Breaking Barriers in Code-Based Cryptography: Cryptanalysis of Decoding, Equivalence and Algebraic Structure

Objetivo

By 2035, national security agencies worldwide, led by NIST and its European counterparts, plan to phase out today’s public-key cryptosystems, requiring a rapid transition to quantum-resistant alternatives. Code-based cryptography is among the most promising candidates, relying on hard problems from algebraic coding theory. Several such proposals were submitted to the NIST post-quantum standardization call, with HQC and Classic McEliece selected for standardization by the US and Germany, respectively.

The ongoing call for signature schemes includes proposals based on novel hardness assumptions and, while adoption is accelerating, the security of these new assumptions are under-explored, limiting our ability to properly assess and compare their robustness.

With CODES, I plan to break this barrier by advancing the cryptanalysis of these emerging problems. We have identified three problems, which underpin the security of all code-based schemes: (1) code equivalence, (2) rank-metric decoding and (3) q-ary decoding.

The main focus is on (1) code equivalence, a structured problem unlikely to be NP-hard. By analyzing its connections to Graph Isomorphism (GI), I aim to reveal its true complexity and potential weaknesses. Several constructions and combinations of closures, power codes and different adjacency matrices will be explored, to finally reduce any instance of code equivalence to GI.

The goal of work package (2) is to develop novel algorithms for decoding in the rank metric. We investigate the associated Hamming-metric code, methods from lattices, and different guessing techniques to reduce the instances.

The investigation of the q-ary setting in (3) is to leverage the extension field structure, through Frobenius
automorphisms and algebraic modelings, ultimately improving on current generic decoding techniques.

This project is high in risk and gain: the proposed improved cryptanalysis will directly impact which signature schemes will ultimately be standardized.