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Ring learning with errors
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<p>In <a href="/facts/Post-quantum_cryptography/1qeqUQve">post-quantum cryptography</a>, ring learning with errors (RLWE) is a <a href="/facts/Computational_problem/jQ7bYUVh">computational problem</a> which serves as the foundation of new cryptographic <a href="/facts/Algorithm/fnl5NmRt">algorithms</a>, such as <a href="/facts/NewHope/B185hwvB">NewHope</a>, designed to protect against <a href="/facts/Cryptanalysis/7CPEQlJI">cryptanalysis</a> by <a href="/facts/Quantum_computers/nbbcgmvq">quantum computers</a> and also to provide the basis for <a href="/facts/Homomorphic_encryption/pbfGnqBn">homomorphic encryption</a>. <a href="/facts/Public-key_cryptography/gowoFTxS">Public-key cryptography</a> relies on construction of mathematical problems that are believed to be hard to solve if no further information is available, but are easy to solve if some information used in the problem construction is known. Some problems of this sort that are currently used in cryptography are at risk of attack if sufficiently large quantum computers can ever be built, so resistant problems are sought. </p><p>RLWE is more properly called <i>learning with errors over rings</i> and is simply the larger <a href="/facts/Learning_with_errors/rUoqf17k">learning with errors</a> (LWE) problem specialized to <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial rings</a> over finite fields. Because of the presumed difficulty of solving the RLWE problem even on a quantum computer, RLWE based cryptography may form the fundamental base for <a href="/facts/Public-key_cryptography/gowoFTxS">public-key cryptography</a> in the future just as the <a href="/facts/Integer_factorization/EjMCTz2t">integer factorization</a> and <a href="/facts/Discrete_logarithm/k8TXh5bL">discrete logarithm</a> problem have served as the base for public key cryptography since the early 1980s. An important feature of basing cryptography on the ring learning with errors problem is the fact that the solution to the RLWE problem can be used to solve a version of the <a href="/facts/Shortest_vector_problem/YHXNNrrG">shortest vector problem</a> (SVP) in a lattice (a polynomial-time reduction from this SVP problem to the RLWE problem has been presented). </p>