On the ideal shortest vector problem over random rational primes

04/21/2020

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by   Yanbin Pan, et al.

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Any ideal in a number field can be factored into a product of prime ideals. In this paper we study the prime ideal shortest vector problem (SVP) in the ring [x]/(x^2^n + 1), a popular choice in the design of ideal lattice based cryptosystems. We show that a majority of rational primes lie under prime ideals admitting a polynomial time algorithm for SVP. Although the shortest vector problem of ideal lattices underpins the security of Ring-LWE cryptosystem, this work does not break Ring-LWE, since the security reduction is from the worst case ideal SVP to the average case Ring-LWE, and it is one-way.