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The Ring-LWE key exchange works in the ring of polynomials of degree n-1 or less modulo a polynomial F(x) (i.e. the ring Z<sub>q</sub>[x]/F(x) for an integer q). The coefficients of the polynomials in this ring are the integers mod q. Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod F(x). This article will closely follow the work of Peikert as further explained by Singh<ref name=":0">{{Cite journal|title = Lattice Cryptography for the Internet|url = http://eprint.iacr.org/2014/070|date = 2014|first = Chris|last = Peikert}}</ref><ref name=":1">{{Cite journal|title = A Practical Key Exchange for the Internet using Lattice Cryptography|url = http://eprint.iacr.org/2015/138|date = 2015|first = Vikram|last = Singh}}</ref>. For this presentation a typical polynomial is expressed as:
p(x) = p<sub>
This cryptographic algorithm uses polynomials which are considered "small" with respect to a measure called the "infinity norm." The infinity norm for a polynomial is simply the value of the largest coefficient of the polynomial when considered as integers in (Z vice Z/pZ). The algorithm will generate random polynomials which are small with respect to the infinity norm. We do this by randomly generating the coefficients of the polynomial (p<sub>n-1</sub>, ..., p<sub>0</sub>) which are guaranteed or very likely to be small. There are two common ways to do this:
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