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Suppose $n = pq$ with $p$ and $q$ both primes.

  1. Suppose that $\gcd(a, pq) = 1$. Prove that if the equation $x^2 ≡ a \bmod n$ has any solutions, then it has four solutions.
  2. Suppose you had a machine that could find all four solutions for some given $a$. How could you use this machine to factor $n$?
Fly by Night
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leeha
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  • You need that $p \neq q$. For $p = q$, there are only two solutions. – Daniel Fischer Sep 28 '13 at 18:55
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If $p\neq q$ then we have an isomorphism of rings $\mathbb Z_n\cong \mathbb Z_p\times\mathbb Z_q$. If $a=x^2$ and $x$ corresponds to $(s,t)$ under this isomorphism, then $(-s,t)$, $(s,-t)$ and $(-s,-t)$ are also solutions. If $gcd(a,pq)=1$ then $s,t$ are non-zero. If $p,q$ are different from $2$ this means that all four solutions are different.

If the machine gave us all 4 solutions, take any two such that their sum is not $0$ (in $\mathbb Z_n$). This sum $r$ is divisible by either $p$ or by $q$. So just compute $gcd(n,r)$ to get $p$ or $q$.

user8268
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