Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography

D. Bailey, Chris­tof Paar

Journal of Cryptology, vol. 14, no. 3, pp. 153-176 , 2001.


Abstract

This contribution focuses on a class of Galois eld used to achieve fast nite eld arithmetic which we call an Optimal Extension Field (OEF), rst introduced in [3]. We extend this work by presenting an adaptation of Itoh and Tsujii's algorithm for nite eld inversion applied to OEFs. In particular, we use the facts that the action of the Frobenius map in GF(pm) can be computed with only m ? 1 sub eld multiplications and that inverses in GF(p) may be computed cheaply using known techniques. As a result, we show that one extension eld inversion can be computed with a logarithmic number of extension eld multiplications. In addition, we provide new extension eld multiplication formulas which give a performance increase. Further, we provide an OEF construction algorithm together with tables of Type I and Type II OEFs along with statistics on the number of pseudo-Mersenne primes and OEFs. We apply this new work to provide implementation results using these methods to construct elliptic curve cryptosystems on both DEC Alpha workstations and Pentium-class PCs. These results show that OEFs when used with our new inversion and multiplication algorithms provide a substantial performance increase over other reported methods.

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Tags: cryptography