Rabin Cryptosystem

MOdular Squaring

Thanks to the Euler's theorem, one can easily extract modular eth roots, until e is co-prime to $\varphi(n)$ (see Euler's Totient function) and the latter value is known: $d = e^{\rm -l}$ mod $varphi(n)$ helps to get it. Unfortunately, $e = 2$ is not co-prime to $varphi(n)$, moreover squaring is not a bijection in the group $\mathbb{Z}_n^\ast$, for $n = p q$ (see also modular arithmetic), and even in $\mathbb{Z}_p^\ast$ for a prime number p: if x is a square root of y in $mathbb{Z}_p^\ast$, then $-x$ is also a square root of y. More formally, the function $f : x \mapsto x^2$ mod p from $\mathbb{Z}_p^\ast$ into $\mathbb{Z}_p^\ast$ is a morphism, whose kernel is $\{-1,+1\}$. As a...