Short generators without quantum computers: the case of multiquadratics.

Finding a short element g of a number field, given the ideal generated by g, is a classic problem in computational algebraic number theory. Solving this problem recovers the private key in cryptosystems introduced by Gentry, Smart-Vercauteren, Gentry-Halevi, Garg-Gentry-Halevi, et al. Work over the last few years has shown that for some number fields this problem has a surprisingly low post-quantum security level. This paper shows, and experimentally verifies, that for some number fields this problem has a surprisingly low pre-quantum security level.