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Related to this question Complexity of finding solutions of trapdoored polynomial. I am trying to build signature scheme based on hardness of finding points on varieties. Let $K$ be field and $M=K[x_1,...

Assume factorization of $N$ is unknown. What is the best complexity we know to find roots of the irreducible equation $$ax^2+bx+c\equiv0\bmod N?$$ Is this problem equivalent to any hardness results?

The reason I ask this question is that cyclotomic polynomial is critical to the construction of lattice-based cryptography. In most of the existing lattice-based cryptographic schemes, $n$ is usually ...

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. $P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and $...

It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...