### Abstract

We describe a new technique for evaluating polynomials over binary finite fields. This is useful in the context of anti-DPA countermeasures when an S-box is expressed as a polynomial over a binary finite field. For n-bit S-boxes, our new technique has heuristic complexity O(2

^{n/2}/√n) instead of O(2^{n/2}) proven complexity for the Parity-Split method. We also prove a lower bound of Ω(2^{n/2}/√n) on the complexity of any method to evaluate n-bit S-boxes; this shows that our method is asymptotically optimal. Here, complexity refers to the number of non-linear multiplications required to evaluate the polynomial corresponding to an S-box. In practice, we can evaluate any 8-bit S-box in 10 non-linear multiplications instead of 16 in the Roy–Vivek paper from CHES 2013, and the DES S-boxes in 4 non-linear multiplications instead of 7. We also evaluate any 4-bit S-box in 2 non-linear multiplications instead of 3. Hence our method achieves optimal complexity for the PRESENT S-boxOriginal language | English |
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Journal | Journal of Cryptographic Engineering |

Volume | 5 |

Issue number | 2 |

Pages (from-to) | 73-83 |

ISSN | 2190-8508 |

DOIs | |

Publication status | Published - 2015 |

## Cite this

Coron, J-S., Roy, A., & Vivek, S. (2015). Fast evaluation of polynomials over binary finite fields and application to side-channel countermeasures.

*Journal of Cryptographic Engineering*,*5*(2), 73-83. https://doi.org/10.1007/s13389-015-0099-9