Exam 4: A: Number Theory and Cryptography

Given that gcd(662, 414) = 2, write 2 as a linear combination of 662 and 414.

find the sum and product of each of these pairs of numbers. Express your answer as a binary expansion. - (11010111100)₂ , (11101110111)₂

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

Convert (101011)₂ to base 8 .

Encrypt the message CANCEL THE ORDER using blocks of seven letters and the transposition cipher based on the permutation of {1,2,3,4,5,6,7} with $\sigma ( 1 ) = 5 , \sigma ( 2 ) = 3 , \sigma ( 3 ) = 6 , \sigma ( 4 ) = 1 , \sigma ( 5 ) = 7 , \sigma ( 6 ) = 2$ and $\sigma ( 7 ) = 4$

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) mod 26, and then translating the numbers back into letters.

Find 50! mod 50.

Find three integers m such that 13 ≡ 7 (mod m).

Find the prime factorization of 1,025.

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. -Suppose the first digit of the student id X123 4566 is illegible (indicated by X). Can you tell what the first digit has to be?

Decrypt the message "AHFXVHFBGZ" that was encrypted using the shift cipher f(x) = (x + 19) mod 26.

Find four integers b (two negative and two positive) such that 7 ≡ b (mod 4).

What is the decryption function for an affine cipher if the encryption function is f(x) = (3x + 7) mod 26?

Solve the linear congruence 5x ≡ 3 (mod 11).

Find a div m and a mod m when $a = 511 , m = 113$

Convert (11010111100)₂ to base 8 .

Find the two's complement of 12.

Use the Euclidean algorithm to find gcd(203, 101).

Find $( 20 ! , 12 ! )$ by directly finding the largest divisor of both numbers.

Find a div m and a mod m when $a = 76 , m = 52$