Improved algorithms for finding low-weight polynomial multiples in $$\mathbb {F}_{2}^{}[x]$$ F 2 [ x ] and some cryptographic applications

Abstract

In this paper we present an improved algorithm for finding low-weight multiples of polynomials over the binary field using coding theoretic methods. The associated code defined by the given polynomial has a cyclic structure, allowing an algorithm to search for shifts of the sought minimum-weight codeword. Therefore, a code with higher dimension is constructed, having a larger number of low-weight codewords and through some additional processing also reduced minimum distance. Applying an algorithm for finding low-weight codewords in the constructed code yields a lower complexity for finding low-weight polynomial multiples compared to previous approaches. As an application, we show a key-recovery attack against [InlineEquation not available: see fulltext.] that has a lower complexity than the chosen security level indicate. Using similar ideas we also present a new probabilistic algorithm for finding a multiple of weight 4, which is faster than previous approaches. For example, this is relevant in correlation attacks on stream ciphers.