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I am stuck at a homework problem to find the square root of a quadratic residue $b$ in $Z_n$ ($n$ is not a prime). Currently, I have figured out that there exists a number $a \in Z_n$ such that $a^2 \equiv 1$. If I understand the theory correctly, that means the order of $QR_n$ is a multiple of $2$. Since $QR_n$ is a subgroup of $Z_n^*$, it means the order of $Z_n^*$ is also a multiple of $2$. I tried to leverage this knowledge to factor $n$ (so that I can apply Chinese remainder theorem and find the quadratic residue more efficiently). However, my computation shows that $n$ does not admit a small factor (I have checked numbers less than $10000$) at all. How can that be? If $2 \;|\; \phi(n)$, that implies at least one of $3, 4, 6$ should divide $n$, right? (I used the table here to save some manual computation.)

Am I going in a wrong direction? Can someone give me a hint in terms of how to find the square roots more efficiently given that we know a number $a \in Z_n$ such that $a^2 \equiv 1$?

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  • $\begingroup$ There always exist at least two $a\in \mathbb{Z}/n\mathbb{Z}$ such that $a^2=1$, namely $a=\pm 1$. If however you now another solution $a\neq \pm1$, then things get interesting because $(a+1)(a-1)=0$ but neither of the factors is zero mod $n$. Taking suitable GCDs should therefore solve your problem. $\endgroup$ Commented Nov 21, 2023 at 16:22
  • $\begingroup$ @MehdiTibouchi I understand how it works now. I can use gcd(a+1, n) and gcd(a-1, n) to find factors of n efficiently. $\endgroup$
    – ark
    Commented Nov 21, 2023 at 18:32

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