The Alaniz Cipher v3: Sheaf-Based Post-Quantum Encryption via Vector-Valued Finite Field Permutations

The Alaniz Cipher v3 is a post-quantum public-key encryption scheme built from cellular sheaves on graphs with nonlinear restriction morphisms defined over extension fields. It achieves IND-CPA security as a PKE and, after applying the Fujisaki-Okamoto transform, IND-CCA security as a KEM.

This is the third substantive revision of the construction, following the cryptanalysis of the two previous versions:

• Version 1 (v1.0.2, 2026) used a component-wise cubing map σ and was broken by Rodríguez Langa through a scaling attack exploiting the identity σ(λy) = λ³σ(y).
• Version 2 introduced an SPN-structured σ to defeat scaling. However, this paper shows that v2 is also broken: any deterministic encryption of the form c = As + Bσ(As) with σ of bounded polynomial total degree admits a chosen-plaintext key-recovery attack via polynomial interpolation followed by Gröbner basis computation. For d = 2, p = 17, the full secret key is recovered in 20 CPA queries and approximately 5.7 seconds of symbolic computation. The attack succeeds in all 10 independently generated random keys tested.
• Version 3 (this work) addresses the vulnerability at the structural level. The encryption is made probabilistic via a nonce-derived pseudorandom stream (SHAKE-256), and the nonlinear component σ is redesigned to operate vectorially in the extension field F_{p^d} rather than component-by-component: σ(y) = y + ι⁻¹(π_e(L · ι(y) + 1)) with π_e(x) = x^e a monomial power permutation. A further design decision restricts one key component to a scalar in F_{p^d}^*, which enables polynomial-time decryption via univariate factorization in F_{p^d}[τ] at cost O(e^{2.5} · d³ · log² p) bit operations. A centralizer argument proves that this restriction does not introduce MinRank-exploitable structure.

Key technical results established in the paper:

• Semi-regularity of the attacker's z-substituted algebraic system, verified empirically in 14 parameter configurations against Hilbert-Poincaré predictions with exact agreement. This is stronger evidence than was available for Rainbow, whose structural weakness was precisely a failure of this property.
• Unique decryption probability bound via a Schwartz-Zippel argument over the cohomological consistency of n nodes, yielding Pr[ambiguity] ≤ 2⁻¹¹²¹⁸ for PQ-128 and 2⁻⁶³⁷⁵² for PQ-256 parameters.
• Centralizer theorem establishing that the scalar restriction of β_v is equivalent to d² − d linear commutativity constraints on a matrix key, without introducing rank-deficient submatrices.
• Nonce-robustness theorem: the XL rank deficit of the attacker's system is invariant under fresh, fixed, or zero nonce policies, showing that the vector-valued σ is the primary algebraic barrier rather than the PRG randomness.
• Weak-key analysis: beyond the manifest exclusions β_v = 1 and L = 0 (both ruled out by key generation), no exploitable weak-key class is detected in an empirical sweep of 69 configurations.

Proposed parameter sets:

• PQ-128: d = 6, p ≈ 2⁶¹, n = 32, e = 17. Public key 64 B, ciphertext 1.5 KB, shared key 32 B. Projected ~24 μs encapsulation, ~100 μs decapsulation on optimized C implementation.
• PQ-256: d = 8, p ≈ 2¹²⁷, n = 64, e = 17. Public key 128 B, ciphertext 8 KB.

Open problem honestly stated. The paper does not provide a formal worst-case-to-average-case reduction for the NL-SMIP problem underlying the scheme. This is a limitation shared with all multivariate cryptographic schemes (HFE, Rainbow, UOV, MAYO, GeMSS): no scheme in this family has a Regev-style reduction. Resolving this would be a contribution of independent significance to multivariate cryptography. The paper delimits the three specific obstacles (absence of q-ary symmetry, absence of Gaussian-Fourier duality, absence of natural worst-case universality) and discusses how Alaniz v3 rests on the same heuristic foundation as the rest of the multivariate family, supplemented by empirical semi-regularity verification that was not performed for broken predecessors.

Reproducibility. A complete Python reference implementation is provided at https://github.com/QuantuSync/alaniz-cipher, including the IND-CCA KEM, the cryptanalysis of v2 (reproducing the 20-query key recovery), and five verification experiments covering semi-regularity, weak keys, nonce robustness, the centralizer theorem, and KEM correctness with implicit rejection. All 39 tests pass, 100/100 round-trip encryption experiments succeed, and the 14 semi-regularity configurations match Hilbert-Poincaré predictions to the digit.