By JEAN-PIERRE DE SCHAMPS,
GERY JEAN ANTOINE BIOUL
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3. (a/g)21 mod n/g. The complete set of solutions within Zn is xk ¼ x0 þ k:(n=g), 8k ¼ 0, 1, . . , g À 1: Observe that if k , g and x0 , (n/g), then xk ¼ n 2 1. 4 (Chinese Remainder Theorem) Consider s pairwise relatively prime integers m1, m2, . . , ms whose product is equal to M. Then the system N ; r1 (mod m1 ), N ; r2 (mod m2 ), ... N ; rs (mod ms ), (2:3) has a unique solution N within ZM (jajm stands for a mod m): N ¼ S 1 Ã Ã i s mi :jri =mi jmi M , (2:4) where M ¼ P1 i s mi ; mÃi ¼ M=mi : (2:5) The ri are called residues modulo mi.
Given g(x) in Fp, then (g(x))p ¼ g(x) (Fermat’s little theorem). Given g(x) in Fq such that (g(x))p ¼ g(x); then g(x) [ Fp. The set of nonzero polynomials of Fq is a cyclic group denoted FÃq. Given g(x) in Fq, then (g(x))q ¼ g(x). Given g(x) and h(x) in Fq, then (g(x) þ h(x))p ¼ (g(x))p þ (h(x))p. If r ¼ (pn 2 1)/(p 2 1), that is, r ¼ 1 þ p þ p2 þ Á Á Á þ pnÀ1 , and g(x) is an element of Fq, then (g(x))r is an element of Fp. 9(3)). 3 FUNCTION APPROXIMATION Numerous techniques are used to evaluate functions.
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