Exam 4: Number Theory and Cryptography

Find $a \operatorname { div } m \text { and } a \bmod m \text { when } a = 511 , m = 113$

find the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion. - $( \mathrm { E } 3 \mathrm {~A} ) _ { 16 } , ( \mathrm {~B} 5 \mathrm {~F} 8 ) _ { 16 }$

Use the Euclidean Algorithm to find gcd(390, 72).

Find the prime factorization of 8,827.

Use the Vigen`ere cipher with key NOW to encrypt the message SUMMER.

Convert (10,000)10 to base 2.

Solve the linear congruence $54 x \equiv 12 ( \bmod 73 ) \text { given that the inverse of } 54 \text { modulo } 73 \text { is } 23 \text {. }$

Prove: if n is an integer that is not a multiple of 4, $\text { then } n ^ { 2 } \equiv 0 \bmod 4 \text { or } n ^ { 2 } \equiv 1 \bmod 4 \text {. }$

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

Find the integer a such that $a = 71 ( \bmod 41 ) \text { and } 160 \leq a \leq 200$

Solve the linear congruence $2 x \equiv 5 ( \bmod 9 )$

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 - $c \equiv 5 b ( \bmod 7 )$

What does a 60-second stop watch read 54 seconds before it reads 19 seconds?

Encrypt the message NEED HELP by translating the letters into numbers (A=0, B=1, . . ., Z=25), applying the encryption function $f ( p ) = ( 3 p + 7 ) \bmod 26$ and then translating the numbers back into letters.

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 - $c \equiv a ^ { 2 } - b ^ { 2 } ( \bmod 7 )$

Prove: if n is an integer that is not a multiple of 3, $\text { then } n ^ { 2 } \equiv 1 \bmod 3$

Express gcd(450, 120) as a linear combination of 120 and 450.

Find the discrete logarithms of 5 and 8 to the base 7 modulo 13.

Encrypt the message BALL using the RSA system with $n = 37 \cdot 73$ and $e = 7$ , translating each letter into integers $( \mathrm { A } = 00 , \mathrm {~B} = 01 , \ldots )$ and grouping together pairs of integers.

Encode the message "stop at noon" using the function $f ( x ) = ( x + 6 ) \bmod 26$