Exam 4: A: Number Theory and Cryptography

Find the integer a such that $a = 71 ( \bmod 47 ) \text { and } - 46 \leq a \leq 0$

Convert (BC1)₁₆ to base 2 .

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - $\text { If } a \equiv b ( \bmod 2 m ) , \text { then } a \equiv b ( \bmod m )$

What sequence of pseudorandom numbers is generated using the pure multiplicative generator $\mathfrak { x } _ { n + 1 }$ = 3x mod 11 with seed $\operatorname {x_0 }$ =2 ?

Show that if a, b, k and m are integers such that $k \geq 1 , m \geq 2 , \text { and } a \equiv b ( \bmod m ) , \text { then } k a \equiv k b$ (mod m).

Use Fermat's little theorem to find $25 ^ { 1202 }$ mod 61 .

find the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion. - $( 2 \mathrm {~A} ) _ { 16 } , ( \mathrm { BF } ) _ { 16 }$

Use the Euclidean Algorithm to find gcd(580, 50).

Here is a sample proof that contains an error. Explain why the proof is not correct. Theorem: If $a \mid b$ and $b \mid c$ , then $a \mid c$ . Proof: Since $a \mid b , b = a k$ . Since $b \mid c , c = b k$ . Therefore $c = b k = ( a k ) k = a k ^ { 2 }$ . Therefore $a \mid c$ .

refer to an 8-digit student id at a large university. The eighth digit is a check digit equal to the sum of the first seven digits modulo 7. -Find the check digit of the student id starting with 3179 822.

Encrypt the message "just testing" using the function f(x) = (5x + 3) mod 26.

Find the integer a such that $a = 89 ( \bmod 19 ) \text { and } - 9 \leq a \leq 9$

Find −88 mod 13.

Prove or disprove: There exist two consecutive primes, each greater than 2.

What does a 60-second stop watch read 82 seconds after it reads 27 seconds?

Encrypt the message WATCH OUT using blocks of four letters and the transposition cipher based on the permutation of {1, 2, 3, 4} with σ(1) = 3, σ(2) = 4, σ(3) = 2, and σ(4) = 1.

Convert (204)₁₀ to base 2 .

A message has been encrypted using the function f(x) = (x + 5) mod 26. If the message in coded form is JCFHY, decode the message.

Convert (10011000011)₂ to base 16 .

Find the first five terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 9361.