Ring learning with errors: Difference between revisions

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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]] (LWE) 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]] in the future just as the [[integer factorization]] and [[discrete logarithm]] problem have served as the base for public key cryptography since the early 1980s.<ref name=":2">{{Cite book|title = Lattice Cryptography for the Internet|url = https://link.springer.com/chapter/10.1007/978-3-319-11659-4_12|publisher = Springer International Publishing|isbn = 978-3-319-11658-7|pages = 197–219|series = Lecture Notes in Computer Science|first = Chris|last = Peikert|editor-first = Michele|editor-last = Mosca|doi = 10.1007/978-3-319-11659-4_12|title = Post-Quantum Cryptography|volume = 8772|year = 2014|chapter = Lattice Cryptography for the Internet|citeseerx = 10.1.1.800.4743}}</ref> An important feature of basing cryptography on the ring learning with errors problem is the fact that the solution to the RLWE problem may be reducible to the [[NP-hard]] [[shortest vector problem]] (SVP) in a lattice.<ref name=":0" />

== Background ==

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Randomly generating a "small" polynomial done by generating the coefficients of the polynomial from <math>\mathbf{F}_q</math> in a way that either guarantees or makes very likely small coefficients. There are two common ways to do this:

# Using Uniform Sampling – The coefficients of the small polynomial are uniformly sampled from a set of small coefficients. Let <math display="inline">b</math> be an integer that is much less than <math display="inline">q</math>. If we randomly choose coefficients from the set: <math display="inline">\{ -b, -b+1, -b+2, \ldots , -2, -1, 0, 1, 2, \ldots , b-2, b-1, b \}</math> the polynomial will be small with respect to the bound (<math display="inline">b</math>).

# Using [[Gaussian function|Discrete Gaussian Sampling]] – For an odd value for <math display="inline">q</math>, the coefficients of the polynomial are randomly chosen by sampling from the set <math display="inline"> \{ -(q-1)/2, \ldots , (q-1)/2 \} </math> according to a discrete Gaussian distribution with mean <math>0</math> and distribution parameter <math display="inline">\sigma</math>. The references describe in full detail how this can be accomplished. It is more complicated than uniform sampling but it allows for a proof of security of the algorithm. The paper "Sampling from Discrete Gaussians for Lattice-Based Cryptography on a Constrained Device" by Dwarakanath and Galbraith provide an overview of this problem.<ref>{{Cite journal|title = Sampling from discrete Gaussians for lattice-based cryptography on a constrained device|url = https://link.springer.com/article/10.1007/s00200-014-0218-3|journal = Applicable Algebra in Engineering, Communication and Computing|date = 2014-03-18|issn = 0938-1279|pages = 159–180|volume = 25|issue = 3|doi = 10.1007/s00200-014-0218-3|first = Nagarjun C.|last = Dwarakanath|first2 = Steven D.|last2 = Galbraith}}</ref>

== The RLWE Problem ==

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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{Z}_q[x]/\Phi(x)</math>.

The α-SVP in regular lattices is known to be [[NP-hard]] due to work by Daniele Micciancio in 2001, although not for values of α required for a reduction to general learning with errors problem.<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|citeseerx = 10.1.1.93.6646}}</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" />

Regarding the difficulty of Shortest Vector Problems in Ideal Lattices, researcher Michael Schneider writes, ''"So far there is no SVP algorithm making use of the special structure of ideal lattices. It is widely believed that solving SVP (and all other lattice problems) in ideal lattices is as hard as in regular lattices."''<ref>{{Cite journal|title = Sieving for Shortest Vectors in Ideal Lattices|url = http://eprint.iacr.org/2011/458|date = 2011|first = Michael|last = Schneider}}</ref> The difficulty of these problems on regular lattices is provably [[NP-hard]].<ref name=":1" /> There are, however, a minority of researchers who do not believe that ideal lattices share the same security properties as regular lattices.<ref>{{Cite web|title = cr.yp.to: 2014.02.13: A subfield-logarithm attack against ideal lattices|url = http://blog.cr.yp.to/20140213-ideal.html|website = blog.cr.yp.to|accessdate = 2015-07-03}}</ref>

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=== Ring learning with errors signature (RLWE-SIG) ===

{{main|Ring learning with errors signature}}

A RLWE version of the classic [[Feige–Fiat–Shamir identification scheme|Feige–Fiat–Shamir Identification protocol]] was created and converted to a digital signature in 2011 by Lyubashevsky.<ref>{{Cite journal|title = Lattice Signatures Without Trapdoors|url = http://eprint.iacr.org/2011/537|date = 2011|first = Vadim|last = Lyubashevsky}}</ref> The details of this signature were extended in 2012 by Gunesyu, Lyubashevsky, and Popplemann in 2012 and published in their paper "Practical Lattice Based Cryptography – A Signature Scheme for Embedded Systems."<ref>{{Cite book|title = Practical Lattice-Based Cryptography: A Signature Scheme for Embedded Systems|url = https://link.springer.com/chapter/10.1007/978-3-642-33027-8_31|publisher = Springer Berlin Heidelberg|date = 2012|isbn = 978-3-642-33026-1|pages = 530–547|series = Lecture Notes in Computer Science|first = Tim|last = Güneysu|first2 = Vadim|last2 = Lyubashevsky|first3 = Thomas|last3 = Pöppelmann|editor-first = Emmanuel|editor-last = Prouff|editor-first2 = Patrick|editor-last2 = Schaumont|doi = 10.1007/978-3-642-33027-8_31}}</ref> These papers laid the groundwork for a variety of recent signature algorithms some based directly on the ring learning with errors problem and some which are not tied to the same hard RLWE problems.<ref>{{Cite web|title = BLISS Signature Scheme|url = http://bliss.di.ens.fr/|website = bliss.di.ens.fr|accessdate = 2015-07-04}}</ref>

=== Ring learning with errors homomorphic encryption (RLWE-HOM) ===

{{main|Homomorphic encryption}}

The purpose of [[homomorphic encryption]] is to allow the computations on sensitive data to occur on computing devices that should not be trusted with the data. These computing devices are allowed to process the ciphertext which is output from a homomorphic encryption. In 2011, Brakersky and Vaikuntanathan, published "Fully Homomorphic Encryption from Ring-LWE and Security for Key Dependent Messages" which builds a homomorphic encryption scheme directly on the RLWE problem.<ref>{{Cite book|title = Fully Homomorphic Encryption from Ring-LWE and Security for Key Dependent Messages|url = https://link.springer.com/chapter/10.1007/978-3-642-22792-9_29|publisher = Springer Berlin Heidelberg|date = 2011|isbn = 978-3-642-22791-2|pages = 505–524|series = Lecture Notes in Computer Science|first = Zvika|last = Brakerski|first2 = Vinod|last2 = Vaikuntanathan|editor-first = Phillip|editor-last = Rogaway|doi = 10.1007/978-3-642-22792-9_29}}</ref>

The various sets of parameters that have been proposed by different groups of researchers for ring learning with errors key exchange and signatures are found at the ring learning with errors information site.<ref>{{Cite web