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Find the square roots of 57 mod 107. 2. (a).Consider Z5[x] mod x8 + x4 + x3 + x + 1. Show that h(x) = 3×5+ 2×3 + x2 + 4x + 4 is reducible.

1. Find the square roots of 57 mod 107.
2. (a).Consider Z5[x] mod x8 + x4 + x3 + x + 1. Show that h(x) = 3×5+
2×3 + x2 + 4x + 4 is reducible.
Hint: consider q(x) = 4×3 + 3×2 + 4 and do a division.
(b). Find the multiplicative inverse of 1 + 2x in Z3[x] mod x2 + 1
3.(a) Let the input message be 100010110101 and the key be 111000111.
Perform two complete rounds of encryption using the simpli…ed DES type
algorithm.
(b). Starting with the results from 3(a), decrypt the ciphertext
by using the algorithm in “reverse”.
4. The ciphertext is 182 and was obtained from the RSA algorithm using
n = 437 and e = 283. Find the plaintext. {Hint: you can use trial and
error (small integers) to determine d or use the Extended Euclidean
Algorithm}
5. Determine L3(6) mod 31, using the “Baby Step-Giant Step”method.

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