**On differences of quadratic residues**

*Guillermo Morales-Luna*

**Abstract: **Factoring an integer is equivalent to express the integer as the difference of two squares. We test that for any odd modulus, in the corresponding ring of remainders, any element can be realized as the difference of two quadratic residues, and also that, for a fixed remainder value, the map assigning to each modulus the number of ways to express the remainder as difference of quadratic residues is non-decreasing with respect to the divisibility ordering in the odd numbers. The reduction to remainders rings of the problem to express a remainder as the difference of two quadratic residues does not diminish the complexity of the factorization problem.

**Category / Keywords: **foundations /

**Date: **received 8 Oct 2008

**Contact author: **gmorales at cs cinvestav mx

**Available format(s): **PDF | BibTeX Citation

**Version: **20081008:154425 (All versions of this report)

**Short URL: **ia.cr/2008/433

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