Exam 4: Number Theory and Cryptography

Encrypt the message "meet me at noon" using the function $f ( x ) = ( 9 x + 1 ) \bmod 26$

suppose that a and b are integers, $a \equiv 4 ( \bmod 7 ) , \text { and } b \equiv 6 ( \bmod 7 ) . \text { Find the integer } c \text { with } 0 \leq c \leq 6$ such that - $c \equiv 2 a + 4 b ( \bmod 7 )$

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

Find the prime factorization of 1,024.

Find the check digit of the student id starting with 2365 415.

Applying the division algorithm $\text { with } a = - 41 \text { and } d = 6 \text { yields what value of } r \text { ? }$

What is the shared key if Alice and Bob use the Diffie-Hellman key exchange protocol with the prime $p = 67$ , the primitive root $a = 7$ of $p = 67$ , with Alice choosing the secret integer $k _ { 1 } = 12$ and Bob choosing the secret integer $k _ { 2 } = 25$ ?

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

Solve the linear congruence $15 x \equiv 31 ( \bmod 47 ) \text { given that the inverse of } 15 \text { modulo } 47 \text { is } 22 \text {. }$

Show that 7 is a primitive root of 13.

Find Imc(20!, 12!) by directly finding the smallest positive multiple of both numbers

Suppose the first digit of the student id $\text { Х923 } 4562$ is illegible (indicated by X). Can you tell what the first digit has to be?

Find the sequence of pseudorandom numbers generated by the power generator $x _ { n + 1 } = x _ { n } ^ { 2 } \bmod 17$ and seed $x _ { 0 } = 5$

(a) Find two positive integers, each with exactly three positive integer factors greater than 1. (b) Prove that there are an infinite number of positive integers, each with exactly three positive integer factors greater than 1.

Convert (101011)₂ to base 8 .

Use Fermat's little theorem to find $25 ^ { 1202 } \bmod 61$

Prove or disprove: If p and q are primes $( > 2 ) \text {, then } p + q \text { is composite. }$

find the sum and product of each of these pairs of numbers. Express your answer as an octal expansion. - $( 4274 ) _ { 8 } , ( 5366 ) _ { 8 }$

Find an inverse of 5 modulo 17.

Encrypt the message NEED HELP by translating the letters into numbers (A=0, B=1, . . ., Z=25), applying the encryption function $f ( p ) = ( p + 3 ) \bmod 26$ and then translating the numbers back into letters.