Cryptology ePrint Archive: Report 2015/971

Attacks on the Search-RLWE problem with small error

Hao Chen and Kristin E. Lauter and Katherine E. Stange

Abstract: The Ring Learning-With-Errors (RLWE) problem shows great promise for post-quantum cryptography and homomorphic encryption. We describe a new attack on the non-dual search RLWE problem with small error widths, using ring homomorphisms to finite fields and the chi-squared statistical test. In particular, we identify a ``subfield vulnerability'' (Section 5.2) and give a new attack which finds this vulnerability by mapping to a finite field extension and detecting non-uniformity with respect to the number of elements in the subfield. We use this attack to give examples of vulnerable RLWE instances in Galois number fields. We also extend the well-known search-to-decision reduction result to Galois fields with any unramified prime modulus $q$, regardless of the residue degree $f$ of $q$, and we use this in our attacks. The time complexity of our attack is $O(n q^{2f})$, where $n$ is the degree of $K$ and $f$ is the {\it residue degree} of $q$ in $K$. We also show an attack on the non-dual (resp. dual) RLWE problem with narrow error distributions in prime cyclotomic rings when the modulus is a ramified prime (resp. any integer). We demonstrate the attacks in practice by finding many vulnerable instances and successfully attacking them. We include the code for all attacks.

Category / Keywords: attacks, RLWE, cryptanalysis

Original Publication (with minor differences): SIAM Journal on Applied Algebra and Geometry (SIAGA)

Date: received 8 Oct 2015, last revised 9 Oct 2017

Contact author: klauter at microsoft com

Available format(s): PDF | BibTeX Citation

Note: Revised version corresponding to updates made for final publication.

Version: 20171009:185818 (All versions of this report)

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