On Schnorr-Adleman lattice

Lattice theory has been found numerous applications in mathematics and computer science. The shortest vector problem (SVP) and the closest vector problem (CVP) are basic problems of lattice theory. It has been known that the CVP is NP-hard under Karp reduction. SVP is NP-hard under randomized reduction within any constant factor less than √2. The reduction from large integer factorization and discrete logarithm to CVP under a reasonable hypothesis. The factoring problem was random polynomial time reducible to SVP. These methods were believed to provide the possibility of factoring large integers and solving discrete logarithms by approximate lattice reduction algorithms. This is called Schnorr-Adleman lattice. We discuss some properties of Schnorr-Adleman lattice, and improve a result in [D. Micciancio et al., (2002)]. Our results can also be viewed as generalization of some results of [L. M. Adleman] and [C. P. Schnorr (1993)] in some sense.