# Affine encryption function

Let $f(x) = ax + b$ be an affine encryption function. $a$ and the cardinal of the field of our keys $K$ (in English, 26) must be coprime so that the encryption function is bijective and the decryption function defined and bijective as well.

Could you please clarify how (in terms of congruences) the two having a $pgcd>1$ make the encryption function not bijective?

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Let $n$ be the modulus and $d = \gcd(a,n)$. Then, $ax \mod n$ will be a multiple of $d$. If $d > 1$, this implies that there is no $x$ such that $ax \equiv 1 \pmod n$, and so there is no $x$ such that $f(x) = ax+b \equiv b+1 \pmod n$. So $f$ is not a bijection since it is not surjective.