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

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## 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 Open image in new window 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.

## Keywords

Low-weight polynomial multiple Low-weight codeword Information-set decoding Public-key cryptography Open image in new window Correlation attacks## Mathematics Subject Classification

11T71 11T06## Notes

### Acknowledgments

We would like to thank the anonymous reviewers in the submission to DCC and WCC for their valuable and insightful comments that helped improve the manuscript. We also want to thank Martin Ågren for helping out with the initial implementation of the algorithm described in Sect. 6. This research was funded by a grant (621-2009-4646) from the Swedish Research Council.

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