Solving equations in finite fields and some results concerning the structure of GF(pm)

Quasi-self-reciprocal polynomials (QSRP's), were introduced in [2] as a generalization of SRP's. They represent a superset of SRP's, since QSRP's are defined f*(x) = *f(x). This extends SRP's over GF (pk), p > 2, to include those for which f*(x) =-f(x). (15) Thus, if f(x) = x' +fr-ixr-' +f,-2x'-2 + a** +f,x + f,,, fnei =-fi, with f. =-1 since f(x) is manic. Polynomials defined by (15) are denoted in the following as QSRP's to distinguish them from SRP's. It is useful to examine these polynomials to determine if they can offer improvements over SRP's. Proof: To satisfy (15) the coefficient of xk in f(x) must be the negative of the coefficient of xrpk. Thus, 1 must be a root of f(x) and x-1 a factor, so x-1 is the only irreducible QSRP. 0 For r even, the coefficient of x'/' in a QSRP must be zero if p > 2, since by definition fr,2 =-f,.,2, which is true only if fr,2 = 0. Division of a QSRP by x-1 produces a polynomial which is self-reciprocal with even degree. Therefore, all QSRP's can be derived by multiplying a SRP by x-1, and the number of QSRP's of degree r = 2 t + 1 is equal to the number of SRP's of degree r = 2 t. Since the exponent of x-1 is 1, the exponent of a QSRP is the same as the underlying SRP, so no gains in exponent are possible by considering a QSRP over a SRP. Like quadratic residues of nonresidues, the product of a QSRP and a QSRP produces a SRP, i.e., QSRP * QSRP = SRP, SRP * QSRP = QSRP and SRP * SRP = SRP, since (x-1)' is a SRP if t is even. The maximum possible exponent for a self-reciprocal polynomial over GF (cJ), 4 a prime power pk, has been derived for 4 odd, and a bound determined for 4 even. A construction method has been given for q even which improves the algorithm given in [8]. quasi-self-reciprocal polynomials have also been examined.