PQC standards alternatives -- reliable semantically secure key encapsulation mechanism and digital signature protocols using the rank-deficient matrix power function

Post-quantum cryptography-PQC- aims to develop public-key primitives that are secure against adversaries using classical and quantum computing technologies. This study introduces novel protocols, a key encapsulation mechanism, a digital signature scheme, and special protection against linear attacks. Our purpose is to create reliable alternatives to current standards, seeking compact, fast, and secure replacements of the key interchange and digital signature in the TLS 1_3 protocol, which safeguards Internet traffic, allowing an easy post-quantum transition to protect current data from the harvest now, decrypt later threat.

💡 Research Summary

The paper proposes two post‑quantum cryptographic primitives built on a novel algebraic construct called the Rank‑Deficient Matrix Power Function (RDMPF). RDMPF takes three matrices (X, W, Y) together with a “sigma‑variant” modification and iteratively applies modular multiplication, exponentiation, and addition to produce an output matrix Q whose closed‑form expression is infeasible. The authors claim that the rank‑deficiency of the underlying matrices thwarts classical linearisation attacks and that no known quantum algorithm can efficiently invert the function.

Using this primitive, the authors design (1) a key‑encapsulation mechanism (FO‑RDMPF‑KEM) and (2) a digital signature scheme (FO‑RDMPF‑DSA). Both constructions employ the Fujisaki‑Okamoto (FO) transform and an implicit‑rejection (IR) technique. In the KEM, the encapsulation process maps a random secret m to a set of (X_r, Y_r) pairs, runs RDMPF twice per round to obtain temporary values TA_r and S_r, encodes them, and derives a shared secret K via a KDF together with hash‑derived masks and tags. Decapsulation recomputes S′_r from the secret key, verifies consistency of TA values and tags, and either returns the correct K or a pseudorandom fallback key derived from a secret‑fallback function. The security claim is IND‑CCA2 in the Random Oracle Model (ROM), assuming the underlying RDMPF‑PKE is IND‑CPA.

The signature scheme follows a similar pattern: a hash‑derived nonce r (via H₁) is used to produce a deterministic base signature σ₀, a tag t is generated with a second hash H₂, and the final signature is (σ₀, t). Verification recomputes the tag and accepts only if both the underlying signature verification succeeds and the tags match; otherwise, a fixed‑length pseudorandom value is returned (implicit rejection). The scheme is claimed UF‑CMA in the ROM, assuming the base signature scheme is UF‑CMA.

Security analysis is largely reductionist: the authors argue that the FO transform lifts IND‑CPA to IND‑CCA2 and that IR eliminates decryption‑oracle leakage. They provide proof sketches but no full reductions, and the core assumption—IND‑CPA security of RDMPF‑PKE—is left unproven. No concrete hard problem (e.g., “matrix‑power inversion”) is formally defined, nor is any reduction to known quantum‑hard problems presented.

Performance evaluation is limited to prototype implementations in Mathematica 12.1+. The authors report microsecond‑scale key generation, encapsulation, and decapsulation, and millisecond‑scale signing/verification, comparing these numbers loosely to Kyber and Dilithium. However, Mathematica is an interpreted, non‑optimised environment; no C/Rust/assembly benchmarks, memory footprints, or cache‑behaviour analyses are provided. The paper estimates security levels by counting unknown matrix entries (≈ 3n² − 4n + 2) and applying Grover’s square‑root speed‑up, yielding NIST level 5 security for n ≥ 7. This estimation ignores potential structure‑exploiting quantum algorithms beyond Grover and lacks rigorous complexity proofs.

The discussion highlights the perceived algebraic simplicity of RDMPF compared to lattice‑based schemes, the elimination of randomness failures via deterministic encapsulation and signing, and the ease of integrating the constructions into TLS 1.3 as hybrid PQC replacements. Nevertheless, the authors acknowledge that side‑channel resistance, fault‑injection robustness, and large‑scale parameter selection remain open research topics.

In summary, the paper introduces an interesting algebraic primitive and demonstrates how to embed it into standard FO‑based KEM and signature frameworks. While the high‑level design is sound and the security claims are plausible under the stated ROM assumptions, the work lacks (i) a concrete hardness foundation for RDMPF, (ii) full security reductions beyond proof sketches, (iii) realistic performance measurements, and (iv) thorough side‑channel analysis. Consequently, the proposals are at a theoretical‑exploratory stage and would require substantial further analysis and engineering before being considered viable candidates for NIST PQC standardisation or deployment in production TLS stacks.