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IRTF Crypto Forum zero knowledge hash 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. About This Document The latest revision of this draft can be found at . Status information for this document may be found at . Discussion of this document takes place on the Crypto Forum Research Group mailing list (), which is archived at . Subscribe at . Source for this draft and an issue tracker can be found at .

Introduction The Fiat-Shamir transformation removes interaction from a public-coin interactive argument by relying on a cryptographic hash function. The non-interactive prover derives each verifier message on its own via a hash function, and serializes the protocol transcript into a non-interactive argument (NARG) string. The verifier recomputes the same verifier messages from the NARG string and checks the interactive verifier's decision. The resulting argument is secure in the random oracle model, where the hash function is treated as an ideal random function (see ). Many non-interactive arguments apply the Fiat-Shamir transformation to a public-coin interactive proof, yet each protocol re-specifies it from scratch, duplicating the security analysis and reopening the same bugs. It is notoriously easy to get the Fiat-Shamir transformation wrong, introducing critical security bugs , , , . This document specifies the duplex sponge Fiat-Shamir transformation, and in particular:

• a non-interactive argument prover (NARG prover), and
• a non-interactive argument verifier (NARG verifier)

The prover is a randomized procedure and generally relies on a cryptographically-secure entropy source; the verifier SHOULD be deterministic. Both the non-interactive prover and verifier rely on:

• a duplex sponge, prescribing how to interact with the cryptographic hash function ();
• a set of codecs, describing how each prover and verifier message talk to the duplex sponge ();
• a serialization and deserialization procedure for the NARG string produced by the prover ().

This transformation is also well-suited for recursive proving, since the in-circuit cost of recomputing the Fiat-Shamir challenges is low. It is compatible with arithmetization-friendly hash functions (e.g. Poseidon2 ) that operate natively on field elements. See for the general construction. Other types of non-interactive transformations (with and without random oracles) are possible, but outside the scope of this specification.

Non-interactive prover for the Fiat-Shamir transformation | Init | | | +------+ | | | | | v | | +------------+ +--------+ | | instance ---------------------------->| encode[0] |-->| Absorb | | | +-------------------+ +------------+ +--------+ | | | Interactive Prover| | | | |(instance, witness)| | | | | | prover_msg[1] v | | | | +------------+ +--------+ | | | +---------------->| encode[1] |-->| Absorb | | | | | +------------+ +--------+ | | | | | | | | | verifier_msg[1] v | | | | +------------+ +---------+ | | | |<----------------| decode[1] |<--| Squeeze | | | | | +------------+ +---------+ | | | | | | | | | prover_msg[2] v | | | | +------------+ +--------+ | | | +---------------->| encode[2] |-->| Absorb | | | | | +------------+ +--------+ | | | | | | | | | verifier_msg[2] v | | | | +------------+ +---------+ | | | |<----------------| decode[2] |<--| Squeeze | | | | | +------------+ +---------+ | | | | . . | | | | . . | | | | . . | | | | | | | | | prover_msg[k-1] v | | | | +------------+ +--------+ | | | +---------------->| encode[k-1]|-->| Absorb | | | | | +------------+ +--------+ | | | | | | | | | verifier_msg[k-1] v | | | | +------------+ +---------+ | | | |<----------------| decode[k-1]|<--| Squeeze | | | | | +------------+ +---------+ | | | | prover_msg[k] | | | +----------------> | | +-------------------+ | | | | narg_string := serialize(prover_msg[..]) | +--------------------------------------------------------------------+ ]]>

Non-interactive verifier for the Fiat-Shamir transformation | Init | | | +------+ | | | | | v | | +-----------+ +--------+ | | instance --->| encode[0] |--------->| Absorb | | | +-----------+ +--------+ | | | | | v | |prover_msg[1] +-----------+ +--------+ | |------------->| encode[1] |--------->| Absorb | | | +-----------+ +--------+ | | | | | v | | verifier_msg[1] +-----------+ +---------+ | | <---------------| decode[1] |<------| Squeeze | | | +-----------+ +---------+ | | | | | v | |prover_msg[2] +-----------+ +--------+ | |------------->| encode[2] |--------->| Absorb | | | +-----------+ +--------+ | | | | | v | | verifier_msg[2] +-----------+ +---------+ | | <---------------| decode[2] |<------| Squeeze | | | +-----------+ +---------+ | | . . | | . . | | . . | |prover_msg[k] +-----------+ +--------+ | |-------------->| encode[k] |------>| Absorb | | | +-----------+ +--------+ | | | | | v | | verifier_msg[k] +-----------+ +---------+ | | <-----------------| decode[k] |<--| Squeeze | | | +-----------+ +---------+ | | | | 3. Run the interactive verifier | | | | +--------------------------------------+ | | | Interactive Verifier | | | | (instance, prover_msg[..], | | | | verifier_msg[..]) | | | +--------------------------------------+ | +-------------------------------------------------------------------+ ]]>

Note that the prover does not need to compute the last verifier message verifier_msg[k]. The security guarantees provided by this transformation are described in .

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 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.

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

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 ), 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 , 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 , 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 and analyzed in .

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 ). 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 () 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.

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 () 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 ().
• Decoding converts the bytes squeezed from the duplex sponge into a uniformly-distributed verifier message ().

Serialization () 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 . 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 and ).

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

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 ; its role in composability and reuse is discussed in .

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 (). 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: 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 ), for efficiency (). The security properties required of the XOF, and the security level attained, are fixed by the suite (); the corresponding requirements on the construction are discussed in .

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

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

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.

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.

Encoding into byte strings The encoding of the instance and of each prover message is its serialization, as described in .

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.

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

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. 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 ).

A similar observation in the context of hashing to the elliptic curve field is available in .

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

Initialization Before any prover message is processed, both parties start the duplex sponge with the session identifier (), and then the instance (). Neither the session identifier nor the instance is part of the NARG string: the verifier holds both as its own inputs.

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:

1. 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.
2. 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.
3. 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 ).
4. the tag SHOULD begin with a fixed identification string that is unique to the application.
5. 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. Above, DS denotes the duplex sponge in use (), instantiated with one of the suites of . 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: 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 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: 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 IO pattern.

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 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 ). The encoding map encode[0] SHOULD reuse the serialization functions of . 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: A valid instance encoding function is: 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 A valid instance encoding function is: where enc is the group element-serialization function described in . 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 .

Non-interactive argument string

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 ().

Byte strings Serialization of an N-byte string is the identity function. 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 ). On such a fixed-length domain the identity is prefix-free, as required of encodings by . 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

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

Unsigned integers An integer modulo M is represented by its unique integer representative in the range [0, M) and serialized via LE. where Ns is the smallest integer with 256^Ns >= M.

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 ). 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 (); 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 , Ed25519 , ristretto255 serialize field elements as a fixed-width little-endian integer, matching the default. Similarly, in Section 7.1 of , the integer coordinates of lattice vectors are serialized least-significant-byte first. Other standards instead fix a big-endian serialization, such as P-256 and BLS12-381 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 () with modulus p. Let Ns be the smallest integer with 256^Ns >= p; a field element serializes to m * Ns bytes. Note that a prime field is the case m = 1, in which case SerializeField is equivalent to SerializeUint.

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

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.

Byte strings For an N-byte string whose length is known from the message's type and the instance, deserialization reads N bytes. DeserializeBytes is the inverse of SerializeBytes (), 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 (). This consumes 4 + N bytes of the NARG string, and fails if fewer bytes remain. The value decoded is the resulting byte string.

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.

Unsigned integers Read the next Ns bytes, with Ns as in , 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 (). = 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).

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 , 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 (). 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 (): where the profile pins a standard's own serialization, that standard's deserialization governs.

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 (), and fail unless the input is the canonical encoding of a valid group element. Note that for elliptic curves defined in , 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.

Efficiency considerations For both codecs and serialization, batch algorithms should be preferred when available, because they amortize per-element cost over a whole sequence. For example, the dominant cost in point compression is a modular inversion, and serializing a batch of compressed elliptic-curve points requires only one modular inversion for the entire batch (via Montgomery's trick) rather than one per point. (Note that deserialization does not batch in the same way, since point decompression requires a per-element square root.) Init(session_id) (see ) can be precomputed. Implementations can therefore start each prover and verifier execution from a copy of the duplex sponge state, instead of initializing it every time. In the XOF duplex sponge (), the padded session identifier fills exactly one rate block (), saving one invocation of the permutation function per execution. Similarly, DeriveSessionID can be precomputed when the session identifier is derived from a tag. The same observation extends to longer shared prefixes: proofs for the same instance can additionally start from a stored copy of the state obtained after absorbing encode[0](instance). XOF evaluations () without copying the XOF state (see the notation ctx.Copy() in the pseudocode) will yield identical bytes, but incur a cost quadratic in the number of rounds. Implementations SHOULD instead maintain the incremental duplex sponge state of .

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

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 . 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.

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.

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.

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

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 .

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 . 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 . 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 (for example, the operating system's getrandom(2) interface), or by deriving them with a pseudorandom function.

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.

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 , 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 . 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 (): 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.

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 . 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 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 .

Suites The suites defined by this document, and the identifiers used by the test vectors, are: Duplex sponge suites

Identifier	Primitive	Alphabet
SHAKE128	SHAKE128	bytes
TurboSHAKE128	TurboSHAKE128	bytes

The suite identifier is a natural component of the tag (), since it fixes the hash instantiation.

SHAKE128 In the SHA-3 family, two extendable-output functions (SHAKEs) are defined over the Keccak-f permutation: SHAKE128 and SHAKE256. A SHAKE is an eXtendable-Output Function (XOF) defined as SHAKE(M, n) where the output is an n-bit string. The corresponding collision and second-preimage-resistance for SHAKE128 are min(n/2,128) and min(n,128) bits, respectively (see Appendix A.1 of ). This instantiation targets 128-bit security. The SHAKE128 state is a 200-byte (1600-bit) string, split into a rate of R = 168 bytes and a capacity of 32 bytes (256 bits).

TurboSHAKE128 TurboSHAKE128 is an eXtendable-Output Function (XOF) built on Keccak-p[1600, 12], the Keccak-f[1600] permutation reduced to its last 12 rounds. Its state is a 200-byte (1600-bit) string, split into a rate of R = 168 bytes and a capacity of 32 bytes (256 bits). The corresponding collision and second-preimage-resistance are min(n/2,128) and min(n,128) bits for an n-bit output string, respectively. This instantiation targets 128-bit security. In this instantiation, every verifier message is the TurboSHAKE128 XOF evaluation TurboSHAKE128(M, D, L), where M is the concatenation of the session identifier, the encoded instance, and the encoded prover messages up to and including the current round, D (the domain-separation byte in the range 0x01 to 0x7F) is fixed to D = 0x1F, the default value, and L is the desired output length in bytes .

IANA Considerations This document has no IANA actions.

Acknowledgments The authors thank Thomas Pornin, Vishruti Ganesh, Brent Zundel, Hart Montgomery, Opal Wright, Giap Vu, David Wong, Théophile Wallez, and Thomas Coratger for their reviews and contributions to this specification.

References Normative References n.d. In many standards track documents several words are used to signify the requirements in the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements. RFC 2119 specifies common key words that may be used in protocol specifications. This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the defined special meanings. This document defines four eXtendable-Output Functions (XOFs), hash functions with output of arbitrary length, named TurboSHAKE128, TurboSHAKE256, KT128, and KT256. All four functions provide efficient and secure hashing primitives, and the last two are able to exploit the parallelism of the implementation in a scalable way. This document is a product of the Crypto Forum Research Group. It builds up on the definitions of the permutations and of the sponge construction in NIST FIPS 202 and is meant to serve as a stable reference and an implementation guide. Informative References National Institute of Standards and Technology (NIST) Trail of Bits Solana Foundation Consensys This document specifies a number of algorithms for encoding or hashing an arbitrary string to a point on an elliptic curve. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. This memo specifies two elliptic curves over prime fields that offer a high level of practical security in cryptographic applications, including Transport Layer Security (TLS). These curves are intended to operate at the ~128-bit and ~224-bit security level, respectively, and are generated deterministically based on a list of required properties. This document describes elliptic curve signature scheme Edwards-curve Digital Signature Algorithm (EdDSA). The algorithm is instantiated with recommended parameters for the edwards25519 and edwards448 curves. An example implementation and test vectors are provided. This memo specifies two prime-order groups, ristretto255 and decaf448, suitable for safely implementing higher-level and complex cryptographic protocols. The ristretto255 group can be implemented using Curve25519, allowing existing Curve25519 implementations to be reused and extended to provide a prime-order group. Likewise, the decaf448 group can be implemented using edwards448. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. GMO CONNECT Inc. GMO CONNECT Inc. Stanford University Pairing-based cryptography, a subfield of elliptic curve cryptography, has received attention due to its flexible and practical functionality. Pairings are special maps defined using elliptic curves and it can be applied to construct several cryptographic protocols such as identity-based encryption, attribute- based encryption, and so on. At CRYPTO 2016, Kim and Barbulescu proposed an efficient number field sieve algorithm named exTNFS for the discrete logarithm problem in a finite field. Several types of pairing-friendly curves such as Barreto-Naehrig curves are affected by the attack. In particular, a Barreto-Naehrig curve with a 254-bit characteristic was adopted by a lot of cryptographic libraries as a parameter of 128-bit security, however, it ensures no more than the 100-bit security level due to the effect of the attack. In this memo, we list the security levels of certain pairing-friendly curves, and motivate our choices of curves. First, we summarize the adoption status of pairing-friendly curves in standards, libraries and applications, and classify them in the 128-bit, 192-bit, and 256-bit security levels. Then, from the viewpoints of "security" and "widely used", we select the recommended pairing-friendly curves considering exTNFS. Security systems are built on strong cryptographic algorithms that foil pattern analysis attempts. However, the security of these systems is dependent on generating secret quantities for passwords, cryptographic keys, and similar quantities. The use of pseudo-random processes to generate secret quantities can result in pseudo-security. A sophisticated attacker may find it easier to reproduce the environment that produced the secret quantities and to search the resulting small set of possibilities than to locate the quantities in the whole of the potential number space. Choosing random quantities to foil a resourceful and motivated adversary is surprisingly difficult. This document points out many pitfalls in using poor entropy sources or traditional pseudo-random number generation techniques for generating such quantities. It recommends the use of truly random hardware techniques and shows that the existing hardware on many systems can be used for this purpose. It provides suggestions to ameliorate the problem when a hardware solution is not available, and it gives examples of how large such quantities need to be for some applications. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements.

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 (). 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.) 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. 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.

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:

1. 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.
2. 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 (). 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 (). 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.

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

Byte-string serialization: SerializeVarLenString

SerializeUint: unsigned-integer serialization.

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

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

Decoding is infallible and distribution-preserving

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

The modulus itself is not accepted as a valid serialization.

Deserialization fails for inputs shorter than Ns bytes

When deserializing field extension elements, all coordinates must be validated

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

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

The example protocol (), where the first prover message is an invalid serialization (p, the modulus, is added to the canonical encoding)

An invalid NARG string for the example protocol (), where a prover message does not satisfy verification

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

Squeeze a 32-byte string after initialization

Absorb the byte string hello world, then squeeze 64 bytes

Absorb is associative: Absorb("abc") is equivalent to Absorb("ab"); Absorb("c")

Squeeze is associative: Squeeze(16 + 16) is equivalent to Squeeze(16) || Squeeze(16)

Absorb of the empty string is a no-op

Absorb and squeeze can be interleaved

Absorbing a byte string of length longer than the rate.

Squeeze at the rate boundary

A zero-length squeeze between absorbs is a no-op

Derive a session identifier from an application tag

Squeeze and reduce a P-256 scalar challenge (DecodeUint)

The sumcheck protocol example () over Mersenne31.

A NARG string with trailing bytes is rejected

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

Squeeze a 32-byte string after initialization

Absorb the byte string hello world, then squeeze 64 bytes

Absorb is associative: Absorb("abc") is equivalent to Absorb("ab"); Absorb("c")

Squeeze is associative: Squeeze(16 + 16) is equivalent to Squeeze(16) || Squeeze(16)

Absorb of the empty string is a no-op

Absorb and squeeze can be interleaved

Absorbing a byte string of length longer than the rate.

Squeeze at the rate boundary

A zero-length squeeze between absorbs is a no-op

Derive a session identifier from an application tag

Squeeze and reduce a P-256 scalar challenge (DecodeUint)

The sumcheck protocol example () over Mersenne31.

A NARG string with trailing bytes is rejected