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{{technical|date=September 2015}}
'''Ring Learning with Errors (RLWE)''' is a [[computational problem]] which serves as the foundation of new cryptographic [[algorithm]]s designed to protect against [[cryptanalysis]] by [[quantum computers]] and also to provide the basis for [[homomorphic encryption]]. RLWE is more properly called Learning with Errors over Rings and is simply the larger [[Learning with errors|Learning with Errors]] problem specialized to [[polynomial rings]] over finite fields.<ref name=":0" /> Because of the presumed difficulty of solving the RLWE problem even on a quantum computer, RLWE based cryptography may form the fundamental base for [[Public-key cryptography|public key cryptography]] in the future just as the [[integer factorization]] and [[
== Background ==
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The RLWE problem can be stated in two different ways. One is called the "Search" version and the other is the "Decision" version. The Search can be stated as follows. Let
* <math>a_i(x)</math> be a set of random but '''known''' polynomials from <math>\mathbf{
* <math>e_i(x)</math> be a set of small random and '''unknown''' polynomials relative to a bound <math>b</math> in the ring <math>\mathbf{
* <math>s(x)</math> be a small '''unknown''' polynomial relative to a bound <math>b</math> in the ring <math>\mathbf{
* <math>b_i(x) = (a_i(x)\cdot s(x)) + e_i(x)</math>
Given the list of polynomial pairs <math>( a_i(x), b_i(x) )</math> find the unknown polynomial <math>s(x)</math>
Using the same definitions, the Decision version of the problem can be stated as follows. Given a list of polynomial pairs <math>( a_i(x), b_i(x) )</math> determine whether the <math>b_i(x)</math> polynomials were constructed as <math>b_i(x) = (a_i(x)\cdot s(x)) + e_i(x)</math> or were generated randomly from <math>\mathbf{
The difficulty of this problem is parameterized by the choice of the quotient polynomial (<math>\Phi(x)</math>), its degree (<math>n</math>), the field (<math>\mathbf{F}_q</math>), and the smallness bound (<math>b</math>). In many RLWE based public key algorithms the private key will be a pair of small polynomials <math>s(x)</math> and <math>e(x)</math>. The corresponding public key will be a pair of polynomials <math>a(x)</math>, selected randomly from <math>\mathbf{
== Security Reduction ==
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:''"... we give a quantum reduction from approximate SVP (in the worst case) on ideal lattices in <math>\mathbf{R}</math> to the search version of ring-LWE, where the goal is to recover the secret <math>s \in \mathbf{R}_q</math> (with high probability, for any <math>s</math>) from arbitrarily many noisy products."''<ref name=":0" />
In that quote, The ring <math>\mathbf{R}</math> is <math>\mathbf{Z}[x]/\Phi(x)</math> and the ring <math>\mathbf{R}_q</math> is <math>\mathbf{
The α-SVP in regular lattices is known to be [[NP-hard]] due to work by Daniele Micciancio in 2001.<ref name=":1">{{Cite journal|title = The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant|url = http://epubs.siam.org/doi/abs/10.1137/S0097539700373039|journal = SIAM Journal on Computing|date = January 1, 2001|issn = 0097-5397|pages = 2008–2035|volume = 30|issue = 6|doi = 10.1137/S0097539700373039|first = D.|last = Micciancio}}</ref> However, there is not yet a proof to show that the difficulty of the α-SVP for ideal lattices is equivalent to the average α-SVP. Rather we have a proof that if there are ''any'' α-SVP instances that are hard to solve in ideal lattices then the RLWE Problem will be hard in random instances.<ref name=":0" />
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