Ring learning with errors: Difference between revisions

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'''Ring Learning with Errors (RLWE)''' is a [[computational problem]] which serves as the foundation of new cryptographic algorithms 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 [[discrete logarithm]] problem have served as the base for public key cryptography since the early 1980's.<ref name=":2">{{Cite book|title = Lattice Cryptography for the Internet|url = http://link.springer.com/chapter/10.1007/978-3-319-11659-4_12|publisher = Springer International Publishing|isbn = 978-3-319-11658-7|pages = 197-219197–219|series = Lecture Notes in Computer Science|language = en|first = Chris|last = Peikert|editor-first = Michele|editor-last = Mosca}}</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|NP-Hard]] [[Shortest vector problem|Shortest Vector Problem]] (SVP) in a Lattice.<ref name=":0" />

== Background ==

The security of modern cryptography, in particular [[Public-key cryptography|Public Key Cryptography]], is based on the difficulty of solving computational problems believed to be intractable if the size of the problem is large enough and the instance of the problem to be solved is chosen randomly.<ref>{{Cite journal|title = Public-key cryptography|url = https://en.wikipedia.org/w/index.php?title=Public-key_cryptography&oldid=670016308|date = 2015|language = en}}</ref> The classic example that has been used since the 1970's is the [[integer factorization]] problem.<ref>{{Cite journal|title = RSA (cryptosystem)|url = https://en.wikipedia.org/w/index.php?title=RSA_(cryptosystem)&oldid=669826149|date = 2015|language = en}}</ref> It is believed that it is computationally intractable to factor the product of two prime numbers if those prime numbers are large enough and chosen at random. As of 2015 research has led to the factorization of the product of two 384-bit primes but not the product of two 512-bit primes. Integer factorization forms the basis of the widely used [[RSA (cryptosystem)|RSA]] cryptographic algorithm.<ref>{{Cite journal|title = Integer factorization records|url = https://en.wikipedia.org/w/index.php?title=Integer_factorization_records&oldid=669003206|date = 2015|language = en}}</ref>

The Ring Learning with Errors (RLWE) problem is built on the arithmetic of [[polynomials]] with coefficients from a [[finite field]].<ref name=":0" /> A typical polynomial a(x) is expressed as:

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<math>a(x) = a_0 + a_1x + a_2x^2 + ... + a_{n-2}x^{n-2} + a_{n-1}x^{n-1}</math>

Polynomials can be added and multiplied in the usual fashion. In the RLWE context the coefficients of the polynomials and all operations involving those coefficients will be done in a finite field, typically the field <math display="inline">Z/qZ = F_q</math> for a prime integer <math display="inline">q</math>. The set of polynomials over a finite field with the operations of addition and multiplication forms an infinite [[polynomial ring]] (<math>F_q[x]</math>).<ref>{{Cite journal|title = Polynomial ring|url = https://en.wikipedia.org/w/index.php?title=Polynomial_ring&oldid=661646453|date = 2015|language = en}}</ref> The RLWE context works with a finite sub-ring of this infinite ring. The sub-ring is typically the finite [[Quotient ring|quotient (factor) ring]] formed by reducing all of the polynomials in <math>F_q[x]</math> modulo an [[irreducible polynomial]] <math>\Phi(x)</math>. This finite quotient ring can be written as <math>F_q[x]/\Phi(x)</math> though many authors write <math>Z_q[x]/\Phi(x)</math> .<ref name=":0" />

If the degree polynomial <math>\Phi(x)</math> is n, the sub-ring becomes the ring of polynomials of degree less than n modulo <math>\Phi(x)</math> with coefficients from <math>F_q</math>. The values n, q, together with the polynomial <math>\Phi(x)</math> partially define the mathematical context for the RLWE problem.

Another concept necessary for the RLWE problem is the idea of "small" polynomials with respect to some norm. The typical norm used in the RLWE problem is known as the [[infinity norm]].<ref>{{Cite journal|title = Norm (mathematics)|url = https://en.wikipedia.org/w/index.php?title=Norm_(mathematics)&oldid=667112150|date = 2015|language = en}}</ref> The infinity norm of a polynomial is simply the largest coefficient of the polynomial when these coefficients are viewed as integers. Hence, <math>||a(x)||_\infty = b</math> states that the [[infinity norm]] of the polynomial <math>a(x)</math> is <math>b</math>. Thus <math>b</math> is the largest coefficient of <math>a(x)</math>.

The final concept necessary to understand the RLWE problem is the generation of random polynomials in <math>Z_q[x]/\Phi(x)</math> . This is easily done by simply randomly sampling the n coefficients of the polynomial from <math>F_q</math> where <math>F_q</math> is typically represented as the set:<math>(-(q-1)/2, ..., -1, 0, 1, ..., (q-1)/2)</math>.

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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>Z_q[x]/\Phi(x)</math> with coefficients from all of <math>F_q</math>.

The difficulty of this problem is parameterized by the choice of the quotient polynomial ( <math>\Phi(x)</math> ), its degree (n), the field (<math>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>Z_q[x]/\Phi(x)</math>, and the polynomial <math>t(x)= (a(x)\cdot s(x)) + e(x)</math>. Given <math>a(x)</math> and <math>t(x)</math>, it should be computationally infeasible to recover the polynomial <math>s(x)</math>

== Security Reduction ==

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In that quote, The ring R is <math>Z[x]/\Phi(x)</math> and the ring R_q is <math>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.<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 = 2008-20352008–2035|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" />

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>

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>

Peikert believes that these security equivalences make the RLWE problem a good basis for future cryptography. He writes: ''"There is a mathematical proof that the''  ''only''  ''way to break the cryptosystem (within some formal attack model) on its random instances is by being able to solve the underlying lattice problem in the''  ''worst case"'' (emphasis in the original).<ref>{{Cite web|title = What does GCHQ's “cautionary"cautionary tale”tale" mean for lattice cryptography?|url = http://www.cc.gatech.edu/~cpeikert/soliloquy.html|website = www.cc.gatech.edu|accessdate = 2015-07-03}}</ref>

== RLWE Cryptography  ==

A major advantage that RLWE based cryptography has over the original [[Learning with errors|Learning With Errors]] (LWE) based cryptography is found in the size of the public and private keys. RLWE keys are roughly the square root of keys in LWE.<ref name=":0" /> For 128-bits of security an RLWE cryptographic algorithm would use public keys around 7000 bits in length.<ref>{{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> The corresponding LWE scheme would require public keys of 49 million bits for the same level of security.<ref name=":0" /> On the other hand, RLWE keys are larger than the keys sizes for currently used public key algorithms like RSA and Elliptic Curve Diffie-Hellman which require public key sizes of 3072 bits and 256 bits, respectively, to achieve a 128-bit level of security.<ref>{{Cite journal|title = Key size|url = https://en.wikipedia.org/w/index.php?title=Key_size&oldid=668301843|date = 2015|language = en}}</ref> From a computational standpoint, however, RLWE algorithms have been shown to be the equal of or better than existing public key systems.<ref>{{Cite journal|title = Efficient Software Implementation of Ring-LWE Encryption|url = http://eprint.iacr.org/2014/725|date = 2014|first = Ruan de Clercq, Sujoy Sinha Roy, Frederik Vercauteren, Ingrid|last = Verbauwhede}}</ref>

Three groups of RLWE cryptographic algorithms exist:

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=== [[Ring learning with errors signature|Ring Learning with Errors Signatures]] (RLWE-SIG) ===

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 = http://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 = 530-547530–547|series = Lecture Notes in Computer Science|language = en|first = Tim|last = Güneysu|first2 = Vadim|last2 = Lyubashevsky|first3 = Thomas|last3 = Pöppelmann|editor-first = Emmanuel|editor-last = Prouff|editor-first2 = Patrick|editor-last2 = Schaumont}}</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>

=== [[Homomorphic encryption|Ring Learning with Errors Homomorphic Encryption]] (RLWE-HOM) ===

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.<ref>{{Cite journal|title = Homomorphic encryption|url = https://en.wikipedia.org/w/index.php?title=Homomorphic_encryption&oldid=668290551|date = 2015-06-23|language = en}}</ref> 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 = http://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 = 505-524505–524|series = Lecture Notes in Computer Science|language = en|first = Zvika|last = Brakerski|first2 = Vinod|last2 = Vaikuntanathan|editor-first = Phillip|editor-last = Rogaway}}</ref>

=== References ===

{{Reflist}}

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