On Codes and Learning with Errors over Function Fields

Bombar, Maxime; Couvreur, Alain; Debris-Alazard, Thomas

doi:10.1007/978-3-031-15979-4_18

Cited by 3 publications

(3 citation statements)

References 28 publications

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“…However, such structured codes are not restricted to encryption schemes. In particular, Bombar, Couteau, Couvreur and Ducros ( [BCCD23]) have recently used the decision version of the Decoding Problem of random quasi-abelian codes, which generalise both random linear codes and quasi-cyclic codes, to build an efficient pseudorandom correlations generator for the OLE correlation (Oblivious Linear Evaluation) over any field F q with q > 2. This allows to design the first efficient silent (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…However, on the security reductions point of view, the situation is even worse than that of the plain Decoding Problem; there is even no complete search-to-decision reduction. The only known reduction for structured variants is the recent work of Bombar, Couvreur and Debris-Alazard [BCD22], via the introduction of the new problem called Function Field Decoding Problem (FFDP), which yield a search-to-decision reduction for some quasi-cyclic codes. This reduction has been extended in [BCCD23] to slightly more general quasi-abelian codes, but the question remains fully open for the codes and parameter sets used in NIST submissions BIKE and HQC.…”

Section: Introductionmentioning

confidence: 99%

“…From lattices to codes. Motivated by this state-of-affairs, a recent trend of research in codebased cryptography has been to take inspiration from the literature on Euclidean lattices to provide new reductions for codes: [DRT21] gives a quantum reduction from the Decoding Problem to the problem of finding a short codeword, in the way of [Reg05,SSTX09]; the reduction of [BCD22] is an average-case to average-case search-to-decision reduction for structured variants of the Decoding Problem, in the spirit of [LPR10], replacing number fields used in the lattice setting by function fields (somehow the analogue in positive characteristics). However, their reduction only works when the irreducible modulus splits completely in the underlying ring of integers, which is not the case with the parameter choice of BIKE and HQC.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pseudorandomness of Decoding, Revisited: Adapting OHCP to Code-Based Cryptography

Bombar,

Couvreur,

Debris-Alazard

2023

Lecture Notes in Computer Science

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Recent code-based cryptosystems rely, among other things, on the hardness of the decisional decoding problem. If the search version is well understood, both from practical and theoretical standpoints, the decision version has been less studied in the literature, and little is known about its relationships with the search version, especially for structured variants. On the other hand, in the world of Euclidean lattices, the situation is rather different, and many reductions exist, both for unstructured and structured versions of the underlying problems. For the latter versions, a powerful tool called the OHCP framework (for Oracle with Hidden Center Problem), which appears to be very general, has been introduced by Peikert et al. (STOC 2017) and has proved to be very useful as a black box inside reductions.In this work, we revisit this technique and extract the very essence of this framework, namely the Oracle Comparison Problem (OCP), to show how to recover the support of the error, solving an Oracle with Hidden Support Problem (OHSP), more suitable for code-based cryptography. This yields a new worst-case to average-case search-to-decision reduction for the Decoding Problem, as well as a new average-case to average-case reduction. We then turn to the structured versions and explain why this is not as straightforward as for Euclidean lattices. If we fail to give a search-to-decision reduction for structured codes, we believe that our work opens the way towards new reductions for structured codes, given that the OHCP framework proved to be so powerful in lattice-based cryptography. Furthermore, we also believe that this technique could be extended to codes endowed with other metrics, such as the rank metric, for which no reduction is known.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%