An Indexing for Quadratic Residues Modulo N and a Non-uniform Efficient Decoding Algorithm
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An indexing of a finite set S is a bijection D : {1,...,|S|}→ S. We present an indexing for the set of quadratic residues modulo N that is decodable in polynomial time on the size of N, given the factorization of N. We derive two information theoretical consequences from this indexing. The first consequence is a procedure for sampling quadratic residues modulo N, when the factorization of N is known, that runs in strict polynomial time and requires the theoretical minimum amount of random bits (i.e., (ϕ(N)/2^r) bits, where ϕ(N) is Euler's totient function and r is the number of distinct prime factors of N). A previously known procedure for this same problem runs in expected (not strict) polynomial time and requires more random bits. The second consequence is a tighter upper bound for the time-bounded Kolmogorov Complexity of strings consisting of multiple quadratic residues modulo N.
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