Cryptology ePrint Archive: Report 2015/1132

Tighter Security for Efficient Lattice Cryptography via the Rényi Divergence of Optimized Orders

Katsuyuki Takashima and Atsushi Takayasu

Abstract: In security proofs of lattice based cryptography, bounding the closeness of two probability distributions is an important procedure. To measure the closeness, the Rényi divergence has been used instead of the classical statistical distance. Recent results have shown that the Rényi divergence offers security reductions with better parameters, e.g. smaller deviations for discrete Gaussian distributions. However, since previous analyses used a fixed order Rényi divergence, i.e., order two, they lost tightness of reductions. To overcome the deficiency, we adaptively optimize the orders based on the advantages of the adversary for several lattice-based schemes. The optimizations enable us to prove the security with both improved efficiency and tighter reductions. Indeed, our analysis offers security reductions with smaller parameters than the statistical distance based analysis and the reductions are tighter than those of previous Rényi divergence based analyses. As applications, we show tighter security reductions for sampling discrete Gaussian distributions with smaller precomputed tables for Bimodal Lattice Signature Scheme (BLISS), and the variants of learning with errors (LWE) problem and the small integer solution (SIS) problem called k-LWE and k-SIS, respectively.

Category / Keywords: public-key cryptography / lattice based cryptography, tight reduction, R\'{e}nyi divergence, sampling discrete Gaussian, BLISS, LWE, SIS

Original Publication (with minor differences): ProvSec 2015

Date: received 23 Nov 2015, last revised 26 Nov 2015

Contact author: a-takayasu at it k u-tokyo ac jp

Available format(s): PDF | BibTeX Citation

Version: 20151127:052758 (All versions of this report)

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