Cryptanalysis

Affine Cipher

The affine function is defined as e(x)=(ax+b) mod 26, where aa and bb are integers and x is an element in $Z_{26}$​, the set of integers modulo 26. To determine when this affine function is injective, you are looking at the congruence ax≡y(mod26) and claiming that this congruence has a unique solution for every y if and only if gcd(a,26)=1.

Let's break down the argument step by step:

• Claim: If gcd(a,26)=1, then the congruence ax≡y(mod26) has a unique solution for every y. Explanation: When gcd(a,26)=1, it implies that a and 26 are relatively prime. This means that there is no integer greater than 1 that divides both a and 26. In this case, a has a modular multiplicative inverse modulo 26. Let $a^{−1}$ be the modular inverse of a modulo 26. The congruence ax≡y(mod26) can then be multiplied by $a^{-1}$ to obtain x ≡ $a^{-1}$ y(mod26). This congruence has a unique solution for every y because $a^{-1}$ exists, and the multiplication is well-defined modulo 26.
• Counterclaim: If gcd(a,26)=d>1, then the congruence ax≡y(mod26) may not have a unique solution for every y. Explanation: When gcd(a,26)=d>1, it implies that aa and 26 have a common divisor greater than 1. In this case, the congruence ax≡0(mod26) has at least two distinct solutions in $Z_{26}$​: x=0 and x=26/d. Since ax≡0(mod26), multiplying both sides by a will give ax≡0(mod26). This implies that ax is a multiple of 26, and therefore, adding any multiple of 26 to the solutions x=0 and x=26/d will still satisfy the congruence. This lack of unique solutions means that the function e(x)=(ax+b)mod 26 is not injective, and it is not a valid encryption function.