## Cryptology ePrint Archive: Report 2021/277

On the Integer Polynomial Learning with Errors Problem

Julien Devevey and Amin Sakzad and Damien Stehlé and Ron Steinfeld

Abstract: Several recent proposals of efficient public-key encryption are based on variants of the polynomial learning with errors problem ($\mathsf{PLWE}^f$) in which the underlying polynomial ring $\mathbb{Z}_q[x]/f$ \ is replaced with the (related) modular integer ring $\mathbb{Z}_{f(q)}$; the corresponding problem is known as Integer Polynomial Learning with Errors ($\mathsf{I-PLWE}^f$). Cryptosystems based on $\mathsf{I-PLWE}^f$ and its variants can exploit optimised big-integer arithmetic to achieve good practical performance, as exhibited by the $\mathsf{ThreeBears}$ cryptosystem. Unfortunately, the average-case hardness of $\mathsf{I-PLWE}^f$ and its relation to more established lattice problems have to date remained unclear.

We describe the first polynomial-time average-case reductions for the search variant of $\mathsf{I-PLWE}^f$, proving its computational equivalence with the search variant of its counterpart problem $\mathsf{PLWE}^f$. Our reductions apply to a large class of defining polynomials $f$. To obtain our results, we employ a careful adaptation of Rényi divergence analysis techniques to bound the impact of the integer ring arithmetic carries on the error distributions. As an application, we present a deterministic public-key cryptosystem over integer rings. Our cryptosystem, which resembles $\mathsf{ThreeBears}$, enjoys one-way (OW-CPA) security provably based on the search variant of $\mathsf{I-PLWE}^f$.

Category / Keywords: foundations / lattices, ring-LWE, average-case reduction, OW-CPA, I-RLWE

Original Publication (with major differences): IACR-PKC-2021

Date: received 4 Mar 2021

Contact author: julien devevey at ens-lyon fr, amin sakzad at monash edu, damien stehle at ens-lyon fr, ron steinfeld at monash edu

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/277

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