hash from the offer has to be salted and converted to the
IBF-Key (as described in Figure 7). The IBF-Key can be matched with
the received _Inquiry_ message.
At the end of the set reconciliation operation after receiving and
sending the _Done_ message, it should be checked that all demands
have been satisfied and all elements have been received.
This is based on [byzantine_fault_tolerant_set_reconciliation],
section 5.3 (Message Control Flow).
8.1.5. Limit Active/Passive Decoding changes
To prevent an attacker from sending a peer into an endless loop
between active and passive decoding, a limitation for active/passive
roll switches is required. Otherwise, an attacker could force the
victim to waste unlimited amount of resources by just transmitting
IBFs that do not decode. This can be implemented by a simple counter
which terminates the operation after a predefined number of switches.
The maximum number of switches needs to be defined in such a way that
it is very improbable that more switches are required in a legitimate
interaction, and hence the malicious behavior of the other peer is
assured.
The question after how many active/passive switches it can be assumed
that the other peer is not honest, depends on the various tuning
parameters of the algorithm. Section 5.4 of
[byzantine_fault_tolerant_set_reconciliation] demonstrates that the
probability of decoding failure is less than 15% for each round. The
probability that there will be n legitimate active/passive changes is
thus less than 0.15^{round number}. Which means that after about 30
active/passive switches it can be said with a certainty of 2^80 that
one of the peers is not following the protocol. Hence, participants
MUST impose a maximum of 30 active/passive changes.
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8.1.6. Full Synchronisation Plausibility Check
An attacker can try to use up a peer's bandwidth by pretending that
the peer needs full synchronisation, even if the set difference is
very small and the attacker only has a few (or even zero) elements
that are not already synchronised. In such a case, it would be ideal
if the plausibility could already be checked during full
synchronisation as to whether the other peer was honest or not with
regard to the estimation of the set size difference and thus the
choice of mode of operation.
In order to calculate this plausibility, section 5.5 of
[byzantine_fault_tolerant_set_reconciliation] describes a formula,
which depicts the probability with which one can calculate the
corresponding plausibility based on the number of new and repeated
elements after each received element.
Besides this approach from probability theory, there is an additional
check that can be made. After the entire set has been transferred to
the other peer, no known elements may be returned by the second peer,
since the second peer should only return the elements that are
missing from the initial peer's set.
This two approaches are implemented in the following pseudocode:
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# Input:
# SECURITY_LEVEL: The security level used e.g. 2^80
# state: The statemachine state
# rs: Estimated remote set difference
# lis: Number of elements in set
# rd: Number of duplicated elements received
# rf: Number of fresh elements received
# Output:
# Returns TRUE if full synchronisation is plausible and FALSE otherwise
FUNCTION full_sync_plausibility_check (state,rs,lis,rd,rf)
security_level_lb = -1 * SECURITY_LEVEL
# Make sure that no element is received double when
# all elements already are transmitted to the oder side.
IF FULL_SENDING == state AND rd > 0 THEN
RETURN FALSE
END IF
# Probabilistic algorithm to check for plausible
# element distribution
IF FULL_RECEIVING == state THEN
# Prevent division by 0
IF 0 <= rs THEN
rs = 1
END IF
# Formula to verify plausibility
base = 1 - (rs / (lis + rs))
exponent = rd - rf * lis / rs
value = exponent * (LOG2(base)/LOG2(2))
IF value < security_level_lb OR value > SECURITY_LEVEL THEN
RETURN FALSE
END IF
END IF
RETURN TRUE
END FUNCTION
Figure 40
8.2. States
In this section the security considerations for each valid message in
all states is described, if any other message is received the peer
MUST terminate the operation.
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8.2.1. Expecting IBF
Security considerations for received messages:
Request Full It needs to be checked that the full synchronisation
mode with receiving peer sending first is plausible according to
the algorithm deciding which operation mode is applicable as
described in Section 7.1.1.
IBF It needs to be checked that the differential synchronisation
mode is plausible according to the algorithm deciding which
operation mode is applicable as described in Section 7.1.1.
Send Full It needs to be checked that the full synchronisation mode
with initiating peer sending first is plausible according to the
algorithm deciding which operation mode is applicable as described
in Section 7.1.1.
8.2.2. Full Sending
Security considerations for received messages:
Full Element When receiving full elements there needs to be checked,
that every element is a valid element, that no element has been
received more than once, and that not more elements have been
received than the other peer has committed to at the beginning of
the operation. The plausibility should also be checked with an
algorithm as described in Section 8.1.6.
Full Done When receiving the _Full Done_ message, it is important to
check that not fewer elements have been received than the other
peer has committed to send at the beginning of the operation. If
the sets differ (the FINAL CHECKSUM field in the Full Done message
does not match to the SHA-512 hash XOR sum of the local set), the
operation has failed and the reconciliation MUST be aborted. It
is a strong indicator that something went wrong (eg. some hardware
bug). This should never occur!
8.2.3. Expecting IBF Last
Security considerations for received messages:
IBF The application should check that the overall size of the IBF
that is being transmitted is within its resource bounds, and abort
the protocol if its resource limits are likely to be exceeded, or
if the size is implausible for the given operation.
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It needs to be checked that the offset (message field "OFFSET")
for every received _IBF_ message is strictly monotonic increasing
and is a multiple of the MAX_BUCKETS_PER_MESSAGE defined in the
Constants section, otherwise the connection MUST be aborted.
Another sanity check is to ensure that the "OFFSET" message field
never is higher than the "IBF SIZE" field in the _IBF_ message.
IBF Last When all _IBF_ messages have been received an _IBF Last_
message should conclude the transmission of the IBF and a change
to the *Active Decoding* phase should be ensured.
To verify that all IBFs have been received, a simple validation
can be made. The number of buckets in the _IBF Last_ message
added to the value in the message OFFSET field should always be
equal to the "IBF SIZE".
Further plausibility checks can be made. One is to ensure that
after each active/passive switch the IBF can never be more than
double in size. Another plausibility check is that an IBF
probably never will be larger than the byzantine upperbound
multiplied by two. The third plausibility check is to take
successfully decoded IBF keys (received offers and demands) into
account and to validate the size of the received IBF with the in
Figure 35 described function get_next_ibf_size(). If any of these
three checks fail the operation must be aborted.
8.2.4. Active Decoding
In the *Active Decoding* state it is important to prevent an attacker
from generating and transmitting an unlimited number of IBFs that all
do not decode, or to generate an IBF constructed to send the peers in
an endless loop. To prevent an endless loop in decoding, loop
detection MUST be implemented. A solution to prevent endless loop is
to limit the number of elements decoded from an IBF. This limit is
defined by the number of buckets in the IBF. It is not possible that
more elements are decoded from an IBF than an IBF has buckets. If
more elements than buckets are in an IBF it is not possible to get
pure buckets. An additional check that should be implemented, is to
store all element IDs that were prior decoded. When a new element ID
is decoded from the IBF it should always be checked that no element
ID is repeated. If the same element ID is decoded more than once,
this is a strong indication for an invalid IBF and the operation MUST
be aborted. Notice that the decoded element IDs are salted as
described in Figure 7 so the described bit rotation needs to be
reverted before the decoded element ID is stored and compared to the
previous decoded element IDs.
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If the IBF decodes more elements than are plausible, the operation
MUST be terminated. Furthermore, if the IBF decoding successfully
terminates and fewer elements were decoded than plausible, the
operation MUST also be terminated. The upper thresholds for decoded
elements from the IBF is the remote set size the other peer has
committed too (Case if the complete remote set is new). The lower
threshold for decoding element is the absolute value of the
difference between the local and remote set size (Case the set
difference is only in the set of a single peer). The other peer's
committed set sizes is transmitted in the the *Expecting IBF* state.
Security considerations for received messages:
Offer If an offer for an element, that never has been requested by
an inquiry or if an offer is received twice, the operation MUST be
terminated. This requirement can be fulfilled by saving lists
that keep track of the state of all sent inquiries and offers.
When answering offers these lists MUST be checked. The sending
and receiving of Offer messages should always be protected with an
Message Flow Control to secure the protocol against missing,
duplicated, out-of-order or unexpected messages.
Element If an element that never has been requested by a demand or
is received twice, the operation MUST be terminated. The sending
and receiving of Element messages should always be protected with
an Message Flow Control to secure the protocol against missing,
duplicated, out-of-order or unexpected messages.
Demand For every received demand an offer has to be sent in advance.
If a demand for an element is received, that never has been
offered or the offer already has been answered with a demand, the
operation MUST be terminated. It is required to implement a list
which keeps track of the state of all sent offers and received
demands. The sending and receiving of _Demand_ messages should
always be protected with an Message Flow Control to secure the
protocol against missing, duplicated, out-of-order or unexpected
messages.
Done The _Done_ message is only received if the IBF has finished
decoding and all offers have been sent. If the _Done_ message is
received before the decoding of the IBF is finished or all open
demands have been answered, the operation MUST be terminated. If
the sets differ (the FINAL CHECKSUM field in the Done message does
not match to the SHA-512 hash XOR sum of the local set), the
operation has failed and the reconciliation MUST be aborted. It
is a strong indicator that something went wrong (eg. some hardware
bug). This should never occur!
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When a _Done_ message is received the
"check_if_synchronisation_is_complete()" function from the Message
Flow Control is required to ensure that all demands have been
satisfied successfully.
8.2.5. Finish Closing
In the *Finish Closing* state the protocol waits for all sent demands
to be fulfilled.
In case not all sent demands have been answered in time, the
operation has failed and MUST be terminated.
Security considerations for received messages:
Element When receiving Element messages it is important to always
check the Message Flow Control to secure the protocol against
missing, duplicated, out-of-order or unexpected messages.
8.2.6. Finished
In this state the connection is terminated, so no security
considerations are needed.
8.2.7. Expect SE
Security considerations for received messages:
Strata Estimator In case the strata estimator does not decode, the
operation MUST be terminated to prevent to get to an unresolvable
state. The set difference calculated from the strata estimator
needs to be plausible, which means within the byzantine boundaries
described in section Byzantine Boundaries.
8.2.8. Full Receiving
Security considerations for received messages:
Full Element When receiving full elements there needs to be checked,
that every element is a valid element, no element has been
received more than once and not more elements are received than
the other peer committed to sending at the beginning of the
operation. The plausibility should also be checked with an
algorithm as described in Section 8.1.6.
Full Done When the _Full Done_ message is received from the remote
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peer, it should be checked that the number of elements received
matches the number that the remote peer originally committed to
transmitting, otherwise the operation MUST be terminated. If the
sets differ (the FINAL CHECKSUM field in the Full Done message
does not match to the SHA-512 hash XOR sum of the local set), the
operation has failed and the reconciliation MUST be aborted. It
is a strong indicator that something went wrong (eg. some hardware
bug). This should never occur!
8.2.9. Passive Decoding
Security considerations for received messages:
IBF In case an IBF message is received by the peer a active/passive
role switch is initiated by the active decoding remote peer. A
switch into active decoding mode MUST only be permitted for a
predefined number of times as described in Section 8.1.5
Inquiry A check needs to be in place that prevents receiving an
inquiry for an element multiple times or more inquiries than are
plausible. The upper thresholds for sent/received inquiries is
the remote set size the other peer has committed too (Case if the
complete remote set is new). The lower threshold for for sent/
received inquiries is the absolute value of the set difference
between the local and remote set size (Case the set difference is
only in the set of a single peer). The other peer's committed set
sizes is transmitted in the the *Expecting IBF* state. Beware
that it is possible to get key collisions and an inquiry for the
same key can be transmitted multiple times, so the threshold
should take this into account. The sending and receiving of
_Inquiry_ messages should always be protected with an Message Flow
Control to secure the protocol against missing, duplicated, out-
of-order or unexpected messages.
Demand Same action as described for _Demand_ message in section
Active Decoding.
Offer Same action as described for _Offer_ message in section Active
Decoding.
Done Same action as described for _Done_ message in section Active
Decoding.
Element Same action as described for _Element_ message in section
Active Decoding.
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8.2.10. Finish Waiting
In the *Finish Waiting* state the protocol waits for all transmitted
demands to be fulfilled.
In case not all transmitted demands have been answered at this time,
the operation has failed and the protocol MUST be terminated with an
error.
Security considerations for received messages:
Element When receiving Element messages it is important to always
check the Message Flow Control to secure the protocol against
missing, duplicated, out-of-order or unexpected messages.
9. Constants
The following table contains constants used by the protocol. The
constants marked with a * are validated through experiments in
[byzantine_fault_tolerant_set_reconciliation].
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Name | Value | Description
----------------------------+------------+-------------------------------
SE_STRATA_COUNT | 32 | Number of IBFs in a strata
estimator.
IBF_HASH_NUM* | 3 | Number of times an element is
hashed to an IBF.
(from section 4.5.2)
IBF_FACTOR* | 2 | The factor by which the size
of the IBF is increased in
case of decoding failure or
initially from the set
difference.
(from section 4.5.2)
MAX_BUCKETS_PER_MESSAGE | 1120 | Maximum bucket of an IBF
that are transmitted in
single message.
IBF_MIN_SIZE* | 37 | Minimal number of buckets
in an IBF. (from section 3.8)
DIFFERENTIAL_RTT_MEAN* | 3.65145 | The average RTT that is
needed for a differential
synchronisation.
SECURITY_LEVEL* | 2^80 | Security level for
probabilistic security
algorithms. (from section 5.8)
PROBABILITY_FOR_NEW_ROUND* | 0.15 | The probability for a IBF
decoding failure in the
differential synchronisation
mode. (from section 5.4)
DIFFERENTIAL_RTT_MEAN* | 3.65145 | The average RTT that is needed
for a differential
synchronisation.
(from section 4.5.3)
MAX_IBF_SIZE | 1048576 | Maximal number of buckets in
an IBF.
AVG_BYTE_SIZE_SE* | 4221 | Average byte size of a single
strata estimator.
(from section 3.4.3)
VALID_NUMBER_SE* | [1,2,4,8] | Valid number of SE's
(from section 3.4)
Figure 41
10. GANA Considerations
GANA is requested to amend the "GNUnet Message Type" [GANA] registry
as follows:
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Type | Name | References | Description
--------+----------------------------+------------+----------------------
559 | SETU_P2P_REQUEST_FULL | [This.I-D] | Request the full set
of the other peer.
710 | SETU_P2P_SEND_FULL | [This.I-D] | Signals to send the
full set to the other
peer.
560 | SETU_P2P_DEMAND | [This.I-D] | Demand the whole
element from the
otherpeer, given
only the hash code.
561 | SETU_P2P_INQUIRY | [This.I-D] | Tell the other peer
to send a list of
hashes that match
an IBF key.
562 | SETU_P2P_OFFER | [This.I-D] | Tell the other peer
which hashes match
a given IBF key.
563 | SETU_P2P_OPERATION_REQUEST | [This.I-D] | Request a set union
operation from a
remote peer.
564 | SETU_P2P_SE | [This.I-D] | Strata Estimator
uncompressed.
565 | SETU_P2P_IBF | [This.I-D] | Invertible Bloom
Filter slices.
566 | SETU_P2P_ELEMENTS | [This.I-D] | Actual set elements.
567 | SETU_P2P_IBF_LAST | [This.I-D] | Invertible Bloom
Filter Last Slices.
568 | SETU_P2P_DONE | [This.I-D] | Set operation is
done.
569 | SETU_P2P_SEC | [This.I-D] | Strata Estimator
compressed.
570 | SETU_P2P_FULL_DONE | [This.I-D] | All elements in
full synchronisation
mode have been sent
is done.
571 | SETU_P2P_FULL_ELEMENT | [This.I-D] | Send an actual
element in full
synchronisation mode.
Figure 42
11. Contributors
The GNUnet implementation of the byzantine fault tolerant set
reconciliation protocol was originally implemented by Florian Dold.
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12. Normative References
[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/info/rfc5869>.
[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/info/rfc2119>.
[RFC3385] Sheinwald, D., Satran, J., Thaler, P., and V. Cavanna,
"Internet Protocol Small Computer System Interface (iSCSI)
Cyclic Redundancy Check (CRC)/Checksum Considerations",
RFC 3385, DOI 10.17487/RFC3385, September 2002,
<https://www.rfc-editor.org/info/rfc3385>.
[RFC1951] Deutsch, P., "DEFLATE Compressed Data Format Specification
version 1.3", RFC 1951, DOI 10.17487/RFC1951, May 1996,
<https://www.rfc-editor.org/info/rfc1951>.
[byzantine_fault_tolerant_set_reconciliation]
Summermatter, E., "Byzantine Fault Tolerant Set
Reconciliation", 2021, <https://summermatter.net/
byzantine-fault-tolerant-set-reconciliation-
summermatter.pdf>.
[GANA] GNUnet e.V., "GNUnet Assigned Numbers Authority (GANA)",
April 2020, <https://gana.gnunet.org/>.
[CryptographicallySecureVoting]
Dold, F., "Cryptographically Secure, Distributed
Electronic Voting",
<https://git.gnunet.org/bibliography.git/plain/docs/
ba_dold_voting_24aug2014.pdf>.
[ByzantineSetUnionConsensusUsingEfficientSetReconciliation]
Dold, F. and C. Grothoff, "Byzantine set-union consensus
using efficient set reconciliation",
<https://doi.org/10.1186/s13635-017-0066-3>.
[Eppstein] Eppstein, D., Goodrich, M., Uyeda, F., and G. Varghese,
"What's the Difference? Efficient Set Reconciliation
without Prior Context",
<https://doi.org/10.1145/2018436.2018462>.
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[GNS] Wachs, M., Schanzenbach, M., and C. Grothoff, "A
Censorship-Resistant, Privacy-Enhancing and Fully
Decentralized Name System", 2014,
<https://doi.org/10.1007/978-3-319-12280-9_9>.
Appendix A. Test Vectors
A.1. Map Function
INPUTS:
k: 3
ibf_size: 300
key1: 0xFFFFFFFFFFFFFFFF (64-bit)
key2: 0x0000000000000000 (64-bit)
key3: 0x00000000FFFFFFFF (64-bit)
key4: 0xC662B6298512A22D (64-bit)
key5: 0xF20fA7C0AA0585BE (64-bit)
Figure 43
OUTPUT:
key1: ["122","157","192"]
key2: ["85","243","126"]
key3: ["208","101","222"]
key4: ["239","269","56"]
key5: ["150","104","33"]
Figure 44
A.2. ID Calculation Function
INPUTS:
element 1: 0xFFFFFFFFFFFFFFFF (64-bit)
element 2: 0x0000000000000000 (64-bit)
element 3: 0x00000000FFFFFFFF (64-bit)
element 4: 0xC662B6298512A22D (64-bit)
element 5: 0xF20fA7C0AA0585BE (64-bit)
Figure 45
OUTPUT:
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element 1: 0x5AFB177B
element 2: 0x64AB557C
element 3: 0xCB5DB740
element 4: 0x8C6A2BB2
element 5: 0x7EC42981
Figure 46
A.3. Counter Compression Function
INPUTS:
counter serie 1: [1,8,10,6,2] (min bytes 4)
counter serie 2: [26,17,19,15,2,8] (min bytes 5)
counter serie 3: [4,2,0,1,3] (min bytes 3)
Figure 47
OUTPUT:
counter serie 1: 0x18A62
counter serie 2: 0x3519BC48
counter serie 3: 0x440B
Figure 48
Authors' Addresses
Elias Summermatter
Seccom GmbH
Brunnmattstrasse 44
CH-3007 Bern
Switzerland
Email: elias.summermatter@seccom.ch
Christian Grothoff
Berner Fachhochschule
Hoeheweg 80
CH-2501 Biel/Bienne
Switzerland
Email: grothoff@gnunet.org
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