Ring learning with errors signature: Difference between revisions

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[[Digital signature]]s are a means to protect [[Digital data|digital information]] from intentional modification and to authenticate the source of digital information. [[Public key cryptography]] provides a rich set of different cryptographic algorithms the create digital signatures. However, the primary public key signatures currently in use ([[RSA (cryptosystem)|RSA]] and [[Elliptic Curve Digital Signature Algorithm|Elliptic Curve Signatures)]] will become completely insecure if scientists are ever able to build a moderately sized [[quantum computer]].<ref name=":2">{{Cite web|title = ETSI - Quantum-Safe Cryptography|url = http://www.etsi.org/technologies-clusters/technologies/quantum-safe-cryptography|website = ETSI|access-date = 2015-07-05|first = Sabine|last = Dahmen-Lhuissier}}</ref> [[Post-quantum cryptography|Post quantum cryptography]] is a class of cryptographic algorithms designed to be resistant to attack by a quantum cryptography. Several post quantum digital signature algorithms based on hard problems in lattices are being created replace the commonly used [[RSA (cryptosystem)|RSA]] and elliptic curve signatures. A subset of these lattice based scheme are based on a problem known as [[Ring learning with errors]]. Ring learning with errors based digital signatures are among the post quantum signatures with the smallest public key and signature sizes

== Background ==

Developments in [[quantum computing]] over the past decade and the optimistic prospects for real quantum computers within 20 years have begun to threaten the basic cryptography that secures the internet.<ref>{{Cite web|title = Quantum computing breakthrough claim from IBM|url = http://www.cio.co.uk/news/r-and-d/quantum-computing-breakthrough-claim-from-ibm-3609914/|access-date = 2015-06-01|first = Agam|last = Shah}}</ref><ref>{{Cite news|title = Researchers Report Milestone in Developing Quantum Computer|url = https://www.nytimes.com/2015/03/05/science/quantum-computing-nature-google-uc-santa-barbara.html|newspaper = The New York Times|date = 2015-03-04|access-date = 2015-07-05|issn = 0362-4331|first = John|last = Markoff}}</ref> A relatively small [[quantum computer]] capable of processing only ten thousand of bits of information would easily break all of the widely used [[Public-key cryptography|public key]] cryptography algorithms used to protect privacy and digitally sign information on the internet.<ref name=":2" /><ref>{{Cite journal|title = Efficient Networks for Quantum Factoring|journal = Physical Review A|date = 1996|issn = 1050-2947|pages = 1034–1063|volume = 54|issue = 2|doi = 10.1103/PhysRevA.54.1034|pmid = 9913575|first1 = David|last1 = Beckman|first2 = Amalavoyal N.|last2 = Chari|first3 = Srikrishna|last3 = Devabhaktuni|first4 = John|last4 = Preskill|arxiv = quant-ph/9602016|bibcode = 1996PhRvA..54.1034B|s2cid = 2231795}}</ref>

One of the most widely used public key algorithm used to create [[digital signatures]] is known as [[RSA (cryptosystem)|RSA]]. 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. 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|journal = Nature|date = July 11, 2013|issn = 0028-0836|pages = 163–165|volume = 499|issue = 7457|doi = 10.1038/nature12290|first1 = John A.|last1 = Smolin|first2 = Graeme|last2 = Smith|first3 = Alexander|last3 = Vargo|pmid=23846653|arxiv = 1301.7007|bibcode = 2013Natur.499..163S|s2cid = 4422110}}</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. 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.

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|access-date = 2015-07-05}}</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 Learning with Errors problem. The use of the general [[Learning with errors|Learning with Errors]] problem in cryptography was introduced by Oded Regev in 2005 and has been the source of several cryptographic designs.<ref>{{Cite web|title = The Learning with Errors Problem|url = http://www.cims.nyu.edu/~regev/papers/lwesurvey.pdf|website = www.cims.nyu.edu|access-date = 2015-05-24}}</ref>

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The first RLWE based signature was developed by Lyubashevsky in his paper "Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures"<ref name=":5">{{Cite book|publisher = Springer Berlin Heidelberg|date = 2009-01-01|isbn = 978-3-642-10365-0|pages = 598–616|series = Lecture Notes in Computer Science|first = Vadim|last = Lyubashevsky|editor-first = Mitsuru|editor-last = Matsui|doi = 10.1007/978-3-642-10366-7_35|title = Advances in Cryptology – ASIACRYPT 2009|volume = 5912|chapter = Fiat-Shamir with Aborts: Applications to Lattice and Factoring-Based Signatures}}</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 the fundamental mathematical structure of RLWE signatures and follows the original Lyubashevsky work and the work of Guneysu, Lyubashevsky and Popplemann ([https://web.archive.org/web/20140518004537/http://www.di.ens.fr/~lyubash/papers/signaturechess.pdf GLP]).<ref name=":0" /> This presentation is based on a 2017 update to the GLP scheme called GLYPH.<ref name=":3">{{Cite web|url=https://eprint.iacr.org/2017/766.pdf|title=GLYPH: A New Instantiation of the GLP Digital Signature Scheme|last=Chopra|first=Arjun|date=2017|website=International Association of Cryptographic Research eprint Archive|archive-url=https://eprint.iacr.org|archive-date=26 August 2017|access-date=26 August 2017}}</ref>

A RLWE-SIG works in the quotient [[ring of polynomials]] modulo a degree n polynomial Φ(x) with coefficients in the [[finite field]] Z_q for an odd prime q ( i.e. the ring Z_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:

<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>

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== Public key generation ==

An entity wishing to sign messages generates its public key through the following steps:

# Generate two small polynomials s(x) and e(x) with coefficients chosen uniformly from the set {-b,...-1, 0, 1, ..., b}

# Compute t(x) = a(x)·s(x) + e(x)

# Distribute t(x) as the entity's public key

The polynomials s(x) and e(x) serve as the private key and t(x) is the corresponding public key. The security of this signature scheme is based on the following problem. Given a polynomial t(x) find small polynomials f₁(x) and f₂(x) such that: a(x)·f₁(x) + f₂(x) = t(x)

If this problem is difficult to solve, then the signature scheme will be difficult to forge. [See the Wikipedia article on [[Ring Learning with Errors]] or [[Ideal lattice cryptography|Ideal Lattice Cryptography]] for more details on the theoretical difficulty of this problem]

== Signature generation ==

Following GLYPH,<ref name=":3" /> to sign a message m expressed as a bit string, the signing entity does the following:

# Generate two small polynomials y₁(x) and y₂(x) with coefficients from the set {-b, ..., 0, ...., b}

# Compute w(x) = a(x)·y₁(x) + y₂(x)

# Map w(x) into a bit string ω

# Compute c(x) = POLYHASH(ω | m) (This is a polynomial with k non-zero coefficients. The "|" denotes concatenation of strings)

# Compute z₁(x) = s(x)·c(x) + y₁(x)

# Compute z₂(x) = e(x)·c(x) + y₂(x)

# Until the infinity norms of z₁(x) and z₂(x) ≤ '''β = ('''B - k) go to step 1. (This is the rejection sampling step noted above)

# The signature is the triple of polynomials c(x), z₁(x) and z₂(x)

# Transmit the message along with c(x), z₁(x) and z₂(x) to the verifier.