**Revisiting the Gentry-Szydlo Algorithm**

*H. W. Lenstra and A. Silverberg*

**Abstract: **We put the Gentry-Szydlo algorithm into a mathematical framework, and show that it is part of a general theory of ``lattices with symmetry''. For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction. This sheds new light on the Gentry-Szydlo algorithm, and the ideas should be applicable to a range of questions in cryptography.

**Category / Keywords: **foundations / lattices, Gentry-Szydlo algorithm, ideal lattices, lattice-based cryptography

**Original Publication**** (in the same form): **IACR-CRYPTO-2014

**Date: **received 8 Jun 2014

**Contact author: **asilverb at uci edu

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

**Version: **20140612:025753 (All versions of this report)

**Short URL: **ia.cr/2014/430

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