Exam 4: Number Theory and Cryptography

What is the shared key if Alice and Bob use the Diffie-Hellman key exchange protocol with the prime $p = 431$ , the primitive root $a = 79$ of $p = 431$ , with Alice choosing the secret integer $k _ { 1 } = 236$ and Bob choosing the secret integer $k _ { 2 } = 334$ ?

Convert (271)₈ to base 2 .

Find the two's complement of 12.

Prove or disprove: For all integers $a , b , c \text {, if } a \mid b \text { and } b \mid c \text { then } a \mid c \text {. }$

Use the Euclidean Algorithm to find gcd(400, 0).

Find four integers $b \text { (two negative and two positive) such that } 7 \equiv b ( \bmod 4 ) \text {. }$

Find an inverse of 17 modulo 19.

find the sum and product of each of these pairs of numbers. Express your answer as a base 3 expansion. - $( 21202 ) _ { 3 } , ( 12212 ) _ { 3 }$

find the sum and product of each of these pairs of numbers. Express your answer as a binary expansion. - $( 110\quad1011\quad1100 ) _ { 2 } , ( 111\quad0111\quad0111 ) _ { 2 }$

Solve the linear congruence $5 x \equiv 3 ( \bmod 11 )$

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 $a \mid c$ .

Prove or disprove: $\text { If } f ( n ) = n ^ { 2 } - n + 17 \text {, then } f ( n ) \text { is prime for all positive integers } n \text {. }$

Prove or disprove: For all integers $a , b , c \text {, if } a \mid ( b + c ) \text {, then } a \mid b \text { and } a \mid c \text {. }$

Convert (11010111100)₂ to base 8 .

Alice has the public key $( n , e ) = ( 2623,13 )$ with corresponding private key $d = 1357$ , and she wants to send the message LAST CALL to her friends so that they know she sent it. What should she send to her friends, assuming she signs the message using the RSA cryptosystem?

Prove or disprove: if p and q are prime numbers, then pq + 1 is prime.

Solve the linear congruence $31 x \equiv 57 ( \bmod 61 )$

Convert (8091)10 to base 2.

Explain why $f ( x ) = ( 2 x + 3 ) \bmod 26$ would not be a good coding function.

The cipher text LTDTLLWW was produced by encrypting a plaintext message using the Vigen`ere cipher with the key TEST. What is the plaintext message?