Question

4. Use index calculus (and the tables on pages 213,216,217 in Apostol) to solve each of the following congruences (a) 9r Ξ 7 (mod 17) (b) r511 (mod 19) (c) 5z Ξ 17 (mod 23) Table 10.1 g(p) is the smallest primitive root of the prime p 109 6 269 2439 15 617 3 3 3 2113 3271 6 443 2619 2821 2 5 2 127 3277 5 449 3631 3823 3 7 3 131 2281 3457 13 641 3827 2 11 2 137 3 283 3462 643 11 8292 13 2139 2293 2 463 3647 5839 11 17 3149 23075 467 2653 28532 19 2151 6311 17 479 13 659 2857 3 23 5157 5313 10 4873 661 2 8592 29 2 163 2317 2 491 2673 58635 167 5 331 3 499 7677 28772 31 3 37 2173 2337 10 5035683 5 881 3 41 6 179 2347 2 509 269 3 883 2 43 3181 2349 25213701 28875 47 5 19 9353 35232709 2907 2 53 2 193 5 359 7 54 2719 1 911 17 59 2197 23676 547 2 727 5919 7 61 2199 33732557 27336 929 3 67 2211 23792563 2 739 3 937 5 71 7223 3 383 5 569 3 7435 94 2 73 5 227 23892571 3 751 3947 2 79 3 229 63975577 5757 2 953 3 83 2 233 3401 3587 2761 6 967 5 89 3 239 7 409 21 97 5 241 7419 2599 7 773 2 977 3 593 3 769 97 6 101 22516 421 2 601 7 787 2 983 5 103 5257 3 4 7607 3797 2 997 7 107 2 263 5 433 5 613 28093 Table 10.2 Indices of all numbers a primes p 50. The base g is the smallest primitive root of p 13 16 12 16 15 12 32 19 13 18 13 16 17 45 25 15 15 13 10 16 10 10 19 18 18 35

Answer 1

9 (mod 16 ) 1 7 Tndy 姪11(mod 17 ) - (b) Selution e 5 and 1 E 12 Cmed 1) hasiue selution. O SE 1 mod 23)