# Why do we search for square roots of 1 in Shor's algorithm unlike the qudratic sieve?

In the quadratic sieve algorithm, the idea is to find $$a$$ and $$a$$ such that $$a^2 \equiv b^2 \bmod n$$. We need that $$a\not\equiv \pm b \bmod n$$. However, there the $$c$$ is not necessarily $$1$$. $$\gcd(b \pm c,n)$$ returns non-trivial factors.

However, in Shor's algorithm, we specifically need to find square roots of $$1$$ (in modulo $$n$$) i.e. we look for $$a$$ such that $$a^2 \equiv 1 \bmod n$$. That is, $$b$$ is specifically $$1$$. Why is this choice necessary?

If you give me a pair $$a, b$$ such that $$a^2 = b^2$$ with $$a \neq \pm b$$ then $$c = a b^{-1}$$ satisfies $$c \neq \pm 1$$ and $$c^2 = 1$$.
If you give me a $$c$$ such that $$c^2 = 1$$ and $$c \neq \pm 1$$, and some target value $$a$$, then $$b = a \cdot c$$ satisfies $$a^2 = b^2$$ and $$a \neq \pm b$$.
You can derive a square root of 1 from an $$a,b$$ pair. You can derive an $$a,b$$ pair with desired $$a$$ from a square root of 1. They're reducible to each other.