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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. 3) observe that every mi is relatively prime with every mj ( j = i) so that every mj is relatively prime with mÃj ¼ M/mj.

The design environment includes virtual components (synthesizable VHDL models, Chapter 9), among others PicoBlaze, an 8-bit microprocessor, and its program memory ([XIL2002]). 5 Hardware– Software Partitioning As mentioned above, the only complex operation is the computation of yx modulo m. All the other operations can be carried out by the processor. 3. 3 . out_port(0) start z Y(1) Y(0) System architecture. PicoBlaze reads the character input at address 0 and the command input at address 1, where command = 0 0 0 0 0 0 sign new_character.

5 Z7 ¼ f0, 1, 2, 3, 4, 5, 6g and ZÃ7 ¼ f1, 2, 3, 4, 5, 6g; 7 is prime and f(7) ¼ 6; 11 ; 1 (mod 7), 23 ; 1 (mod 7), 36 ; 1 (mod 7), 43 ; 1 (mod 7), 56 ; 1 (mod 7), 62 ; 1 (mod 7). There are two generators: 3 and 5. For example, 31 ; 3 (mod 7), 32 ; 2 (mod 7), 33 ; 6 (mod 7), 34 ; 4 (mod 7), 35 ; 5 (mod 7), 36 ; 1 (mod 7). 10 A group (G, Ã, 1) consists of a set G with a binary operation Ã and an identity element 1 satisfying the following three axioms: 1. x Ã (y Ã z) ¼ (x Ã y) Ã z,8x, y, z [ G (associativity); 2.