Sparse polynomials, redundant bases, gauss periods, and efficient exponentiation of primitive elements for small characteristic finite fields

Authors:
Soonhak Kwon;Chang Hoon Kim;Chun Pyo Hong
Affiliations:
Department of Mathematics and Institute of Basic Science, Sungkyunkwan University, Seoul, Korea;Department of Computer and Communication Engineering, Daegu University, Daegu, Korea;Department of Computer and Communication Engineering, Daegu University, Daegu, Korea
Venue:
Designs, Codes and Cryptography
Year:
2006

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Abstract

Gauss periods give an exponentiation algorithm that is fast for many finite fields but slow for many other fields. The current paper presents a different method for construction of elements that yield a fast exponentiation algorithm for finite fields where the Gauss period method is slow or does not work. The basic idea is to use elements of low multiplicative order and search for primitive elements that are binomial or trinomial of these elements. Computational experiments indicate that such primitive elements exist, and it is shown that they can be exponentiated fast.