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Conference Papers Year : 2021

On the Integer Polynomial Learning with Errors Problem

Abstract

Several recent proposals of efficient public-key encryption are based on variants of the polynomial learning with errors problem (PLWE f ) in which the underlying polynomial ring $Zq[x]/f $ is replaced with the (related) modular integer ring $Zf(q)$ ; the corresponding problem is known as Integer Polynomial Learning with Errors (I-PLWE$^f$ ). Cryptosystems based on I-PLWE$^f$ and its variants can exploit optimised big-integer arithmetic to achieve good practical performance, as exhibited by the ThreeBears cryptosystem. Unfortunately, the average-case hardness of 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 I-PLWE$^f$ , proving its computational equivalence with the search variant of its counterpart problem 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 ThreeBears, enjoys one-way (OW-CPA) security provably based on the search variant of I-PLWE$^f $
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Dates and versions

hal-03475737 , version 1 (11-12-2021)

Identifiers

Cite

Julien Devevey, Amin Sakzad, Damien Stehlé, Ron Steinfeld. On the Integer Polynomial Learning with Errors Problem. PKC 2021 - 24th edition of the International Conference on Practice and Theory of Public-Key Cryptography, May 2021, Edinburgh, United Kingdom. pp.184-214, ⟨10.1007/978-3-030-75245-3_8⟩. ⟨hal-03475737⟩
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