A Code-specific Conservative Model for the Failure Rate of Bit-flipping Decoding of LDPC Codes with Cryptographic Applications

TL;DR

This paper introduces a conservative statistical model for the failure rate of bit-flipping decoding in LDPC/MDPC codes, enhancing cryptosystem security by identifying weak keys and ensuring reliable decoding.

Contribution

It provides a worst-case failure rate analysis for an improved bit-flipping decoder, aiding secure cryptosystem design with predictable decoding performance.

Findings

Derived a code-specific failure rate bound

Identified weak keys with high failure probability

Enabled secure cryptosystem construction with guaranteed correctness

Abstract

Characterizing the decoding failure rate of iteratively decoded Low- and Moderate-Density Parity Check (LDPC/MDPC) codes is paramount to build cryptosystems based on them, able to achieve indistinguishability under adaptive chosen ciphertext attacks. In this paper, we provide a statistical worst-case analysis of our proposed iterative decoder obtained through a simple modification of the classic in-place bit-flipping decoder. This worst case analysis allows both to derive the worst-case behaviour of an LDPC/MDPC code picked among the family with the same length, rate and number of parity checks, and a code-specific bound on the decoding failure rate. The former result allows us to build a code-based cryptosystem enjoying the $\delta$-correctness property required by IND-CCA2 constructions, while the latter result allows us to discard code instances which may have a decoding failure rate significantly different from the average one (i.e., representing weak keys), should they be picked during the key generation procedure.