Science:Math Exam Resources/Courses/MATH312/December 2010/Question 06/Statement

Recall that an affine cryptosystem with encryption key $K_{E} = (26, a, b)$ and decryption key $K_{D} = (26, c, d)$ works as follows. Each letter is assigned a numerical value, in alphabetical order:

$A \mapsto 0, B \mapsto 1, . . ., Z \mapsto 25$

These numbers are then viewed modulo 26: They are encrypted by the formula

$x \to y \equiv a x + b mod 26$

and decrypted by the formula

$y \to x \equiv c y + d mod 26$

It is known that the most frequently occurring letter in the English language is E (with numerical value 4) and the second most frequently occurring letter is T (with numerical value 19). Suppose these letters are encoded as F (numerical value 5) and G (numerical value 6), respectively.

(a) Find the encryption key $K_{E}$

(b) Find the decryption key $K_{D}$