Ring learning with errors: Difference between revisions

Revision as of 02:08, 5 July 2015

Revision as of 12:47, 5 July 2015

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In cases where the polynomial <math>\Phi(x)</math> is a cyclotomic polynomial, the difficulty of solving the search version of RLWE problem is equivalent to finding a short vector (but not necessarily the shortest) vector in an ideal lattice formed from elements of <math>Z[x]/\Phi(x)</math> represented as integer vectors.<ref name=":0">{{Cite journal|title = On Ideal Lattices and Learning with Errors Over Rings|url = http://eprint.iacr.org/2012/230|date = 2012|first = Vadim|last = Lyubashevsky|first2 = Chris|last2 = Peikert|first3 = Oded|last3 = Regev}}</ref> This problem is commonly known as the [[Shortest vector problem|Approximate Shortest Vector Problem (α-SVP)]] and it is the problem of finding a vector shorter than α times the shortest vector. The authors of the proof for this equivalence write:

''"... we give a quantum reduction from approximate SVP (in the worst case) on ideal lattices in R to the search version of ring-LWE, where the goal is to recover the secret s ∈ R_q (with high probability, for any s) from arbitrarily many noisy products."''<ref name=":0" />

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>.

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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" />

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>

ThesePeikert believes that these security equivalences make the RLWE problem a good basis for future cryptography. ChrisHe Peikert, one of the authors of the proof of this security reduction, states the followingwrites: ''"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 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:

While there are a large number of different RLWE cryptographic algorithms that have been invented since their first appearance in 2012 but three classes of algorithms are noteworthy:

==== [[Ring learning with errors key exchange|Ring Learning with Errors Key Exchanges]] (RLWE-KEX) ====

A RLWE version of the classic Diffie-Hellman key exchange was designed by Peikert and published in early 2014.<ref name=":2" /> An RLWE version of the classic MQV variant of a Diffie-Hellman key exchange was later published by Zhang et. al.<ref>{{Cite journal|title = Authenticated Key Exchange from Ideal Lattices|url = http://eprint.iacr.org/2014/589|date = 2014|first = Jiang|last = Zhang|first2 = Zhenfeng|last2 = Zhang|first3 = Jintai|last3 = Ding|first4 = Michael|last4 = Snook|first5 = Özgür|last5 = Dagdelen}}</ref> The security of both key exchanges is directly related to the problem of finding approximate short vectors in an ideal lattice.

==== [[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 = 530-547|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>