Ring learning with errors signature: Difference between revisions

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One of the most widely used public key algorithm used to create [[digital signatures]] is known as [[RSA]].<ref>{{Cite journal|title = RSA Cryptosystem|url = https://en.wikipedia.org/wiki/RSA_%2528cryptosystem%2529}}</ref> Its security is based on the classical difficulty of factoring the product of two large and unknown primes into the constituent primes. The [[integer factorization problem]] is believed to be intractable on any conventional computer if the primes are chosen at random and are sufficiently large.<ref>{{Cite journal|title = Integer factorization records|url = https://en.wikipedia.org/w/index.php?title=Integer_factorization_records&oldid=669003206|date = 2015}}</ref> However, to factor the product of two n-bit primes, a quantum computer with roughly 6n bits of logical [[qubit]] memory and capable of executing a program known as [[Shor's algorithm|Shor’s algorithm]] will easily accomplish the task.<ref>{{Cite journal|title = Oversimplifying quantum factoring|url = http://www.nature.com/nature/journal/v499/n7457/full/nature12290.html|journal = Nature|date = July 11, 2013|issn = 0028-0836|pages = 163–165|volume = 499|issue = 7457|doi = 10.1038/nature12290|first = John A.|last = Smolin|first2 = Graeme|last2 = Smith|first3 = Alexander|last3 = Vargo}}</ref> Shor's algorithm can also quickly break digital signatures based on what is known as the [[discrete logarithm]] problem and the more esoteric [[Elliptic curve cryptography|elliptic curve discrete logarithm]] problem.<ref name=":3" /> In effect, a relatively small quantum computer running Shor's algorithm could quickly break all of the digital signatures used to ensure the privacy and integrity of information on the internet today.<ref name=":3" />

Even though we do not know when a quantum computer to break RSA and other digital signature algorithms will exist, there has been active research over the past decade to create cryptographic algorithms which remain secure even when an attacker has the resources of a quantum computer at their disposal.<ref name=":2" /><ref name=":4">{{Cite web|title = Introduction|url = http://pqcrypto.org/|website = pqcrypto.org|accessdate = 2015-07-05}}</ref><ref>{{Cite journal|title = Post-quantum cryptography|url = https://en.wikipedia.org/w/index.php?title=Post-quantum_cryptography&oldid=668208432|date = 2015}}</ref> This new area of cryptography is called [[Post-quantum cryptography|Post Quantum]] or [[Quantum Safe Cryptography|Quantum Safe]] cryptography.<ref name=":2" /><ref name=":4" /> This article is about one class of these algorithms: digital signatures based on the Ring [[LearningRing learning with errors signature|Ring Learning with Errors]] problem. The use of this problem in cryptography was introduced by Oded Regev in 2005 and has been the source of several cryptographic designs.<ref>{{Cite web|title = http://www.cims.nyu.edu/~regev/papers/lwesurvey.pdf|url = http://www.cims.nyu.edu/~regev/papers/lwesurvey.pdf|website = www.cims.nyu.edu|accessdate = 2015-05-24}}</ref>

The creators of the [[Ring Learning with Errors|RLWE]] basis for cryptography believe that an important feature of these algorithms based on Ring-Learning with Errors is their provable reduction to known hard problems.<ref>{{Cite journal|title = On ideal lattices and learning with errors over rings|url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.297.6108|publisher = Springer|journal = In Proc. of EUROCRYPT, volume 6110 of LNCS|date = 2010|pages = 1–23|first = Vadim|last = Lyubashevsky|first2 = Chris|last2 = Peikert|first3 = Oded|last3 = Regev}}</ref><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 = 2015-07-05}}</ref> The signature described below has a provable reduction to the [[Shortest vector problem|Shortest Vector Problem]] in an [[Ideal lattice cryptography|ideal lattice]].<ref name=":0" /> This means that if an attack can be found on the Ring-LWE cryptosystem then a whole class of presumed hard computational problems will have a solution.<ref>{{Cite journal|title = The shortest vector in a lattice is hard to approximate to within some constant|url = http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.7305|journal = in Proc. 39th Symposium on Foundations of Computer Science|date = 1998|pages = 92–98|first = Daniele|last = Micciancio}}</ref>

The first RLWE-SIG was developed by Lyubashevsky in his paper "Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures"<ref name=":5">{{Cite book|title = Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures|url = http://link.springer.com/chapter/10.1007/978-3-642-10366-7_35|publisher = Springer Berlin Heidelberg|date = 2009-01-01|isbn = 978-3-642-10365-0|pages = 598-616|series = Lecture Notes in Computer Science|language = en|first = Vadim|last = Lyubashevsky|editor-first = Mitsuru|editor-last = Matsui}}</ref> and refined in "Lattice Signatures Without Trapdoors" in 2011.<ref name=":1">{{Cite journal|title = Lattice Signatures Without Trapdoors|url = http://eprint.iacr.org/2011/537|date = 2011|first = Vadim|last = Lyubashevsky}}</ref> A number of refinements and variants have followed. This article highlights some features of a RLWE-SIG which closely follows the original Lyubashevsky work and is due to the work of Guneysu, Lyubashevsky and Popplemann ([http://www.di.ens.fr/~lyubash/papers/signaturechess.pdf GLP]).<ref name=":0" /> A later section of this article provides references to other signatures based on Lyubashevsky's original work.

A RLWE-SIG works in the quotient [[ring of polynomials]] modulo a degree n polynomial Φ(x) with coefficients in the [[finite field]] F_q for an odd prime q ( i.e. the ring F_q[x]/Φ(x) ).<ref name=":1" /> Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod Φ(x). For this presentation a typical polynomial is expressed as:

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<math>a(x) = a_0 + a_1x + a_{2}x^2 + \ldots + a_{n-3}x^{n-3} + a_{n-2}x^{n-2} + a_{n-1}x^{n-1}</math>

The field F_q has its representative elements in the set { -(q-1)/2, ...-1, 0, 1, ... (q-1)/2 }. The polynomial Φ(x) will be the cyclotomic polynomial xⁿ + 1.

=== Generating "Small" Polynomials. ===

A RLWE-SIG uses polynomials which are considered "small" with respect to a measure called the "[[infinity norm]]". The [[infinity norm]] for a polynomial is simply the largest absolute value of the coefficients of the polynomial .<ref name=":0" /> The signature algorithm will create random polynomials which are small with respect to a particular infinity norm bound. This is easily done by randomly generating all of the [[Coefficient|coefficients of the polynomial]] (a₀, ..., a_n-1) in a manner which is guaranteed or very likely to be less than or equal to this bound. In the literature on Ring Learning with Errors, there are two common ways to do this:<ref name=":1" />

* Using [[Uniform distribution (discrete)|Uniform Sampling]] - The coefficients of the small polynomial are uniformly sampled from a set of small coefficients. Let b be an integer that is much less than q. If we randomly choose polynomial coefficients from the set: { -b, -b+1, -b+2. ... -2, -1, 0, 1, 2, ... , b-2, b-1, b} the infinity norm of the polynomial will be ≤ (b).

* Using Discrete Gaussian Sampling - For an odd integer q, the coefficients are randomly chosen by sampling from the set { -(q-1)/2 to (q-1)/2 } according to a discrete Gaussian distribution with mean 0 and distribution parameter σ. The references provide more details on this method.

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=== Rejection Sampling ===

An key feature of RLWE-SIG algorithms is the use of a technique known as [[rejection sampling]].<ref name=":1" /><ref name=":5" /> In this technique, if the [[infinity norm]] of a signature polynomial exceeds a fixed bound, '''β,''' that polynomial will be discarded and the signing process will begin again. This process will be repeated until the infinity norm of the signature polynomial is less than or equal to the bound. Rejection sampling ensures that the output signature is not exploitably correlated with the signer's secret key values.

In the example which follows, the bound, '''β,''' will be (b - k), where b is the range of the uniform sampling described above.<ref name=":0" />

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Following GLP and as noted above, the maximum degree of the polynomials will be n-1 and therefore have n coefficients.<ref name=":0" /> Typical values for n are 512, and 1024.<ref name=":0" /> The coefficients of these polynomials will be from the field F_q where q is an odd prime congruent to 1 mod 4. For n =512 the authors of GLP set q to be a 22 bit prime and the corresponding b value to be 2¹⁴. For n=1024, GLP sets q to be a 23-bit prime and b to be 2¹⁵.<ref name=":0" /> The number of non-zero coefficients produced by the hash function (k) is equal to 32 for both cases.<ref name=":0" /> The security of the signature scheme is closely tied to the relative sizes of n, q, b, and k. Details on setting these parameters can be found in references 5 and 6 below.<ref name=":1" /><ref name=":0" />

TheAs noted above, the polynomial Φ(x) which defines the ring of polynomials used will be xⁿ + 1. Finally, a(x) will be a randomly chosen and fixed polynomial with coefficients from the set { -(q-1)/2 to (q-1)/2 }. All signers and verifiers of signatures will know n, q, b, k, Φ(x), a(x) and '''β''' = b-k

== Public Key Generation ==

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This brief derivation demonstrates that the verification process will have c'(x) = c(x) if the signature was not tampered with.

== Other Approaches ==

There are a number of other cryptographic signatures based on lattices over polynomial rings though not specifically in the Ring Learning With Errors construct explained by Lyubashevsky in his "Lattice Signatures without Trapdoors".<ref name=":1" /> Among these are the [https://eprint.iacr.org/2013/838.pdf Learning with Errors Approach] by Bai and Galbraith (which has a Ring-LWE variant) and the Trapdoor Lattice approach by Ducas, Durmas, Lepoint and Lyubashevsky.<ref>{{Cite journal|title = An improved compression technique for signatures based on learning with errors|url = http://eprint.iacr.org/2013/838|date = 2013|first = Shi|last = Bai|first2 = Steven D.|last2 = Galbraith}}</ref><ref>{{Cite journal|title = Lattice Signatures and Bimodal Gaussians|url = http://eprint.iacr.org/2013/383|date = 2013|first = Léo|last = Ducas|first2 = Alain|last2 = Durmus|first3 = Tancrède|last3 = Lepoint|first4 = Vadim|last4 = Lyubashevsky}}</ref>

== Further reading ==