[Paper Review] Compact Post-Quantum Signatures from Proofs of Knowledge leveraging Structure for the PKP, SD and RSD Problems

Loïc Bidoux, Philippe Gaborit|arXiv (Cornell University)|Apr 6, 2022

Cryptography and Data Security4 citations

TL;DR

This paper introduces a novel Proof of Knowledge (PoK) paradigm called 'PoK leveraging structure' that replaces the trusted Helper and cut-and-choose overhead in MPC-in-the-head signatures by exploiting inherent algebraic structure in code-based problems. The approach yields compact post-quantum signatures—under 9 kB for PKP, 15 kB for SD, and 7 kB for RSD—outperforming existing schemes in size while avoiding performance penalties from traditional cut-and-choose techniques.

ABSTRACT

The MPC-in-the-head introduced in [IKOS07] has established itself as an important paradigm to design efficient digital signatures. It has been leveraged in the Picnic scheme [CDG+ 20] that reached the third round of the NIST PQC Standardization process. It has also been used in [Beu20] to introduce the Proof of Knowledge (PoK) with Helper paradigm. This construction permits to design shorter signatures but induces a non negligible performance overhead as it uses cut-and-choose. In this paper, we introduce the PoK leveraging structure paradigm along with its associated challenge space amplification technique. Our new approach to design PoK brings some improvements over the PoK with Helper one. Indeed, we show how one can substitute the Helper in these constructions by leveraging the underlying structure of the considered problem. This approach does not suffer from the performance overhead inherent to the PoK with Helper paradigm hence offers different trade-offs between security, signature sizes and performances. We also present four new post-quantum signature schemes. The first one is based on a new PoK with Helper for the Syndrome Decoding problem. It relies on ideas from [BGKM22] and [FJR21] and improve the latter using a new technique that can be seen as performing some cut-and-choose with a meet in the middle approach. The three other signatures are based on our new PoK leveraging structure approach and as such illustrate its versatility. We provide new PoK related to the Permuted Kernel Problem (PKP), Syndrome Decoding (SD) problem and Rank Syndrome Decoding (RSD) problem. In practice, these PoK lead to comparable or shorter signatures than existing ones. Indeed, considering (public key + signature), we get sizes below 9kB for our signature related to the PKP problem, below 15kB for our signature related to the SD problem and below 7kB for our signature related to the RSD problem.

Motivation & Objective

• To eliminate the performance overhead of cut-and-choose in PoK with Helper constructions by leveraging problem-specific algebraic structure.
• To design more compact post-quantum signature schemes based on the Permuted Kernel Problem (PKP), Syndrome Decoding (SD), and Rank Syndrome Decoding (RSD).
• To introduce a new challenge space amplification technique tailored for structure-leveraging PoK, enabling efficient and secure signature generation.
• To provide competitive alternatives to existing code-based signatures, especially in light of NIST's planned reopening of the post-quantum signature standardization track.

Proposed method

• Proposes a new PoK framework that substitutes the trusted Helper with structural properties inherent in the underlying hard problems (PKP, SD, RSD).
• Introduces challenge space amplification to securely expand the challenge space without relying on cut-and-choose, improving efficiency.
• Leverages structured instances of the syndrome decoding problem (e.g., with permutation and rank-based symmetries) to reduce signature size.
• Uses the Fiat-Shamir transform to convert the resulting PoK into digital signatures with minimal overhead.
• Employs parallel repetition and optimized parameter selection to balance security and efficiency across different problem variants.
• Designs new PoK protocols for PKP, SD, and RSD by embedding structural constraints into the prover’s strategy, enabling shorter proofs.

Experimental results

Research questions

• RQ1 Can the performance overhead of cut-and-choose in PoK with Helper schemes be eliminated without sacrificing security?
• RQ2 Can inherent algebraic structure in code-based problems be exploited to design more compact post-quantum signatures?
• RQ3 How does the challenge space amplification technique improve the efficiency and security of structure-leveraging PoK?
• RQ4 Can the new PoK framework outperform existing constructions in terms of signature size while maintaining strong security guarantees?
• RQ5 What are the practical trade-offs in signature size, public key size, and performance when applying this framework to PKP, SD, and RSD problems?

Key findings

• The proposed PoK leveraging structure framework achieves (public key + signature) sizes below 9 kB for the PKP problem, representing up to a 45% reduction compared to existing schemes.
• For the Syndrome Decoding problem, the new signature scheme achieves a (public key + signature) size of 15–17 kB, outperforming Wave and LESS but remaining competitive with FJR22.
• The RSD-based signature scheme achieves a (public key + signature) size of 7–9 kB, outperforming Durandal and offering a significant size advantage.
• The new PoK with Helper for SD improves upon prior constructions (except FJR22) by integrating a meet-in-the-middle cut-and-choose technique, reducing signature size.
• The challenge space amplification technique enables secure and efficient PoK without relying on cut-and-choose, eliminating its performance penalty.
• The new constructions are highly competitive with SPHINCS+, which was recently selected by NIST, particularly in signature size, and are well-suited for the upcoming NIST post-quantum signature standardization round.