Self-reciprocal polynomials and generalized Fermat numbers

Self-reciprocal polynomials (SRPs) over GF(q), where q is a prime power, q=p^k, are investigated. The maximum possible component for these polynomials is found for q odd. The construction of Fermat maximum exponent self-reciprocal polynomials (MRPs) over GF(2) is extended to GF(2^k ) with the aid of generalized Fermat numbers. These polynomials leads to a bound on the maximum possible exponent of SRPs over GF(2^k), and a simplified algorithm for finding these MRPs. Self-reciprocal polynomials have applications in cryptography, error-correction coding, and the synthesis of linear feedback shift registers. They are advantageous when available memory or hardware is restricted or when data can be read in either direction. Some results on quasi-self-reciprocal polynomials are also presented