Choose as common parameters the curve E d e fin e d by y2 = x 3 + 5x + 1 (mod 1367), and the point P = (951,512) which has order 173. With private key a = 100, and with k = 125, sign the message m =10

Choose as common parameters the curve E d e fin e d by y2 = x 3 + 5x + 1 (mod 1367), and the point P = (951,512) which has order 173. With private key a = 100, and with k = 125, sign the message m =1000 , and verify the signature.

Suppose the  entropy  of  English  was  r  =  1.3.   Determine  the  unicity distances of  the   Caesar , Vigenere and  general substitution  ciphers. Repeat  using  r  = 1.1.

Develop a simple Playfair-like cipher with the following encryption: for a pair of letters XY in a Polybius square, if the indices of X and Y are ( a, b) and ( c, d) respectively, then the ciphertext is the pair of letters whose indices are (a, d) and (b, c).

Use Fermat's theorem to reduce the power in the following:(a) 47 1000 (mod 53), (b) 1712781 (mod 61), (c) 23 58 11 (mod 71),(d) 51 9219 (mod 97).

Using Euclid's algorithm, find the following modular inverses: (a) 29 -1 (mod 47), (b) 39 -l (mod 315), (c) 105 -1 (mod 143), (d) 54 -1 (mod 81).Determine the following proverbs, from which alternate fourth and fifth characters  (treating  the  space as a character)  have been  removed: (a)  A  STCHN  TE  SES  NE (b) MANHAN MA LIT WK

Express each of  the  following  numbers as a  product  of prime  powers: (a)  100,        (b)  10000,        (c)  1728,        (d) 3025,         (e)  10829.

Use the ADFGVX cipher with the array given on page 17 and the keyword CODE to (a) Encrypt BRING A FRIEND - FGGAFDXDDVAGAAGAFDGXXDDD(b) Decrypt XAAGGX GAAAXX XDVXGG ADAADG -- SAERF9NX8LF3

Sage Exercises Solve  the  following simultaneous  congruences  (all moduli  are  primes): x  = 1000    (mod  2 13  -   1) x  = 2000    (mod  3 8 + 2) x  = 5000    (mod  5 6 + 4).

By trial division, test each of the following numbers for primality: (a)      289, (b) 541, (c) 2813, (d) 1583, (e) 14803,(f) 7919.1