1. (a). Using the a¢ne function Ee(x) = 5x + 8 mod 26 encipher the plain text "its cool". (b). Determine the corresponding decryption function. 2. (a). Using the Hill method with matrix M= the text "movies". (b). Using the inverse matrix M the encrypted message and hence show that it is "movies" as expected. State clearly what is required for M 3. The encrypted message "eznwxÖzpwlprzdcmugablxyqlhvjvpkgftkkwchru fmrycgvqduswkuevvkcviiyjkvtzwrplbppzvigfhrviuvvgtiramtctebvgqatzxygdp ymfwapskftqioqjcztpqgnmblrkgldugfoxjaiiutpykvkvilkvtapyilumsmsiumrbvvg vrycgvqdu" was obtained using the Vigenere method. Determine the original plain text which should be "readable" and make sense. You may use Maple providing key steps are clearly explained. Copy and paste the given encripted message directly into your Maple worksheet. 4. Consider the recurrence relation: xn+3 = xn + xn+1 + xn+2 mod 2. (a). Determine the sequence of terms with respective period, for each of the following initial conditions: (i). x1 = 1; x2 = 0; x3 = 1 (ii) . x1 = 1; x2 = 1; x3 = 1 (iii). x1 = 0; x2 = 0; x3 = 1 (b). Show that the seqence of terms in (a) (i) can also be given by a length 2 recurrence relation? (c). How do your results in (b) support the proposition on page 48 of the text (Trappe and Washington). Note that if one row of a matrix is congruent mod 2 to a linear combination of the other rows, then the determinant (mod 2) of the matrix is zero. 5. (a). Using the Euclidean algorthm, determine the gcd(3084; 1424): (b). Using the result in (a), solve 3084x + 1424y = gcd(3084; 1424) 6. (a). Using the Chinese Remainder Theorem, solve the system fx = 3 mod 5; x = 7 mod 8; x = 5 mod 7g (b). What can you say about the system {x = 3 mod 4; x = 0 mod 6g ? 7. 1 5 3 6 mod 26 encrypt 1 ; apply the inverse transformation to 1 to exist. 1 The contrapositive form of Fermatís Little Theorem is: If m is a positive integer and a some a not congruent to 0 mod m; then m is not prime. Use this result to prove that 48703 must be composite. 8. Use Eulerís theorem to determine the remainder of 29198 when divided by 20. 9. Let a and n > 1 be integers with gcd(a; n) = 1. The order of a mod n is the smallest positive integer r such that a Denote r = ordn(a): (a). Show that r (n): (b). Show that if m = rk is a multiple of r then a 1 is not congruent to 1 mod m for m r 1 mod n: m = 1 mod n