Starting with a [[Prime number|prime]] integer q, the [[Ring learning with errors|Ring-LWE]] key exchange works in the [[ring of polynomials]] modulo a polynomial <math>\Phi(x)</math> with coefficients in the field of integers mod q (i.e. the ring <math>R_q := Z_q[x] / \Phi(x)</math>). Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod <math>\Phi(x)</math>.
The idea of using LWE and Ring LWE for key exchange was first proposed and filed at the University of Cincinnati in 2011 by Jintai Ding. The idea comes from the associativity of matrix multiplications, and the errors are used to provide the security. The paper<ref name=":0">{{Cite journalbook|last=Lin|first=Jintai Ding, Xiang Xie, Xiaodong|dateurl=https://eprint.iacr.org/2012-01-01/688.pdf|title=A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem|urllast=Ding|first=Jintai|last2=Xie|first2=Xiang|last3=Lin|first3=Xiaodong|publisher=|year=https://eprint.iacr.org/2012/688|isbn=|___location=|pages=|via=}}</ref> appeared in 2012 after a provisional patent application was filed in 2012. The security of the protocol is proven based on the hardness of solving the LWE problem. In 2014, Peikert<ref>{{Cite journal|last=Peikert|first=Chris|date=2014-01-01|title=Lattice Cryptography for the Internet|url=http://eprint.iacr.org/2014/070|journal=|volume=|issue=|doi=|pmid=|access-date=|via=}}</ref> presented a key transport scheme based on Ring-LWE. For somewhat greater than 128 [[bits of security]], Singh presents a set of parameters which have 6956-bit public keys for the Peikert's scheme.<ref name=":1">{{Cite journal|last=Singh|first=Vikram|date=2015|title=A Practical Key Exchange for the Internet using Lattice Cryptography|url=http://eprint.iacr.org/2015/138}}</ref> The corresponding private key would be roughly 14,000 bits. An RLWE version of the classic MQV variant of a Diffie–Hellman key exchange was later published by Zhang et al. in 2014. The security of both key exchanges is directly related to the problem of finding approximate short vectors in an ideal lattice. This article will closely follow the RLWE work of Ding in "A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem".<ref name=":0">{{Cite book|url=https://eprint.iacr.org/2012/688.pdf|title=A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem|last=Ding|first=Jintai|last2=Xie|first2=Xiang|last3=Lin|first3=Xiaodong|publisher=|year=2012|isbn=|___location=|pages=|via=}}</ref> For this presentation a typical polynomial is expressed as:
