What is a fast way of finding a solution M to a quadratic modulus of the form
a == M (M + b) mod n
where a and n are very large, and b << a
n is the product of two large primes (RSA modulus)
2026-03-26 23:01:06.1774566066
Quadratic modulus
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One can complete the square so that this is simply asking for a quadratic residue. Now knowing how to find a quadratic residue gives an efficient (and currently unknown) way to factor $n$. Thus, assuming RSA-type numbers where no factorization is known, it can't be done.
See here for an algorithm assuming the prime factorization is known.