Symplectic Lattice Reduction and NTRU
NTRU is a very efficient public-key cryptosystem based on polynomial arithmetic. Its security is related to the hardness of lattice problems in a very special class of lattices. This article is motivated by an interesting peculiar property of NTRU lattices. Namely, we show that NTRU lattices are proportional to the so-called symplectic lattices. This suggests to try to adapt the classical reduction theory to symplectic lattices, from both a mathematical and an algorithmic point of view. As a first step, we show that orthogonalization techniques (Cholesky, Gram-Schmidt, QR factorization, etc.) which are at the heart of all reduction algorithms known, are all compatible with symplecticity, and that they can be significantly sped up for symplectic matrices. Surprisingly, by doing so, we also discover a new integer Gram-Schmidt algorithm, which is faster than the usual algorithm for all matrices. Finally, we study symplectic variants of the celebrated LLL reduction algorithm, and obtain interesting speed ups.
KeywordsReduction Algorithm Symplectic Group Cholesky Factorization Lattice Reduction Symplectic Transformation
- 4.Cohen, H.: A Course in Computational Algebraic Number Theory, 2nd edn. Springer, Heidelberg (1995)Google Scholar
- 9.Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press (1996)Google Scholar
- 14.Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Darnell, M.J. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)Google Scholar
- 16.IEEE P1363.1 Public-Key Cryptographic Techniques Based on Hard Problems over Lattices. IEEE (June 2003), available from: http://grouper.ieee.org/groups/1363/lattPK/index.html.
- 17.LaMacchia, B.A.: Basis reduction algorithms and subset sum problems. Technical Report AITR-1283 (1991)Google Scholar
- 19.Mackey, D.S., Mackey, N., Tisseur, F.: Structured factorizations in scalar product spaces. SIAM J. of Matrix Analysis and Appl. (to appear, 2005)Google Scholar
- 28.Weyl, H.: The classical groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997); Their invariants and representations, Fifteenth printing, Princeton PaperbacksGoogle Scholar