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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 <InlineEquation ID=\"IEq1\" <EquationSource Format=\"TEX\"${\\cal O}2^{n/2}/\\sqrt{n}$</EquationSource> </InlineEquation> instead of <InlineEquation ID=\"IEq2\" <EquationSource Format=\"TEX\"${\\cal O}2^{n/2}$</EquationSource> </InlineEquation> proven complexity for the Parity-Split method. We also prove a lower bound of <InlineEquation ID=\"IEq3\" <EquationSource Format=\"TEX\"${\\Omega}2^{n/2}/\\sqrt{n}$</EquationSource> </InlineEquation> 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-box.

Fast Evaluation of Polynomials over Binary Finite Fields and Application to Side-Channel Countermeasures