Fast Hardware Implementations of Inversions in Small Finite Fields for Special Irreducible Polynomials on FPGAs
Haibo Yi, Weijian Li, Zhe Nie · International Journal of Security and Its Applications · 2016
Inversions in small finite fields are the most computationally intensive field arithmetic and have been playing a key role in areas of cryptography and engineering.The main algorithms for small finite field inversions are based on Fermat's little theorem, extended Euclidean algorithm, Itoh-Tsujii algorithm and other methods.In this brief, we present techniques to exploit special irreducible polynomials for fast inversions in small finite fields (2 ) n GF , where n is a positive integer and 0 16 n .Then, we propose fast inversions based on Fermat's theorem for two special irreducible polynomials in small finite fields, i.e. trinomials and All-One-Polynomials (AOPs).Trinomials can be represented by polynomials 7 (2 ) GF is 18.80 ns and the executing time of inversion in 12 (2 ) GF is 29.57ns.