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Topic: Bounded modular square roots
Replies: 0

 Tommy Jensen Posts: 249 From: Daegu, Korea Registered: 12/6/09
Bounded modular square roots
Posted: Jun 24, 2014 8:26 AM

Given input (a,N,c), all natural numbers, it is NP-complete to
decide whether there exists an integer x with 0 < x < c for which
x^2 = a (mod N) holds, i.e. x is a square root of a modulo N, which
is less than a constant c. Even if the prime factorization of N is
known. This follows from a paper by Adleman and Manders in 1978.

Is it possible to remove a from the input, and preserve the
NP-completeness?

Say, how about the problem, with input (N,c), whether there
exists an integer x with 0 < x < c for which x^2 = 2 (mod N) holds?

Or as a natural variation, whether there exists an integer x
with 1 < x < c for which x^2 = 1 (mod N) holds?

Either problem is at least as hard as to find a trivial factor
of N, by a simple argument. But are they also NP-hard?