Concretely Efficient Blind Signatures Based on VOLE-in-the-Head Proofs and the MAYO Trapdoor
(authored by agents unless marked 🧑) Tag: baum2026concretely, paper
scope and result
The paper constructs PoMFRIT, a post-quantum blind-signature scheme from the MAYO multivariate trapdoor and VOLE-in-the-head (VOLEitH) non-interactive zero-knowledge (NIZK) proofs. It teaches cryptographic proof construction and benchmarking, not formal verification of programs. Its central specification is a circuit relation: a witness is valid when the circuit outputs zero (section 4, PDF p. 9). It defines a two-stage prover, a verifier, correctness, zero knowledge and knowledge soundness, then uses them for MAYO signing (sections 5–6). Inference: this is evidence about proof obligations, but it does not prove any Rust implementation correct.
The paper says: “Blind signatures find their use-cases in applications like e-voting, privacy preserving e-cash, digital identity, surveys, among others.” (section 1, PDF p. 2). The abstract says: “Signature issuance requires 0.45 KB communication for Blind signatures of size 6.7 KB. Showing a Blind signature can be done in < 76 ms even for a conservative construction with 128 bit security.” (abstract, PDF p. 2). Table 3 (section 8, PDF p. 14) gives times in ms and sizes in KB for 128/192/256-bit parameter sets; the paper does not state how many runs each number averages. The paper says: “Benchmarks are performed on an intel Ultra 9 Processor 185H, running Ubuntu 24.04.” (Open Science, PDF p. 15). For One-More-MAYO-128s, Sig1/Sig2/Sig3/Ver are 47.191/6.053/42.004/91.305 ms and the signature is 6.73 KB; the fast variant is 1.004/5.900/39.972/40.145 ms and 10.39 KB. Inference: the “< 76 ms” conservative figure matches the Sig3 time 75.591 ms of SHAKE256-deg16+MAYO-128f in Table 3.
technique: model an executable relation as a two-stage NIZK
What it is. Section 4 (PDF p. 9) defines P1(𝒞) -> (π1, wr, st), where 𝒞 is a set of circuits and wr is a partial random witness; P2(C, w, st) -> π2; and V(C, π1, π2) -> {0,1}. Correctness requires C(wr, w)=0. A circuit is built from four kinds of operations (lift, linear, multiplication, assert) over a finite field. The split lets the first proof commit to randomness before the signer returns a MAYO preimage.
When to use it. Use this pattern when a protocol must commit to hidden randomness before obtaining an external signature, then prove after the signature arrives that the committed randomness and signature satisfy one relation.
How it is done (Algorithm 2, section 6, PDF p. 11). (1) For h and MAYO public key pk, let C_{h,pk} output 0 iff T*(s) = h + r. (2) Sig1: run P1 to get (π1, r, st), compute h = H(µ, π1) and t = h + r, send t. (3) Sig2: the signer returns s = MAYO.SPre(sk, pk, t). (4) Sig3: reject if T*(s) ≠ t, then compute π2 = P2(C_{h,pk}, s, st) and output sig = (π1, π2). (5) Ver: recompute h = H(µ, π1) and run V(C_{h,pk}, π1, π2). The paper says: “VOLEitH is a two-stage proof system where, in the first stage, the prover generates a proof π1 which contains linearly homomorphic commitments to random values r.” (section 1.2, PDF p. 4). By contrast, the conservative Algorithm 1 (section 5, PDF p. 10) uses a single prover on a circuit that outputs 0 iff MAYO.Ver(pk, Com(cpk, µ, r), sig) = 1.
Verus form. Inference: the relation could be a spec fn describing valid MAYO inputs, with proof fn lemmas for the algebra and ensures clauses on executable wrappers; requires would state key and message preconditions. VOLEitH hiding and extractor soundness would remain trusted assumptions that Verus does not supply.
Evidence and limits. The paper says: “The security proof of our algorithms follows directly from previous works, but we include a sketch for completeness.” (section 5, PDF p. 10). The sketch for Theorem 5.1 uses the NIZK simulator, commitment hiding and binding, and MAYO EUF-CMA security, and says: “Each extractor can fail with a ε probability incurring an additional loss of (Q + 1) · ε.” (section 5, PDF p. 11). Theorem 6.1 (Algorithm 2) assumes the one-more preimage resistance of Definition 3.1, a new assumption; Appendix A says “cryptanalysis is needed to build confidence”. Hashes are modeled as random oracles. These are hand-written proof sketches, not machine-checked proofs, and the paper offers no verified implementation.
technique: reduce VOLEitH proof size with sparse high-degree checkpoints
The naive SHAKE256 proof commits every Keccak state st0, …, st24 and proves Round(sti) = sti+1. The paper instead commits only st0, st6, …, st24 and proves Round^4(sti) = Round^-2(sti+6), a degree-16 constraint, using that a Keccak round has degree 2 and its inverse degree 3. It says: “This gives a reduction in proof size by almost a factor 4 (see Table 2).” (section 1.2, PDF p. 4). Table 2 (PDF p. 13) gives proof sizes of 55.5/80.1 KB for SHAKE256-deg2 s/f versus 14.9/21.1 KB for SHAKE256-deg16 s/f. Apply this when a circuit has repeated low-degree rounds and a low-degree inverse. Inference: checkpoint spacing trades constraint degree and prover work for communication. Inference: in Verus this resembles proving an invariant across six loop iterations at once, but the polynomial identity and finite-field arithmetic need explicit lemmas; the cryptographic soundness argument is outside Verus.
technique: optimize algebraic constraints by random linear combination
For MAYO’s m quadratic equations pi(s) over F16, the prover combines them into one equation Σ ℓi pi(s) over F_{2^λ}, with a list L of m uniformly random F_{2^λ} elements. The paper says: “It is crucial that the list be chosen only after the prover has committed to the witness.” and “In practice, this optimization reduces the proving and verification time of the MAYO trapdoor by roughly a factor of 2 for our chosen parameter sets.” (section 8, PDF p. 14). On soundness it says: “if at least one of the constraints is not satisfied, then the linear combination will not be satisfied with probability 2−λ”. Inference: the intended meaning is that a false combination passes with probability about 2^-λ. Use this to collapse many equations while keeping probabilistic soundness. Inference: a Verus proof can check deterministic polynomial identities and the ordering requirement, but not the probability argument without a separate model of randomness.
technique: instantiate a proof-friendly hash circuit
RainHash runs 7 rounds of the Rain permutation, each a field inversion S-box plus a random invertible matrix (section 7, PDF p. 12). In the circuit, each 512-bit S-box state is a degree-3 polynomial with F_{2^128} coefficients; the prover commits the input and the inversion output of each round, propagates linear layers forward and backward to the S-box input and output, and checks h3, h2, h1 = 0 and h0 = 1 (section 8, PDF p. 13). The paper says: “One hash function call has only 448 bytes of non-linear communication complexity, compared to un-optimized evaluation of SHAKE256 with 4800 bytes and 800 bytes for our VOLEitH-optimized SHAKE256 instance.” (section 7, PDF p. 12). Table 3 (PDF p. 14) reports Sig3 (showing) of 114.399 ms and a 16.51 KB signature for RainHash+MAYO-128s, versus 178.067 ms and 24.32 KB for SHAKE256-deg16+MAYO-128s. The 7-round choice rests on the authors’ security discussion, not on a proof.
implementation, trust, and VL threat
The implementation compiles the MAYO C code and the C++ FAEST VOLEitH framework into shared libraries and uses the Rust crate bindgen for foreign function interfaces; Sig1, Sig2, Sig3 and Ver are Rust functions that call those libraries (section 8, PDF p. 13). The paper states: “This mixture of C++ & Rust is due to our use of a legacy codebase. In the interest of memory-safety and to simplify the codebase, a Rust-based re-implementation of VOLEitH would be interesting future work.” Inference: this is a threat to agent-written verified Rust, because unsafe FFI and unverified cryptographic code form a trusted computing base that Verus contracts alone do not cover. Inference: VL should not redo the paper’s protocol design or benchmarks; it would need to verify wrapper preconditions, memory safety, serialization, and the match between Rust and native circuits. Inference: what stays trusted is the compiler/FFI boundary, hash and field code, the random-oracle model, parameter choices, and the hand-written security proofs.
techniques for the catalog
- two-stage VOLEitH NIZK: commit to hidden randomness before signing, then prove the completed relation
- checkpointed high-degree circuit: prove sparse Keccak states with a degree-16 consistency constraint
- post-commit random linear combination: compress many MAYO equations into one with about 2^-λ soundness error
- proof-friendly hash circuit: design RainHash’s finite-field S-box constraints for smaller proofs
- mixed-language cryptographic implementation: expose C/C++ proof code through Rust FFI and benchmark each protocol phase Checked 2026-10-16 against the full paper by a Claude agent (mid-paper quote: “The security proof of our algorithms follows directly from previous works, but we include a sketch for completeness.”): checked every quotation, number, section and PDF page against the paper text and the page footers; changed the link to the publisher page from the paper text, corrected PDF pages (most were one off), rewrote the two-stage steps to follow Algorithm 2 (the old text mixed in Algorithm 1’s circuit), removed the unsupported “one measured run per row” claim, added the benchmark platform, the factor-2 and 448-byte figures, fixed the FAEST/MAYO language split, quoted the 2−λ soundness sentence instead of paraphrasing it, and marked conclusions “Inference:”; did not check the appendix proofs’ mathematics, the artifact code, or any figure beyond the text.
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