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 Stepmethod.

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