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Irreducible Pentanomials and their Applications to Effective Implementations of Arithmetic in Binary Fields

Paszkiewicz A.

Electronics and Telecommunications Quarterly

2009

Vol. 55, No 2

363-375

There are only few main classes of irreducible polynomials which are used for designing arithmetic in Galois Fields with characteristic two. These are: irreducible trinomials, pentanomials, all-one polynomials (AOP) and equally spaced irreducible polynomials (ESP). The most critical and time consuming arithmetical operations in Galois Fields are multiplication and modular reduction. A special structure of the modular polynomial defining the arithmetic allows significant speedup of these operations. The best class of binary irreducible polynomials are trinomials, but for about one half of degrees below 30000 an irreducible trinomial does not exist. By exhaustive computation we established that for all degrees n between 4 and 30000 an irreducible pentanomial always exists. Therefore using irreducible pentanomials for defining the arithmetic of Galois Fields have practical interest. In the paper we investigate a function describing the number of binary irreducible pentanomials of a given degree n greater than 3 and study its properties. We also analyze the complexity of a circuit (the number of XOR and AND gates) implementing multiplication in the finite field represented by general irreducible pentanomials.