Exam 4: Number Theory and Cryptography

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 $\sigma ( 7 ) = 4$

Convert (100 1100 0011)₂ to base 16 .

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

Use Fermat's little theorem to $\text { find } 9 ^ { 45 } \bmod 23 \text {. }$

find the sum and product of each of these pairs of numbers. Express your answer as a base 3 expansion. - $( 202 ) _ { 3 } , ( 122 ) _ { 3 }$

Express gcd(84, 18) as a linear combination of 18 and 84.

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

Use the Euclidean algorithm to find gcd(44, 52).

Find $50 ! \bmod 50$

Use the Vigen`ere cipher with key LOCK to encrypt the message NEXT FALL.

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

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

Eitherno suchfindinteger.an integer x such that x ≡ 2 (mod 6) and x ≡ 3 (mod 9) are both true, or else prove that there is

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