Fiat-Shamir Transformation

Internet-Draft	Fiat-Shamir Transformation	August 2026
Orrù	Expires 18 February 2027	[Page]

Abstract

This document describes the Fiat-Shamir transformation, which allows making a public-coin protocol non-interactive by means of a cryptographic hash function.¶

It specifies how the hash function is employed, how prover messages are encoded as hash-function input, and how verifier messages are decoded from the hash function's output, as well as the serialization and deserialization of the non-interactive argument string.¶

2. Terminology and conventions in this document

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶

The algorithms and procedures in this document are specified using Python-like pseudocode. Each function accepts defined inputs and parameters and returns one or more output values. Once a protocol variant and ciphersuite are selected, all associated parameters are treated as constants.¶

The following notation is used throughout this document.¶

2.1. Bytes and integers

A byte is an 8-bit unsigned integer (an octet), and a byte string is a finite sequence of bytes. An N-byte string is a byte string of length N. The empty byte string is written ""; x || y is the concatenation of the byte strings x and y. len(x) denotes the length in bytes of the byte string x. zeros(N) denotes the N-byte string of zero bytes.¶

Byte strings are indexed from zero. For integers 0 <= i <= j <= len(x), x[i : j] denotes the (j - i)-byte substring of x consisting of the bytes at positions i, i+1, ..., j-1. In particular, x[0 : N] is the first N bytes of x, x[i : i] is the empty byte string "", and x[0 : len(x)] is x itself.¶

A byte string x is a prefix of a byte string y if y == x || z for some byte string z (even empty). An encoding is prefix-free if, for any two distinct values, the encoding of one is never a prefix of the encoding of the other. A simple prefix-free encoding of a byte string b is LE(len(b), 4) || b as described in Section 6.1.1.¶

LE(n, w) and LE2IP(x) are the integer/byte-string conversion primitives used throughout this document, in little-endian byte order. LE(n, w) converts a non-negative integer n less than 256^w into a w-byte, little-endian byte string, and fails if n >= 256^w. LE2IP(x) converts a byte string x into a non-negative integer using the little-endian byte order.¶

LE(n, w)

Inputs:

- n, a non-negative integer with 0 <= n < 256^w
- w, the output length in bytes

Output: out, a w-byte string

1. fail if n < 0 or n >= 256^w
2. out = zeros(w)
3. for i in 0, ..., w-1:
4.    out[i] := n mod 256
5.    n := floor(n / 256)
6. return out

LE2IP(x)

Input: x, a byte string of length w

Output: n, a non-negative integer with 0 <= n < 256^w

1. n = 0
2. for i in 0, ..., len(x)-1:
3.    n = n + x[i] * 256^i
4. return n

The set of integers between 0 and N-1 is denoted [0, N).¶

2.2. Duplex sponge interface

The Fiat-Shamir transformation relies on a cryptographic hash function, modeled as a random oracle. This is implemented as a stateful interface called a duplex sponge (defined in Section 3), which Absorbs prover messages into an evolving internal state and Squeezes from that state the bytes from which verifier messages are derived.¶

The interface generalizes the sponge [SPONGE], which maps a variable-length input to a variable-length output by absorbing all of its input and then squeezing all of its output, to the duplex setting [DUPLEX], in which absorbing and squeezing may be arbitrarily interleaved while retaining the same state. The state is split into a rate, the portion through which bytes are absorbed and squeezed, and a capacity, which is never read or written directly. Security relies on the size of the capacity. The properties a concrete instantiation must satisfy for this to hold, and the resulting security loss, are given in Section 8 and analyzed in [CO25].¶

2.3. Proof systems terminology

The session identifier is a 32-byte string that identifies the application context and the specific non-interactive argument in use; it is held by both the prover and the verifier (see Section 5.1).¶

A prover message is a message sent by the interactive prover, and a verifier message is a message sent by the interactive verifier (a uniformly random value, sometimes called challenge). A message can be a value of any type for which a codec (Section 4) is defined, such as a byte string, an unsigned integer, or a group element. The transcript is the ordered sequence of prover and verifier messages. In particular, the transcript does not include the instance and session identifier.¶

The instance specifies the statement being proven and is held by both the prover and the verifier. The (encoded) instance MUST be non-empty.¶

The witness is the prover's private input. It is known only to the prover and is never revealed. It appears neither in the transcript nor in the NARG string.¶

For an NP language, the instance is a word, the witness is a proof of its membership in the language, and the resulting non-interactive argument proves that the instance is indeed in the language. This claim is also referred to as the statement. Proof systems might support different statements or express the same language in different ways.¶

The NARG string (non-interactive argument string) is the serialized output of the non-interactive prover.¶

The notation in this document is for an interactive argument with k rounds in which the prover moves first (that is, sends the first message) and the verifier moves last. Other types of interactions can be expressed in the same notation by setting the unused messages to the empty string: a protocol whose verifier moves first (such as a batch argument) sets its first prover message to "", and one whose prover moves last (such as a sigma protocol) sets its final verifier message to "". Prover and verifier round messages 2, ..., k-1 MUST be non-empty.¶

2.4. Codec and serialization

A prover message is processed in two independent ways: it is absorbed into the hash function to derive the verifier messages, and it is written into the NARG string sent to the verifier. This document keeps the two separate.¶

A codec (Section 4) is the pair of maps between messages and the hash function's alphabet (bytes, in this document):¶

Encoding converts the instance and each prover message into the bytes absorbed by the duplex sponge (Section 4.1).¶
Decoding converts the bytes squeezed from the duplex sponge into a uniformly-distributed verifier message (Section 4.2).¶

Serialization (Section 6) is concerned with mapping prover messages to and from the NARG string:¶

Serialization writes the prover messages into the NARG string produced by the non-interactive prover.¶
Deserialization reads the prover messages back from the NARG string, and returns an error if the message is invalid.¶

For a prover message, the encoded bytes coincide with the serialized bytes: the encoding maps are the serialization functions of Section 6.1. However, codecs and serialization serve different purposes. Codecs must maintain the soundness of the transformation, whereas deserialization keeps the NARG string unambiguous and rejects malformed proofs (see Section 4.2 and Section 6.2).¶

3. Duplex sponge

This section defines the duplex sponge instantiations used in this document.¶

3.1. Interface

Prover and verifier messages are handled via three operations:¶

Init(session_id) -> state: create a new duplex sponge state, seeded by the 32-byte string session_id.¶
state.Absorb(x): absorb x into the state.¶
state.Squeeze(n) -> buf: produce n elements from the state.¶

In the duplex sponge interface, messages can be absorbed incrementally, and insert no separators: state.Absorb(x) followed by state.Absorb(y) (with no state.Squeeze in between) is equivalent to state.Absorb(x || y).¶

Each state.Squeeze(n) is uniformly distributed, and consecutive state.Squeeze calls continue one output stream.¶

Guidance on how to produce a 32-byte session_id, and its security requirements, is given in Section 5.1; its role in composability and reuse is discussed in Section 8.3.¶

3.2. XOF duplex sponge

This section implements the duplex sponge interface using an eXtendable-Output Function (XOF). An XOF is a hash function mapping a byte string to an output string of any desired length. It has the following operations:¶

XOF.New() -> xof_state, a fresh XOF state;¶
xof_state.Update(x), absorbing the byte string x into the state;¶
xof_state.Copy() -> xof_state, a copy of the XOF state;¶
xof_state.Finalize() -> reader, finalizing the XOF state and returning a reader over the output stream;¶
reader.Read(n) -> buf, the next n bytes of the output stream.¶

R (the rate) is the block size at which the XOF processes absorbed input. It MUST be R >= 32. XOF(M, len) denotes the XOF evaluation of the byte string M, producing len bytes of output. The concrete XOF is fixed by the suite (Section 9). Consecutive Read calls continue the output stream.¶

Every verifier message is the XOF evaluation over the session identifier, the encoded instance, and the encoded prover messages up to and including the current round. That is, the i-th verifier message (for 1 <= i <= k) of byte length len_i is computed as:¶

verifier_msg[i] := decode[i](XOF(
                       session_id || zeros(R - 32)
                       || encode[0](instance)
                       || encode[1](prover_msg[1])
                       || ...
                       || encode[i](prover_msg[i]),
                   len_i))

The session identifier is padded with R - 32 zero bytes so that the instance and prover messages begin on a fresh rate-block boundary (see Section 3.2.1), for efficiency (Section 7).¶

The security properties required of the XOF, and the security level attained, are fixed by the suite (Section 9); the corresponding requirements on the construction are discussed in Section 8.4.¶

3.2.1. Init

Seed the state by absorbing the session identifier, padded with zeros to fill the rate (the remaining R - 32 bytes).¶

Init(session_id)

Input: session_id, a byte array

Output: a duplex sponge state

1. assert len(session_id) == 32
2. ctx = XOF.New()
3. ctx.Update(session_id || zeros(R - 32))
4. return state := (ctx, reader = None)

3.2.2. Absorb

Feed a byte string x into the state. Absorbing the empty string leaves the state unchanged.¶

state.Absorb(x)

Input: x, a byte string

1. state.ctx.Update(x)
2. if len(x) != 0:
3.    state.reader = None

3.2.3. Squeeze

Returns the next n bytes of the XOF output stream computed over the absorbed input. If the duplex sponge is in the absorbing phase, it finalizes a copy of the absorbing context as an XOF reader. Consecutive Squeeze calls continue the same output stream.¶

state.Squeeze(n)

Input: n, the number of bytes to be squeezed

Output: a uniformly-distributed random n-byte string

1. if state.reader == None:
2.    state.reader = state.ctx.Copy().Finalize()
3. return state.reader.Read(n)

4. Codecs

A codec is a set of functions that map prover messages to, and verifier messages from, the hash function's alphabet.¶

Encoding maps convert the instance and the prover messages into the bytes absorbed by the duplex sponge. The only security requirement on encoding functions is that they be prefix-free.¶
Decoding maps turn squeezed elements into verifier messages. Decoding MUST preserve the uniform distribution in the verifier message space (up to a small codec error). Decoding is infallible.¶

4.1. Encoding into byte strings

The encoding of the instance and of each prover message is its serialization, as described in Section 6.1.¶

4.2. Decoding from byte strings

Decoding converts a uniformly-distributed Squeeze output into a verifier message. Each verifier message type fixes the number of bytes to squeeze.¶

Decoding is not deserialization, and need not invert encoding nor even be injective; its only requirement is to be distribution-preserving: if its input is a uniformly random byte string, then its output is (statistically close to) uniformly distributed over the verifier message type.¶

4.2.1. Byte arrays

The decoding function for fixed-length byte arrays is the identity.¶

DecodeBytes(buf, N)

Inputs:

- buf, a byte string
- N, the expected byte length

Output: out, a byte string of length N

1. assert len(buf) == N
2. return buf

4.2.2. Unsigned integers

To sample a uniformly random element modulo M, let Ns be the smallest integer with 256^Ns >= M: squeeze Ns + 16 bytes, interpret them as a little-endian non-negative integer via LE2IP, and reduce modulo M.¶

DecodeUint(buf, M)

Inputs:

- buf, a byte string of length Ns + 16
- M, the modulus

Output: out, an integer in the range [0, M)

1. assert len(buf) == Ns + 16
2. return LE2IP(buf) mod M

Decoding always interprets bytes in little-endian order via LE2IP.¶

The 16 extra bytes bound the statistical distance between the reduced value and the uniform distribution over [0, M) to 2^-128. More generally, sampling n extra bytes bounds the bias to 2^-8n. An instantiation targeting a security level of lambda bits SHOULD squeeze lambda/8 extra bytes.¶

In three cases this approach is inefficient:¶

if M is a power of two, since the decoding bias is always 0;¶
if M is only slightly below a power of 256 (for example, the secp256k1 scalar field order) where squeezing just Ns bytes and reducing with a single conditional subtraction already has bias of approximately 2^-128;¶
if the soundness error of the interactive argument is much smaller than the bias introduced, for example a protocol with 30-bit challenges does not require such a big modular reduction.¶

In such cases, applications MAY use an alternative decoding function, provided it meets the following security requirements:¶

The function MUST have bias at most the soundness error of the interactive argument. The bias adds to the soundness error of the resulting non-interactive argument, and this requirement asks that the resulting non-interactive soundness error stays within a small multiple of the interactive soundness error.¶
The function SHOULD be amenable to straight-line implementations. In particular, rejection sampling SHOULD NOT be used (see Section 8.2).¶

A similar observation in the context of hashing to the elliptic curve field is available in Section 5 of [RFC9380].¶

4.2.3. Field elements

A field element of a field of order p^m is decoded coordinate by coordinate, via DecodeUint (Section 4.2.2), starting from the least-significant. With Ns as in Section 6.1.4, this consumes m * (Ns + 16) bytes. A prime field is the case m = 1.¶

DecodeField(buf, p, m)

Inputs:

- buf, a byte string of length m * (Ns + 16)
- p, the prime characteristic of the field
- m, the extension degree

Output: out, an element of the field of order p^m, given by its
        coordinates (a[0], ..., a[m-1]) over the prime field

1. assert len(buf) == m * (Ns + 16)
2. for i in 0, ..., m-1:
3.    chunk = buf[i * (Ns + 16) : (i + 1) * (Ns + 16)]
4.    a[i] = DecodeUint(chunk, p)
5. return (a[0], ..., a[m-1])

For m = 1, DecodeField is DecodeUint, and the same efficiency remarks apply. Applications MAY substitute a more efficient alternative, subject to the same security requirements described in Section 4.2.2.¶

For m > 1, decoding relies on 16 * m additional randomness bytes. Applications with big-integer arithmetic available MAY use a more randomness-efficient decoding algorithm, by instead sampling Nm + 16 bytes, where Nm is the smallest integer with 256^Nm >= p^m, interpreting them as an integer via LE2IP, reducing modulo p^m, and recovering the coordinates (a[0], ..., a[m-1]) as the base-p digits of the result (least-significant digit first). This consumes Nm + 16 bytes, with the same 2^-128 bias bound.¶

5. Initialization

Before any prover message is processed, both parties start the duplex sponge with the session identifier (Section 5.1), and then the instance (Section 5.2).¶

Neither the session identifier nor the instance is part of the NARG string: the verifier holds both as its own inputs.¶

5.1. Session identifiers

The session identifier is a 32-byte string that identifies the context in which the non-interactive argument is used. The procedure DeriveSessionID below is the RECOMMENDED way to obtain a session identifier from a human-meaningful variable-length tag. An application MAY use any 32-byte string it derives by its own means.¶

For a duplex sponge operating over bytes, the session identifier is derived from a tag via the procedure DeriveSessionID. The tag is a byte string whose encoding as a sequence of bytes MUST be specified unambiguously, so that every implementation reproduces identical bytes. It is RECOMMENDED the tag be a US-ASCII string, without byte-order mark at the beginning, nor 0x00 byte termination.¶

When the tag is composed of several fields, those fields MUST be combined unambiguously, so that no two distinct tuples of field values yield the same byte string. For example, concatenating ("SV1", "22") and ("SV12", "2") both yield SV122 and so would share the same session identifier. Using fixed-width fields or an unambiguous delimiter is sufficient.¶

The tag has the following security requirements:¶

the tag MUST uniquely identify the non-interactive argument used, including the interactive argument system, the types of prover and verifier messages, the hash suite, and the language associated with the interactive argument.¶
the tag MUST uniquely identify the codecs used: the order and types of encodings and decodings at each round. For example, implementations that sample verifier messages differently must have different session identifiers.¶
the tag MUST identify the application context in which the non-interactive argument is used, so that a NARG string produced for one application cannot be accepted in another. A namespace string under the application's control is sufficient, such as the URL https://example.com/login/v2. Freshness information, such as an epoch number or timestamp, MAY additionally be included when the application requires proofs not to be replayed within its own context (see Section 8.3).¶
the tag SHOULD begin with a fixed identification string that is unique to the application.¶
the tag SHOULD include a version number.¶

An application that bypasses DeriveSessionID, and sets the 32-byte session_id directly MUST ensure it satisfies the same requirements.¶

DeriveSessionID(tag)

Input: tag, an application-chosen byte string (see above)

Output: session_id, a 32-byte string

1. duplex_sponge := DS.Init("irtf-cfrg-fiat-shamir/session-id")
2. duplex_sponge.Absorb(tag)
3. return duplex_sponge.Squeeze(32)

Above, DS denotes the duplex sponge in use (Section 3), instantiated with one of the suites of Section 9. The 32-byte string "irtf-cfrg-fiat-shamir/session-id" is a domain separator for this derivation.¶

As an example, consider a fictional application named Foo that implements sigma protocols over elliptic curves for encrypted messages shared during a time epoch tttt. A reasonable choice of tag is:¶

FOO-SV{xx}-{tttt}-DSFS-{hashID}-SIGMA-PROOFS-{yy}

where xx is the two-digit number indicating the version, yy is the two-digit number indicating the elliptic-curve ciphersuite, hashID is the hash identifier, and tttt is the epoch number written in decimal US-ASCII digits.¶

As another example, consider a fictional application named Bar that implements an ad-hoc zero-knowledge virtual machine for correct execution of circuits. A reasonable choice of tag is¶

BAR-COM{cc}

where {cc} is the commit hash of the associated version of the cryptographic specification of the protocol.¶

Yet another reasonable choice for the session identifier is to append a description of the interactive argument system together with the length of each prover and verifier message, after the version string. For instance:¶

BAZ-SV{xx}-DSFS-{hashID}-sumcheck-{ff}-A2round-messageS1challenge

where xx is the two-digit version number, hashID is the hash identifier, and ff is the two-digit identifier of the finite field over which the proof is computed. The suffix A2round-messageS1challenge describes one sumcheck round where the prover absorbs (A) two field elements round-message, and the verifier squeezes (S) one field element challenge. This is similar to the SAFE API [SAFE] IO pattern.¶

5.2. Instance

The instance is input to the non-interactive prover and the non-interactive verifier; it fixes the specific statement being proven.¶

The instance is the first value absorbed after Init(session_id) and before any prover message. The prover and verifier MUST absorb encode[0](instance), where encode[0] is the first encoding map. The encoded instance MUST be non-empty. While the session identifier of the previous section Section 5.1 fixes the language, the instance selects one of its members.¶

As for every encoding map, encode[0] MUST be prefix-free, else a malicious prover may be able to satisfy the verification equations on a statement it cannot prove (see Section 8.5). The encoding map encode[0] SHOULD reuse the serialization functions of Section 6.1.¶

As an example, consider the sumcheck relation for multilinear polynomials in N variables over the field of size p^m. For a polynomial committed using the polynomial commitment scheme COM, the relation consists of:¶

instance (S, C): C the commitment, and S the target sum;¶
witness (F, r): F is the multilinear polynomial in N variables, and r is the commitment opening information.¶

such that:¶

COM.Open(C, F, r)  = 1,
sum(F(b1, ..., bN) for (b1, ..., bN) in {0, 1}^N) = S

A valid instance encoding function is:¶

SerializeField(S, p, m) || SerializeUint(N, 2^32) || COM.Serialize(C)

where COM.Serialize is the commitment-serialization function of the scheme COM (the opening check COM.Open is used above).¶

As another example, consider, in the discrete logarithm setting, the Chaum-Pedersen relation over an additive elliptic curve group with generators G, H (for which the relative discrete logarithm is not known). The relation consists of:¶

the instance (C, D), a pair of Pedersen commitments;¶
the witness (x, r, s), scalar field elements with x the commitment message and r, s independent random commitment openings¶

such that¶

C = xG + rH, D = xG + sH.

A valid instance encoding function is:¶

enc(G) || enc(H) || enc(C) || enc(D)

where enc is the group element-serialization function described in Section 6.1.5.¶

Omitting public statement data from the transformation, such as N in the first example or the group generators G, H in the second, can compromise soundness of the proof system. See Section 8.5.¶

6. Non-interactive argument string

6.1. Serialization

The NARG string is the concatenation of the serialization of each prover message, as defined below. The same bytes are absorbed into the duplex sponge to derive the verifier messages (Section 4.1).¶

6.1.1. Byte strings

Serialization of an N-byte string is the identity function.¶

SerializeBytes(s)

Input: s, an N-byte string

Output: out, an N-byte string

1. return s

SerializeBytes carries no length information of its own: it is the identity. It can therefore be used only when N is fixed and known by the message's type and the instance, that is: prover and verifier both agree on N before the NARG string is parsed (see Section 6.2.1). On such a fixed-length domain the identity is prefix-free, as required of encodings by Section 4.1.¶

When the length is not fixed in advance, SerializeBytes MUST NOT be used. Instead, when the length is below 2^32 bytes, a prefix-free serialization is given by¶

SerializeVarLenString(s)

Input: s, an N-byte string

Output: out, an (N+4)-byte string

1. return LE(len(s), 4) || s

6.1.2. Sequences and tuples

A fixed-length array or a tuple is serialized as the concatenation of the serializations of its elements, with no separators.¶

6.1.3. Unsigned integers

An integer modulo M is represented by its unique integer representative in the range [0, M) and serialized via LE.¶

SerializeUint(x, M)

Inputs:

- x, an integer modulo M
- M, the order of the integer ring

Output: out, an Ns-byte string

1. assert 0 <= x < M
2. return LE(x, Ns)

where Ns is the smallest integer with 256^Ns >= M.¶

6.1.4. Field elements

This section specifies the default serialization of a finite field of order q = p^m, where p is the prime characteristic and m >= 1 is the extension degree.¶

The choice of field serialization MUST be reflected in the session tag (see Section 5.1). The field serialization function MUST be prefix-free, and the matching deserialization MUST reject invalid encodings (for instance, reject elements larger than or equal to p). The default is the serialization specified below, which encodes each prime-field coordinate as a fixed-width little-endian integer via SerializeUint (Section 6.1.3); if the standard the application builds on already fixes a canonical serialization for the field, that serialization SHOULD be pinned in place of the default.¶

For example, Curve25519 [RFC7748], Ed25519 [RFC8032], ristretto255 Section 4.4 of [RFC9496] serialize field elements as a fixed-width little-endian integer, matching the default. Similarly, in Section 7.1 of [FIPS204], the integer coordinates of lattice vectors are serialized least-significant-byte first. Other standards instead fix a big-endian serialization, such as P-256 [SEC1] and BLS12-381 [I-D.irtf-cfrg-pairing-friendly-curves] via I2OSP.¶

With respect to a fixed basis, a field element is represented by its m coordinates in the prime field, each an integer in [0, p). It is serialized as the concatenation of the per-coordinate serializations produced by SerializeUint (Section 6.1.3) with modulus p. Let Ns be the smallest integer with 256^Ns >= p; a field element serializes to m * Ns bytes.¶

SerializeField(a, p, m)

Inputs:

- a, an element of the field of order p^m, given by its
     coordinates (a[0], ..., a[m-1]) over the prime field
- p, the prime characteristic of the field
- m, the extension degree

Output: out, an (m * Ns)-byte string

1. out := ""
2. for i in 0, ..., m-1:
3.    out := out || SerializeUint(a[i], p)
4. return out

Note that a prime field is the case m = 1, in which case SerializeField is equivalent to SerializeUint.¶

6.1.5. Elliptic curve group elements

A group element is serialized using the group's element-serialization function.¶

For many prime-order elliptic-curve groups, this is the compressed Elliptic-Curve-Point-to-Octet-String conversion of [SEC1]. All non-trivial group elements have exactly one Ne-byte representation. The value of Ne and the concrete conversion are fixed by the ciphersuite.¶

The [SEC1] serialization of the identity element (the single byte 0x00) SHOULD be rejected to facilitate deserialization (Section 6.2). The ristretto255 and decaf448 [RFC9496] identity encodings have a distinct, fixed-length Ne-byte encoding.¶

6.2. Deserialization

Deserialization of the NARG string consists of reading the prover messages: each message is read by consuming, from the front of the input, a byte string whose length is determined by its type and the instance. Each deserialization function below returns the decoded value together with the unread remainder of its input; the remainder is the input to the next read.¶

Verification MUST fail if any of the prover messages cannot be deserialized successfully. After the last expected prover message has been read, the verifier MUST verify that no bytes remain. Bytes that are never read will cause the proof to be malleable: an adversary will be able to maul a valid proof to obtain a second, distinct accepting proof for the same statement.¶

6.2.1. Byte strings

For an N-byte string whose length is known from the message's type and the instance, deserialization reads N bytes.¶

DeserializeBytes(input, N)

Inputs:

- input, the unread remainder of the NARG string
- N, the expected byte length

Output: an N-byte string, and the unread remainder of input

1. fail if len(input) < N
2. return (input[0 : N], input[N : len(input)])

DeserializeBytes is the inverse of SerializeBytes (Section 6.1.1), and consumes N bytes of the NARG string, and fails if fewer bytes remain.¶

A byte string whose length is not known in advance is deserialized by reading a 4-byte length N via LE2IP, then reading the next N bytes; this is the inverse of SerializeVarLenString (Section 6.1.1).¶

DeserializeVarLenString(input)

Input: input, the unread remainder of the NARG string

Output: an N-byte string, and the unread remainder of input

1. fail if len(input) < 4
2. N := LE2IP(input[0 : 4])
3. fail if len(input) - 4 < N
4. return (input[4 : 4 + N], input[4 + N : len(input)])

This consumes 4 + N bytes of the NARG string, and fails if fewer bytes remain. The value decoded is the resulting byte string.¶

6.2.2. Sequences and tuples

Deserialize each element in order. Fail if any element fails to deserialize. The number and types of the elements are fixed by the protocol.¶

6.2.3. Unsigned integers

Read the next Ns bytes, with Ns as in Section 6.1.3, and interpret them as a little-endian integer x = LE2IP(.). If x >= M, fail: non-canonical integer encodings MUST be rejected. The value returned is x. This is the inverse of SerializeUint (Section 6.1.3).¶

DeserializeUint(input, M)

Inputs:

- input, the unread remainder of the NARG string
- M, the modulus

Output: x, an integer in the range [0, M), and the unread
        remainder of input

1. fail if len(input) < Ns
2. x := LE2IP(input[0 : Ns])
3. fail if x >= M
4. return (x, input[Ns : len(input)])

This consumes Ns bytes of the NARG string. It fails if fewer bytes remain, or if the integer read is not in the range [0, M).¶

6.2.4. Field elements

A field element of a field of order p^m is deserialized coordinate by coordinate: read m * Ns bytes, with Ns as in Section 6.1.4, and deserialize each Ns-byte coordinate as an integer modulo p using the unsigned-integer deserialization above. A prime field is the case m = 1. This is the inverse of SerializeField (Section 6.1.4).¶

DeserializeField(input, p, m)

Inputs:

- input, the unread remainder of the NARG string
- p, the prime characteristic of the field
- m, the extension degree

Output: a, an element of the field of order p^m, given by its
        coordinates (a[0], ..., a[m-1]) over the prime field, and
        the unread remainder of input

1. for i in 0, ..., m-1:
2.    (a[i], input) := DeserializeUint(input, p)
3. return ((a[0], ..., a[m-1]), input)

This consumes m * Ns bytes of the NARG string, and fails if fewer bytes remain or if any coordinate is non-canonical.¶

The deserialization MUST match the pinned serialization (Section 6.1.4): where the profile pins a standard's own serialization, that standard's deserialization governs.¶

6.2.5. Elliptic-curve group elements

Read the next Ne bytes and convert them to a group element using the group's element-deserialization function. Deserialization MUST perform the ciphersuite's input-validation steps, SHOULD reject the identity element (Section 6.1.5), and fail unless the input is the canonical encoding of a valid group element.¶

Note that for elliptic curves defined in [SEC1], decoding is the Octet-String-to-Elliptic-Curve-Point conversion, which checks that the encoding is well-formed and that the point lies on the curve, and returns "invalid" otherwise. The single-byte 0x00 encoding of the identity is not a valid Ne-byte input and SHOULD be rejected.¶

8. Security considerations

This section contains additional security considerations about the Fiat-Shamir transformation.¶

8.1. Codecs

Encoding maps are inverted only in the security analysis (by the knowledge extractor), never by the prover or verifier. The proof relies on a left inverse existing and being efficiently computable, which the knowledge-soundness extractor uses to recover prover messages from the absorbed bytes [CO25].¶

Decoding preserves the uniform distribution only when its input is uniform. Verifier messages SHOULD therefore be derived from Squeeze output and never from prover-controlled, or non-uniform bytes: decoding a non-uniform input yields a verifier message that is distinguishable from uniform, which would break the public-coin property the transformation depends on.¶

8.2. Constant-time requirements

While the protocol operates on "public coins", the instance can contain private information, such as verification keys not meant to be shared, or messages meant to be private between prover and verifier. Therefore, constant-time implementation of all the functions in this document is RECOMMENDED, to avoid leaking information via side channels.¶

For example, in the case of keyed-verification anonymous credentials, the non-interactive argument verifier will compute an instance that depends on the issuer's secret key and therefore the instance is not meant to be public.¶

8.3. Session identifiers

The purpose of session identifiers is to ensure composability and mitigate protocol confusion.¶

A session identifier uniquely identifies one session of a protocol, so that messages and state belonging to concurrent applications or proof systems are not confused. It MAY be reused, and reuse is expected whenever several proofs share the same application context: the identifier names that context, and identical contexts are meant to share one. Applications requiring proofs to be unique, non-replayable, or fresh can achieve this by adding, for example, a counter or timestamp to the session identifier.¶

8.4. Security of the transformation

The Fiat-Shamir transformation carries over the soundness and zero-knowledge properties of the interactive proof. The random oracle instantiation MUST be extraction-friendly and simulation-friendly indifferentiable to preserve soundness and zero-knowledge of the transformation. Both properties are stronger than indifferentiability alone [CO25].¶

Completeness of the non-interactive argument is preserved: if the statement being proven is true, then the resulting non-interactive argument string is valid.¶

8.4.1. Knowledge soundness

If the interactive proof is state-restoration knowledge sound, then so is the non-interactive proof. In particular, valid proofs cannot be generated without the corresponding statement being true (in the random oracle model).¶

Knowledge soundness carries over to the non-interactive argument, with a loss error quadratic in the number of queries the adversary makes to the random oracle [CO25].¶

8.4.2. Zero-Knowledge

If the interactive proof is honest-verifier zero-knowledge, then so is the non-interactive proof. In particular, the resulting argument string does not reveal any information beyond what can be directly inferred from the statement being valid.¶

The additive zero-knowledge loss introduced by the transformation is linear in the number of queries the adversary makes to the random oracle [CO25].¶

Zero-knowledge holds only when the prover's random number generator is indistinguishable from fresh uniform randomness to any party that does not know the witness, as noted in Section 1. Reusing the same randomness (or correlated randomness) across two distinct proofs will compromise zero-knowledge: for example, two Schnorr proofs sharing the same commitment nonce reveal the witness. This can be obtained by relying on a cryptographically secure random number generator meeting the requirements of [RFC4086] (for example, the operating system's getrandom(2) interface), or by deriving them with a pseudorandom function.¶

8.4.3. Quantum adversaries

If the interactive proof is state-restoration sound against quantum adversaries, then the non-interactive proof after the Fiat-Shamir transformation in the random oracle model is also secure against quantum adversaries.¶

The loss introduced by a quantum adversary is polynomial (larger than quadratic) in the number of quantum random-oracle queries.¶

8.5. Instance encoding

Incorrect encoding of the instance has historically led to a number of critical security vulnerabilities, often grouped under the term weak Fiat-Shamir transformation. In each of them, the cryptographic hash function was not provided the full statement being proven. A malicious prover can then compute the verifier message first, and choose the omitted part of the instance afterwards so that the verification equation is satisfied on a statement whose witness it does not hold.¶

As an example [BPW16], Chaum-Pedersen proof of equality for an instance (G, H, X, Y) proves knowledge of a witness x such that X = x * G and Y = x * H. The prover sends commitments (A, B), obtains a challenge c, and replies with a scalar f. The verifier accepts if the verification equations hold: f * G == A + c * X and f * H == B + c * Y. Suppose the challenge c is derived only by absorbing (A, B) and omitting the instance (G, H, X, Y). A malicious prover can pick A, B, H, and f at random, derive c, and then set X and Y to satisfy the verification equations. Verification passes, yet no single x satisfies both X = x * G and Y = x * H. A false statement has been proven. Other examples are available in [DMWG23] [CVE-2022-29566].¶

The transformation does not itself validate the instance. The verifier MUST therefore validate the syntax of the instance before use, exactly as it validates prover messages during deserialization (Section 6.2): for example, checking that each claimed group element lies in the prime-order group, and that integers are in canonical range.¶

Completeness and zero-knowledge are guaranteed only for valid instances: if the prover is invoked on an instance-witness pair outside the relation, no guarantee is provided on its output.¶

8.6. Implementation guidance

The Fiat-Shamir transformation has historically led to a number of critical security vulnerabilities.¶

Some incorrect implementations involve out-of-order (or missing) prover messages [CVE-2024-45039] [CVE-2026-46654]. Absorbing a prover message and serializing it to (or reading it from) the NARG string should be performed within the same function call, to ensure that prover messages are both hashed and serialized, and to prevent them from being skipped or reordered. A byte-level interface, as described in this document, is advisable in place of proof data structures whose fields are randomly addressable. A sequential interface, by contrast, enforces in-order processing. An end-of-input check is necessary to prevent malleability.¶

Test vectors can help confirm that honestly-generated proofs verify, but such tests exercise only completeness. Negative testing will help exercise the rejection paths too. Some such examples are: tampering with a valid NARG string to cause verification to fail, by flipping, appending, or prepending bytes, and by replacing each prover message in turn with a different value.¶

The NARG string must be treated as untrusted input. Therefore, non-interactive verifiers MUST check that length indicators are correct, that integers fall within their expected range, and that the proof length is correct. For example, in Section 6.2.1 the 4-byte length prefix read by LE2IP in DeserializeVarLenString is attacker-controlled, and can be as large as 2^32 - 1, so computing 4 + N can overflow 32-bit integers. As another example, a crafted length indicator can make verification checks trivial, or exhaust memory on deserialization before any cryptographic check runs [GNARK-OOM].¶

Table 1: Duplex sponge suites
Identifier	Primitive	Alphabet
`SHAKE128`	SHAKE128 [SHA3]	bytes
`TurboSHAKE128`	TurboSHAKE128 [RFC9861]	bytes

Appendix A. Example protocol: sumcheck

This appendix describes the Fiat-Shamir transformation for the sumcheck protocol. This protocol is not meant for standalone use; it is a toy example where the verifier's final check would still require the evaluation y = f(r[1], ..., r[v]), which is normally obtained via a polynomial commitment scheme opening.¶

The protocol is parameterized by a prime p and a number of variables v. The witness is the table w of the 2^v evaluations of a multilinear polynomial f on the hypercube: entry w[j] is f(j_0, ..., j_{v-1}), where j_0 is the least-significant bit of j. The instance is (v, S): the number of variables and the claimed sum S of all table entries. The application context is bound through the session identifier (Section 5.1). All field arithmetic below is modulo p.¶

In each round the prover message is the coefficient pair (a0, a1) of the round polynomial g(X) = a0 + a1 * X of the lowest unbound variable: g(0) and g(1) are the even- and odd-indexed half-sums of the table. Each verifier message is one field element, decoded from Ns squeezed bytes as LE2IP(Squeeze(Ns)) mod p. (For the Mersenne31 instantiation below, the bias of the reduction is approximately 2^-31, less than the soundness error of the interactive argument.)¶

SumcheckProve(session_id, v, w)

Inputs:

- session_id, a 32-byte string
- v, the number of variables
- w, a table of 2^v field elements

Outputs: narg_string; y, the final folded evaluation
f(r[1], ..., r[v])

 1. S := w[0] + w[1] + ... + w[2^v - 1]
 2. state := Init(session_id)
 3. state.Absorb(SerializeUint(v, 2^32) || SerializeField(S, p, 1))
 4. narg_string := ""
 5. for i in 1, ..., v:
 6.    a0 := w[0] + w[2] + ... + w[len(w) - 2]
 7.    a1 := (w[1] + w[3] + ... + w[len(w) - 1]) - a0
 8.    msg := SerializeField((a0, a1), p, 2)
 9.    state.Absorb(msg)
10.    narg_string := narg_string || msg
11.    r := LE2IP(state.Squeeze(Ns)) mod p
12.    w := (w[0] + r * (w[1] - w[0]), w[2] + r * (w[3] - w[2]), ...)
13. return (narg_string, w[0])

After v rounds the single remaining entry w[0] is f(r[1], ..., r[v]), where r[i] is the challenge of round i. The NARG string is the concatenation of the round messages.¶

SumcheckVerify(session_id, v, S, narg_string, y)

Inputs:

- session_id, v, as in SumcheckProve
- S, the claimed sum
- narg_string, the NARG string
- y, the evaluation f(r[1], ..., r[v]), supplied by the caller

Output: accept or reject

 1. state := Init(session_id)
 2. state.Absorb(SerializeUint(v, 2^32) || SerializeField(S, p, 1))
 3. for i in 1, ..., v:
 4.    ((a0, a1), narg_string) := DeserializeField(narg_string, p, 2)
 5.    fail if 2 * a0 + a1 != S
 6.    state.Absorb(SerializeField((a0, a1), p, 2))
 7.    r := LE2IP(state.Squeeze(Ns)) mod p
 8.    S := a0 + a1 * r
 9. fail if narg_string != ""
10. fail if S != y
11. return accept

The test vectors instantiate p = 2^31 - 1, v = 4, and the witness w = (1, 2, 4, ..., 2^15), giving S = 65535. The vectors report the NARG string as Narg and f(r[1], ..., r[v]) as FinalEvaluation.¶

Appendix B. Test Vectors

Each test vector is a block of lines of the form Key = Value, and no key repeats within a vector. A value is either an integer, written in decimal or in hexadecimal with the prefix 0x, or a byte string, written in lowercase hexadecimal. The empty byte string is written "".¶

Two rules govern how a value is laid out, and they are the whole grammar:¶

A value is written inline after Key = when it fits the document width. Otherwise it is written on the lines that follow, indented by two spaces, and the value is the concatenation of those lines with no separator. Byte strings wrap at a whole 32 bytes per line, so that a 32-byte value occupies exactly one line and a 64-byte value exactly two.¶
A sequence-valued field always uses the indented form, one item per line, each item introduced by - . An item too long for a line is itself wrapped, and its continuation lines carry a further two spaces of indentation.¶

So an indented line beginning - starts a new item, and any other indented line continues the value above it.¶

Every vector carries Id, a stable name of the form fiat-shamir/<suite>/<vector> by which this document and a test harness refer to it, and Function, the operation the remaining keys describe. The hash suite is identified with key Hash (Section 9). A vector carrying Expected = reject indicates a negative test; a vector with no Expected is a functional test, whose expectation is the output value it carries. The key ByteOrder marks the vectors exercising a non-default serialization (Section 6.1.4).¶

A machine-readable (JSON) copy of every vector below is part of this specification's repo. The two carry the same records: because no key repeats and sequences are explicit, each ~~~ block corresponds to one JSON object, key for key.¶

B.1. Codec test vectors

This section contains vectors for the encoding, decoding, serialization, and deserialization functions.¶

B.1.1. Byte-string serialization: `SerializeVarLenString`

Id = fiat-shamir/codec/serialize_varlen
Function = SerializeVarLenString
Input = 70726f6f66
Output = 0500000070726f6f66

B.1.2. `SerializeUint`: unsigned-integer serialization.

Id = fiat-shamir/codec/serialize_uint
Function = SerializeUint
Modulus =
  0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  43
Value = 0xdeadbeef
Output =
  efbeadde00000000000000000000000000000000000000000000000000000000

B.1.3. `DeserializeField`, used to deserialize a degree-2 element of the field of characteristic `2^256 - 189` (the default, little-endian serialization).

Id = fiat-shamir/codec/deserialize_field
Function = DeserializeField
Modulus =
  0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  43
ExtensionDegree = 2
Input =
  efbeadde00000000000000000000000000000000000000000000000000000000
  42ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
Coordinates =
  - 0xdeadbeef
  - 0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff42

B.1.4. The empty byte string may be encoded as a variable-length string.

Id = fiat-shamir/codec/varlen_empty
Function = SerializeVarLenString
Input = ""
Output = 00000000

B.1.5. Decoding is infallible and distribution-preserving

Id = fiat-shamir/codec/decode_uint_wraparound
Function = DecodeUint
Modulus =
  0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc6325
  51
Input =
  512563fcc2cab9f3849e17a7adfae6bcffffffffffffffff00000000ffffffff
  00000000000000000000000000000000
Challenge = 0x00

B.1.6. Field serialization of the P-256 scalar field happens via `I2OSP`.

Id = fiat-shamir/codec/serialize_field_be
Function = SerializeField
ByteOrder = big-endian
Modulus =
  0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc6325
  51
Value = 0xdeadbeef
Output =
  00000000000000000000000000000000000000000000000000000000deadbeef

B.1.7. The modulus itself is not accepted as a valid serialization.

Id = fiat-shamir/codec/deserialize_uint_reject_modulus
Function = DeserializeUint
Modulus =
  0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  43
Input =
  43ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
Expected = reject

B.1.8. Deserialization fails for inputs shorter than `Ns` bytes

Id = fiat-shamir/codec/deserialize_uint_reject_short
Function = DeserializeUint
Modulus =
  0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  43
Input =
  efbeadde000000000000000000000000000000000000000000000000000000
Expected = reject

B.1.9. When deserializing field extension elements, all coordinates must be validated

Id = fiat-shamir/codec/deserialize_field_reject_second_coordinate
Function = DeserializeField
Modulus =
  0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  43
ExtensionDegree = 2
Input =
  42ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
Expected = reject

B.1.10. A payload one byte shorter than its length prefix is rejected.

Id = fiat-shamir/codec/deserialize_varlen_reject_truncated
Function = DeserializeVarLenString
Input = 0500000070726f6f
Expected = reject

B.1.11. The maximal length prefix 2^32 - 1 is rejected.

Id = fiat-shamir/codec/deserialize_varlen_reject_overflow
Function = DeserializeVarLenString
Input = ffffffffdeadbeef
Expected = reject

B.1.12. The example protocol (Appendix A), where the first prover message is an invalid serialization (`p`, the modulus, is added to the canonical encoding)

Id = fiat-shamir/codec/sumcheck_reject_noncanonical_coefficient
Function = Sumcheck
Modulus = 0x7fffffff
NumVariables = 4
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
ClaimedSum = 0xffff
Narg =
  5455008055550000b8eefc2728ccf677b7aabd44c1001d074205d5576c3d307d
Expected = reject

B.1.13. An invalid NARG string for the example protocol (Appendix A), where a prover message does not satisfy verification

Id = fiat-shamir/codec/sumcheck_reject_round_identity
Function = Sumcheck
Modulus = 0x7fffffff
NumVariables = 4
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
ClaimedSum = 0xffff
Narg =
  5655000055550000b8eefc2728ccf677b7aabd44c1001d074205d5576c3d307d
Expected = reject

B.2. SHAKE128 test vectors

This section contains vectors for the XOF duplex sponge instantiated with the SHAKE128 suite (Section 9.1).¶

B.2.1. Squeeze a 32-byte string after initialization

Id = fiat-shamir/shake128/init_squeeze
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - squeeze 32
Output =
  63e1b3543377fab6fb8cf0f7698a9980ca0211d5bc4aba213dd7a6ef7dd63cfa

B.2.2. Absorb the byte string `hello world`, then squeeze 64 bytes

Id = fiat-shamir/shake128/absorb_squeeze
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 68656c6c6f20776f726c64
  - squeeze 64
Output =
  f627ff348dfee50d2aa5918a2621a0c1daf74c7ef930d49b5ea6eae73455e8c7
  56d433cbde0ade711bdd55d7ed5de38bb9adea8b2eec4402a0df090c16371413

B.2.3. Absorb is associative: `Absorb("abc")` is equivalent to `Absorb("ab"); Absorb("c")`

Id = fiat-shamir/shake128/absorb_split
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 6162
  - absorb 63
  - squeeze 32
Output =
  a629c32a309dda7605798fd07ce20ab14c76635446868eb46e20b6dfd1dd9e41

B.2.4. Squeeze is associative: `Squeeze(16 + 16)` is equivalent to `Squeeze(16) || Squeeze(16)`

Id = fiat-shamir/shake128/stream
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 616263
  - squeeze 16
  - squeeze 16
Output =
  a629c32a309dda7605798fd07ce20ab14c76635446868eb46e20b6dfd1dd9e41

B.2.5. Absorb of the empty string is a no-op

Id = fiat-shamir/shake128/empty_absorb
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 616263
  - squeeze 32
  - absorb ""
  - squeeze 32
Output =
  a629c32a309dda7605798fd07ce20ab14c76635446868eb46e20b6dfd1dd9e41
  d88e36c20e053248b90967a90051ba319688a10783c2ce174602eccc02e8d1a6

B.2.6. Absorb and squeeze can be interleaved

Id = fiat-shamir/shake128/interleave
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 00010203040506070809
  - squeeze 16
  - absorb 6d6f72652064617461
  - squeeze 16
Output =
  2da3c7e3a65c6e92901e8b668c43917eb9f02e9988e66d5ce2fbd833a0ecb93e

B.2.7. Absorbing a byte string of length longer than the rate.

Id = fiat-shamir/shake128/multiblock
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb ababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    bababababababababababababababababababababababababab
  - squeeze 600
Output =
  d0dc63443a117b76b09845af3347a6dbc29d0ff381ad093e3cfea3a326abbd0d
  a81bfc7dd6220f785900a6d04d508439cc65107c8eb75909f277a6f2740ae55f
  c684b851b66662c22252b6bb8028b5f7402b0beafb391835613a6c3d8116323d
  bdcb4494a198ae886821fca3d2af345227ea06c5c2cdb131c90d3fe58eedf090
  a55bb5a8edc614ab99da6c4ed8fe95a6c289c18dc61918a9abaa4f3ed2358711
  7ced29b33bfb87351d4ffc562add96d384fffbcb7cfb4d4d2125bf809cb85b33
  a3f9b7541c5d3d3f435f7d0a837f92f6878276ca3c833ecc1691f923602e9b8c
  8adb9528d8857d7189384eabbab50f0706b82b53db1c92857c2aa84a3527ce4b
  fcfbdbe02ad953b8517c4b91d36b45f81df67e10e4e9a7c7c064aa9e7f593710
  10eab4fd71c7aebcf00a793e469a78c658dd9f2c1d5ed2e3110939c11e916c1f
  51c47553b1bbceeea92649c9bcc7e5538dab18ca95c298b540b6798c065cd2b4
  13fe3915534a5dc6e7100e012b8c53fccc1018cc24570ca09c8e1c8f6e4e523d
  db7b6dcf313b6e98bb4abc94b8063eac8fdbbce945c35dd021a91e8227aeb165
  02a5a6e9e1d85fbd13b6e1e523e8a24040bbb9ddab5e29315780a57d9ef0d758
  b66e0076704f456dec1fc11577fe3644e53ff70a99a912758c289f225ad64e24
  6af9d895d91e6972b18421ccaa2aed6f843870b9dda07d1e975e51c04d58e7ba
  d60c0b8934fec80d4468816793879bc34d831e2689b77525439837fd15f797ed
  3b78ccdfdec6a5574111de3e243464c77c1d7fa76fb170e3722315e70fbf855a
  33281e8b6c15029202d0bb34749b962cc314515c748b74f4

B.2.8. Squeeze at the rate boundary

Id = fiat-shamir/shake128/rate_block
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1
    e1f202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f4
    04142434445464748494a4b4c4d4e4f505152535455565758595a5b5c5d5e5f60616
    2636465666768696a6b6c6d6e6f707172737475767778797a7b7c7d7e7f808182838
    485868788898a8b8c8d8e8f909192939495969798999a9b9c9d9e9fa0a1a2a3a4a5a
    6a7
  - squeeze 167
  - squeeze 2
Output =
  edae25852c909e4acea18d96ddd407e475eeaa7070ff591b49450c7a3ed21b7d
  0bd0ee62ab0c242e636c435b37c38ae6804a179ff434bed773c8d596cd66b928
  b0429247b19cbfc246bb1abd3b741841b21ad0234ba7738abe64ab93914bf5ad
  58a362d86f64d72b8f8603a888421a29769bb77579185409013a271ac258cd71
  a71aedf2801ba6eb4784636e9bfacca229a78aa8dc72af770380a1a981120b37
  16595564c520292578

B.2.9. A zero-length squeeze between absorbs is a no-op

Id = fiat-shamir/shake128/squeeze_zero
Function = DuplexSponge
Hash = SHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 616263
  - squeeze 0
  - absorb 646566
  - squeeze 32
Output =
  fc876a5ffbdc960106af16ca50e3b17b14a172f985f3a6f5df09c9a649ebf588

B.2.10. Derive a session identifier from an application tag

Id = fiat-shamir/shake128/derive_sid
Function = DeriveSessionID
Hash = SHAKE128
Tag = 696e7465726f702d746573742d763030
Output =
  b508aca89eecac56cd33e4a28f817f43f849d035922f354173ae8466628308cf

B.2.11. Squeeze and reduce a P-256 scalar challenge (`DecodeUint`)

Id = fiat-shamir/shake128/decode_uint
Function = DecodeUint
Hash = SHAKE128
Modulus =
  0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc6325
  51
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 08000000696e7374616e6365
  - squeeze 48
Output =
  7124d02b7cdfec99c4033dfd05624cfe2ff3af2c0e71656f770e676bd36de622
  8f85fcb39f34f7bfc24c9f54ab35ddba
Challenge =
  0xf860997c65f8dabecbcc3459a7b89bf69301b19fa1a0e036eb0d132724436d
  4f

B.2.12. The sumcheck protocol example (Appendix A) over Mersenne31.

Id = fiat-shamir/shake128/sumcheck
Function = Sumcheck
Hash = SHAKE128
Modulus = 0x7fffffff
NumVariables = 4
Tag = 73756d636865636b
SessionId =
  0568cefdf774622a3854d82934915fb3e38bc89dc44b6d673fc91b972c886fc2
Witness =
  - 1
  - 2
  - 4
  - 8
  - 16
  - 32
  - 64
  - 128
  - 256
  - 512
  - 1024
  - 2048
  - 4096
  - 8192
  - 16384
  - 32768
ClaimedSum = 0xffff
Narg =
  555500005555000023e362696ba9283c90a3362a74953379afc3b041d3eb126f
FinalEvaluation = 0x3ebfb3b3

B.2.13. A NARG string with trailing bytes is rejected

Id = fiat-shamir/shake128/sumcheck_reject_trailing_bytes
Function = Sumcheck
Hash = SHAKE128
Modulus = 0x7fffffff
NumVariables = 4
Tag = 73756d636865636b
SessionId =
  0568cefdf774622a3854d82934915fb3e38bc89dc44b6d673fc91b972c886fc2
ClaimedSum = 0xffff
Narg =
  555500005555000023e362696ba9283c90a3362a74953379afc3b041d3eb126f
  00
Expected = reject

B.3. TurboSHAKE128 test vectors

This section contains vectors for the XOF duplex sponge instantiated with the TurboSHAKE128 suite (Section 9.2).¶

B.3.1. Squeeze a 32-byte string after initialization

Id = fiat-shamir/turboshake128/init_squeeze
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - squeeze 32
Output =
  7ad8a3af35a3083c055e4a953ff001cdd9eeb1198f4be7a3a9ec5a209434619b

B.3.2. Absorb the byte string `hello world`, then squeeze 64 bytes

Id = fiat-shamir/turboshake128/absorb_squeeze
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 68656c6c6f20776f726c64
  - squeeze 64
Output =
  8b804d7a8c524c242a94e86f7ddfec329d90e29c1f584a98812e63029a0bdb07
  5b12c545bf9e2aa17c88673b6d9df4b08e728dc47f7d7094cee59a0d7d989634

B.3.3. Absorb is associative: `Absorb("abc")` is equivalent to `Absorb("ab"); Absorb("c")`

Id = fiat-shamir/turboshake128/absorb_split
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 6162
  - absorb 63
  - squeeze 32
Output =
  51acee1ee6f0c6a0c5a33b625ac9eaea54bc6b9b1cb85f9b2ef843e73631792e

B.3.4. Squeeze is associative: `Squeeze(16 + 16)` is equivalent to `Squeeze(16) || Squeeze(16)`

Id = fiat-shamir/turboshake128/stream
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 616263
  - squeeze 16
  - squeeze 16
Output =
  51acee1ee6f0c6a0c5a33b625ac9eaea54bc6b9b1cb85f9b2ef843e73631792e

B.3.5. Absorb of the empty string is a no-op

Id = fiat-shamir/turboshake128/empty_absorb
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 616263
  - squeeze 32
  - absorb ""
  - squeeze 32
Output =
  51acee1ee6f0c6a0c5a33b625ac9eaea54bc6b9b1cb85f9b2ef843e73631792e
  599e1dfb1bf60638046f82f5bfa28bcfcabf1404b200647d184ead03e51fbf01

B.3.6. Absorb and squeeze can be interleaved

Id = fiat-shamir/turboshake128/interleave
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 00010203040506070809
  - squeeze 16
  - absorb 6d6f72652064617461
  - squeeze 16
Output =
  f2745534347564bed146c95655122f14636bcc58f768296be8494208db29b6be

B.3.7. Absorbing a byte string of length longer than the rate.

Id = fiat-shamir/turboshake128/multiblock
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb ababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    babababababababababababababababababababababababababababababababababa
    bababababababababababababababababababababababababab
  - squeeze 600
Output =
  0ef6787d725638e19364c25a2ba879af55a4e1c97bff52919e64cd37a218db44
  a475f9d7ef0e88d87b443fe6590f12fc90c41c546c92cbeb3480e679d6e1ea37
  fc1e10e3e41b15e80aea4745ad9ad31c22857ae360fb66848daab7df8f6135ef
  cc940c4cc6431617cca6c05aa9b8bdf041538206ef912c0861dc07e219875509
  24c848cb788532637cf6e7fb1a3ea233fc335077e4910e8d7da15fe89250f284
  58ec17ecf448e05cf660ca4ffd1fdcb4379570d3a871aa759bb4d763d60d3e1b
  36ab8efcd8cd6fcca8fa67a5071ab21a1d31e4610f7ca8825b5e81f22433c495
  862b642c009823c312fd60fcdcd888af9c6c554da3a370946fa7bfa66ac54480
  b9a36d73fa6ee73ccb78b425de22814667742864608a06fa7534b053a68905ff
  532089b6fc0d597671ae4da685b96b1ac5d2513c0fc944d11155ed43461559a1
  2b984fb0cb45b105d9f2391a137d104f7da6c82fcfe375143bf512824685913f
  2bf61613b1a2a8f55f86d1282aa36c02384381335927259361c9e5875dbe1314
  af82aa65264ff009f525d4f0aedcf80cb908e308132113311d0e9a6783f27a13
  93ae28c10914018e263020dc97f219ebdae4118a79c318bef2d3766452075e35
  79f4b1cddbb80a5bca83a2f1fbec44e9d487925ba23cbfce111ba865e0fdb164
  589c66cbb757865d1bfb4540c01dece5b4180eef4262efc24c4ae3f76c4497a5
  3fb7f38a0de18a7053ab59b59b180ad53e3d2318661783682298d54b9f9ee6e3
  3b2be660295ccdd7d6cc976e21827c66880a0fbcc202743cd1ea0b7351755ee4
  b8a0b140bbc4f6a45eb6b05798721094cebe00e08f4f7bde

B.3.8. Squeeze at the rate boundary

Id = fiat-shamir/turboshake128/rate_block
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1
    e1f202122232425262728292a2b2c2d2e2f303132333435363738393a3b3c3d3e3f4
    04142434445464748494a4b4c4d4e4f505152535455565758595a5b5c5d5e5f60616
    2636465666768696a6b6c6d6e6f707172737475767778797a7b7c7d7e7f808182838
    485868788898a8b8c8d8e8f909192939495969798999a9b9c9d9e9fa0a1a2a3a4a5a
    6a7
  - squeeze 167
  - squeeze 2
Output =
  dc4b6f89697ec7e56ad210e6244a3ff25dab91ebd60981761db8f83db3a3a781
  ab43ffd7f325c98b912746b65d233fb5b99fd923cbff5e327b75436afd035c42
  ac9953ca4d686e30e5729aa460b813adf96c16917471679a4de36b19c452aef4
  47f93f4574ddcf09bdf6410774b426ffe415ff8eb2be44c8301ae071b534895b
  0ab66aa0136ffe9656c607bb6e1acaea4069454e297ec0ad4eab437c25455c88
  aeb77d6f33f7b578e9

B.3.9. A zero-length squeeze between absorbs is a no-op

Id = fiat-shamir/turboshake128/squeeze_zero
Function = DuplexSponge
Hash = TurboSHAKE128
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 616263
  - squeeze 0
  - absorb 646566
  - squeeze 32
Output =
  61d3fecc576c3faafb92db1cb22b60794075a024df9626436394c7b852ade899

B.3.10. Derive a session identifier from an application tag

Id = fiat-shamir/turboshake128/derive_sid
Function = DeriveSessionID
Hash = TurboSHAKE128
Tag = 696e7465726f702d746573742d763030
Output =
  4326208c9e56ae847be9356ca7c4447c752a9d7326a44a6cbee0c0dfc69505ac

B.3.11. Squeeze and reduce a P-256 scalar challenge (`DecodeUint`)

Id = fiat-shamir/turboshake128/decode_uint
Function = DecodeUint
Hash = TurboSHAKE128
Modulus =
  0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc6325
  51
SessionId =
  000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
Operations =
  - absorb 08000000696e7374616e6365
  - squeeze 48
Output =
  82a031e31b103ac01253ba3aae215f650a06a4bb24963a05ee1f8c0b82eae90d
  7bd36ea8cb560a91604ae8a97eb0d564
Challenge =
  0xc2088b455016d0126fcdd76335a79566e7fd8379db1de019871d459bfee955
  8b

B.3.12. The sumcheck protocol example (Appendix A) over Mersenne31.

Id = fiat-shamir/turboshake128/sumcheck
Function = Sumcheck
Hash = TurboSHAKE128
Modulus = 0x7fffffff
NumVariables = 4
Tag = 73756d636865636b
SessionId =
  abcbcae1f2f90d02b7e6417dbb2ffe162ab00477453eac3ce83d4e7e61000280
Witness =
  - 1
  - 2
  - 4
  - 8
  - 16
  - 32
  - 64
  - 128
  - 256
  - 512
  - 1024
  - 2048
  - 4096
  - 8192
  - 16384
  - 32768
ClaimedSum = 0xffff
Narg =
  55550000555500006ff9a71d4decf758430dfb69f9c6b5359d8ab2744b13d83d
FinalEvaluation = 0x654028db

B.3.13. A NARG string with trailing bytes is rejected

Id = fiat-shamir/turboshake128/sumcheck_reject_trailing_bytes
Function = Sumcheck
Hash = TurboSHAKE128
Modulus = 0x7fffffff
NumVariables = 4
Tag = 73756d636865636b
SessionId =
  abcbcae1f2f90d02b7e6417dbb2ffe162ab00477453eac3ce83d4e7e61000280
ClaimedSum = 0xffff
Narg =
  55550000555500006ff9a71d4decf758430dfb69f9c6b5359d8ab2744b13d83d
  00
Expected = reject