Journal of Cryptology

, Volume 14, Issue 3, pp 153–176 | Cite as

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

  • Daniel V. Bailey
  • Christof Paar


This contribution focuses on a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF), first introduced in [3]. We extend this work by presenting an adaptation of Itoh and Tsujii's algorithm for finite field inversion applied to OEFs. In particular, we use the facts that the action of the Frobenius map in GF (p m ) can be computed with only m-1 subfield multiplications and that inverses in GF (p) may be computed cheaply using known techniques. As a result, we show that one extension field inversion can be computed with a logarithmic number of extension field multiplications. In addition, we provide new extension field 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.

Key words. Finite fields, Fast arithmetic, Binomials, Modular reduction, Elliptic curves, Inversion. 


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Copyright information

© International Association for Cryptologic Research 2001

Authors and Affiliations

  • Daniel V. Bailey
    • 1
  • Christof Paar
    • 2
  1. 1.Computer Science Department, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, U.S.A. bailey@cs.wpi.eduUS
  2. 2.Electrical and Computer Engineering and Computer Science Departments, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, U.S.A. christof@ece.wpi.eduUS

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