FN-DSA

FN-DSA is a new upcoming post-quantum signature scheme, currently being defined by NIST as part of their Post-Quantum Cryptography Standardization project. FN-DSA is based on the Falcon scheme.

WARNING: As this file is being written, no FN-DSA draft has been published yet, and therefore what is implemented here is not the "real" FN-DSA; such a thing does not exist yet. When FN-DSA gets published (presumably as a draft first, but ultimately as a "final" standard), this implementation will be adjusted accordingly. Correspondingly, it is expected that backward compatiblity will NOT be maintained, i.e. that keys and signatures obtained with this code may cease to be accepted by ulterior versions. Only version 1.0 will provide such stability, and it will be published only after publication of the final FN-DSA standard.

Sizes

FN-DSA (Falcon) nominally has two standard "degrees" n, equal to 512 and 1024, respectively. The implementation also supports "toy" versions with lower degrees 4 to 256 (always a power of two); these variants are meant for research and test purposes only. The API rejects use of such toy versions unless the caller asks for them explicitly. In the API, the degree is provided as parameter in a logarithmic scale, under the name logn, with the rule that n = 2^logn (hence, logn is equal to 9 for degree 512, 10 for degree 1024). Two relevant constants are defined, FN_DSA_LOGN_512 and FN_DSA_LOGN_1024, with values 9 and 10, respectively.

Sizes of signing (private) keys, verifying (public) keys, and signatures are as follows (depending on degree):

    logn    n     sign-key  vrfy-key  signature  security
   ------------------------------------------------------------------
      9    512      1281       897       666     level I (~128 bits)
     10   1024      2305      1793      1280     level V (~256 bits)

      2      4        13         8        47     none
      3      8        25        15        52     none
      4     16        49        29        63     none
      5     32        97        57        82     none
      6     64       177       113       122     none
      7    128       353       225       200     very weak
      8    256       641       449       356     presumed weak

Note that the sizes are fixed. Moreover, all keys and signatures use a canonical encoding which is enforced by the code, i.e. it should not be feasible to modify the encoding of an existing public key or a signature without changing its mathematical value.

Optimizations, Platforms and Features

The base implementation uses only integer computations, which are presumed safe, and constant-time to the extent that such things are possible (this should be considered a "best-effort" implementation, since recent LLVM versions have become pretty good at inferring that values are really Booleans in digsuise, and of course the same could apply to any JIT compilation layer, either hidden in-silicon, or as part of a virtual machine implementation, e.g. is using WASM). On some architectures, some optimizations are applied:

• x86 and x86_64: if SSE2 support is detected at compile-time, then SSE2 opcodes are used for floating-point computations in signature generation. Note that SSE2 is part of the ABI for x86_64, but is also enabled by default for x86.

• aarch64: on ARMv8 (64-bit), if NEON support is detected at compile-time, then NEON opcodes are used, for a result similar to what is done on x86 with SSE2. NEON is part of the ABI for aarch64.

• riscv64: on 64-bit RISC-V systems, we assume that the target implements the D extension (double-precision floating-point). Note that the compiler, by default, assumes RV64GC, i.e. that I, M, A, F, D and C are supported.

On top of that, on x86 and x86_64, a second version of the code is compiled and automatically used if a runtime test shows that the current CPU supports AVX2 (and it was not disabled by the operating system). AVX2 optimizations improve performance of keygen, signing, and verifying (the performance boost over SSE2 for signing is rather slight, but for keygen and verifying it almost halves the cost).

The following features, which are not enabled by default, can be used to modify the code generation:

• no_avx2: do not include the AVX2-optimized code. Using this option also removes the runtime test for CPU support of AVX2 opcodes; it can be used if compiled code footprint is an issue. SSE2 opcodes will still be used.

• div_emu: on riscv64, do not use the hardware implementation for floating-point divisions. This feature was included because some RISC-V cores, in particular the SiFive U74, have constant-time additions and multiplications, but division cost varies depending on the input operands.

• sqrt_emu: this is similar to div_emu but for the square root operation. On the SiFive U74, enabling both div_emu and sqrt_emu increases the cost of signature generation by about 25%.

• small_context: reduce the in-memory size of the signature generator context (SigningKey instance); for the largest degree (n = 1024), using small_context shrinks the context size from about 114 kB to about 82 kB, but it also increases signature cost by about 25%.

Of these options, only no_avx2 has any impact on keygen and verifying.

Performance

This implementation achieves performance similar to that obtained with C code. The key pair generation code is a translation of the ntrugen implementation. On x86 CPUs, AVX2 opcodes are used for better performance if the CPU is detected to support them (the non-AVX2 code is still included, so that the compiled binaries can still run correctly on non-AVX2 CPUs). On 64-bit x86 (x86_64) and ARMv8 (aarch64 and arm64ec) platforms, the native floating-point type (f64) is used in signature generation, because on such platforms the type maps to the hardware support which follows the correct strict IEEE-754 rounding rules; on other platforms (including 32-bit x86 and 32-bit ARM), an integer-only implementation is used, which emulates the expected IEEE-754 primitives. Key pair generation and signature verification use only integer operations.

On an Intel i5-8259U ("Coffee Lake", a Skylake variant), the following performance is achieved (in clock cycles):

    degree    keygen      sign     +sign    verify  +verify
   ---------------------------------------------------------
      512    11800000    840000    645000    75000    52000
     1024    45000000   1560000   1280000   151000   105000

+sign means generating a new signature on a new message but with the same signing key; this allows reusing some computations that depend on the key but not on the message. Similary, +verify is for verifying additional signatures relatively to the same key. We may note that this is about as fast as RSA-2048 for verification, but about 2.5x faster for signature generation, and many times faster for key pair generation.

On an ARM Cortex-A76 CPU (Broadcom BCM2712C1), performance is as follows:

    degree    keygen      sign     +sign    verify  +verify
   ---------------------------------------------------------
      512    21500000   1070000    747000   145000   104000
     1024    80000000   2120000   1500000   298000   212000

These figures are very close to what can be achieved on the Intel Coffee Lake when compiling with no_avx2, i.e. the Cortex-A76 with NEON achieves about cycle-to-cycle parity with the Intel with SSE2, but the Intel can get an extra edge with AVX2. Some newer/larger ARM CPUs may implement the SVE or SVE2 opcodes, with extended register size, but they seem to be a rarity (apparently, even the Apple M1 to M4 CPUs stick to NEON and do not support SVE).

On a 64-bit RISC-V system (SiFive U74 core), which is much smaller/low-end than the two previous, the following is achieved:

    degree    keygen      sign     +sign    verify  +verify
   ---------------------------------------------------------
      512    55000000   4530000   3370000   387000   274000
     1024   198000000   9170000   7050000   801000   566000

To put things into perspective, FN-DSA/512 is substantially faster than RSA-2048 on all these systems (RSA is especially efficient for signature verification, and OpenSSL's implementation on x86 has been very optimized along the years, so that on the x86_64 RSA-2048 verification is about as fast as FN-DSA/512; for all other operations, FN-DSA/512 is faster, sometimes by a large amount, e.g. on the ARM Cortex-A76 signature generation with this code is about 8 times faster than OpenSSL's RSA-2048).

Specific Variant

In the original Falcon scheme, the signing process entails generation of a random 64-byte nonce, and that nonce is hashed together with the message to sign with SHAKE256; the output is then converted to a polynomial hm with rejection sampling:

    hm <- SHAKE256( nonce || message )

This mode is supported by the implementation (using the custom HASH_ID_ORIGINAL_FALCON hash identifier); this is an obsolescent feature and support of the original Falcon design is expected to be dropped at some point. For enhanced functionality (support of pre-hashed messages) and better security in edge cases, the implementation currently implements what is my best guess of how FN-DSA will be defined, using the existing ML-DSA (FIPS 204) as a template. The message is either "raw", or pre-hashed with a collision-resistant hash function. If the message is "raw" then the hm polynomial is obtained as:

    hm <- SHAKE256( nonce || hpk || 0x00 || len(ctx) || ctx || message )

where:

• hpk is the SHAKE256 hash (with a 64-byte output) of the encoded public key
• 0x00 is a single byte
• ctx is an arbitrary domain separation context string of up to 255 bytes in length
• len(ctx) is the length of ctx, encoded over a single byte

The message may also be pre-hashed with a hash function such as SHA3-256, in which case only the hash value is provided to the FN-DSA implementation, and hm is computed as follows:

    hm <- SHAKE256( nonce || hpk || 0x01 || len(ctx) || ctx || id || hv )

where:

• id is the DER-encoded ASN.1 OID that uniquely identifies the hash function used for pre-hashing the message
• hv is the pre-hashed message

Since SHAKE256 is a "good XOF", adding hpk and ctx to the input, with an unambiguous encoding scheme, cannot reduce security; therefore, the "raw message" variant as shown above is necessarily at least as secure as the original Falcon design. In the case of pre-hashing, this obviously adds the requirement that the pre-hash function must be collision resistant, but it is otherwise equally obviously safe. Note that ASN.1/DER encodings are self-terminated, thus there is no ambiguousness related to the concatenation of id and hv.

Adding hpk to the input makes FN-DSA achieve BUFF security, a property which is not necessarily useful in any given situation, but can be obtained here at very low cost (no change in size of either public keys or signatures, and only some moderate extra hashing). The hpk value is set to 64 bytes just like in ML-DSA.

As an additional variation: the Falcon signature generation works in a loop, because it may happen, with low probability, that either the sampled vector is not short enough, or that the final signature cannot be encoded within the target signature size (which is fixed). In either case, with the original Falcon, the process restarts but reuses the same nonce (hence the same hm value). In the variant implemented here (outside of the "original Falcon" mode), a new nonce is generated when such a restart happens. Though the original Falcon method is not known to be unsafe in any way, this nonce regeneration has been recently argued to make it much easier to prove some security properties of the scheme. Since restarts are rare, this nonce regeneration does not imply any noticeable performance hit. In any case, regenerating the nonce cannot harm security.

Usage

The code is split into five crates:

• fn-dsa is the toplevel crate; it re-exports all relevant types, constants and functions, and most applications will only need to use that crate. Internally, fn-dsa pulls the other four crates as dependencies.

• fn-dsa-kgen implements key pair generation.

• fn-dsa-sign implements signature generation.

• fn-dsa-vrfy implements signature verification.

• fn-dsa-comm provides some utility functions which are used by all three other crates.

The main point of this separation is that some applications will need only a subset of the features (typically, only verification) and may wish to depend only on the relevant crates, to avoid pulling the entire code as a dependency (especially since some of the unit tests in the key pair generation and signature generation can be somewhat expensive to run).

An example usage code looks as follows:

use rand_core::OsRng;
use fn_dsa::{
    sign_key_size, vrfy_key_size, signature_size, FN_DSA_LOGN_512,
    KeyPairGenerator, KeyPairGeneratorStandard,
    SigningKey, SigningKeyStandard,
    VerifyingKey, VerifyingKeyStandard,
    DOMAIN_NONE, HASH_ID_RAW,
};

// Generate key pair.
let mut kg = KeyPairGeneratorStandard::default();
let mut sign_key = [0u8; sign_key_size(FN_DSA_LOGN_512)];
let mut vrfy_key = [0u8; vrfy_key_size(FN_DSA_LOGN_512)];
kg.keygen(FN_DSA_LOGN_512, &mut OsRng, &mut sign_key, &mut vrfy_key);

// Sign a message with the signing key.
let mut sk = SigningKeyStandard::decode(&sign_key).or_else(...);
let mut sig = vec![0u8; signature_size(sk.get_logn())];
sk.sign(&mut OsRng, &DOMAIN_NONE, &HASH_ID_RAW, b"message", &mut sig);

// Verify a signature with the verifying key.
match VerifyingKeyStandard::decode(&vrfy_key) {
    Some(vk) => {
        if vk.verify(&sig, &DOMAIN_NONE, &HASH_ID_RAW, b"message") {
            // signature is valid
        } else {
            // signature is not valid
        }
    }
    _ => {
        // could not decode verifying key
    }
}