Ring learning with errors key exchange: Difference between revisions

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{{technical|date=June 2015}}

In [[cryptography]], a [[key exchange|public key exchange]] algorithm is a [[cryptographic algorithm]] which allows two parties to create and share a secret key, which they can use to encrypt messages between themselves. The '''[[ring learning with errors]] key exchange''' ('''RLWE-KEX''') is one of a new class of public key exchange algorithms that are designed to be secure against an adversary that possesses a [[quantum computer]]. This is important because some [[public key algorithm]]s in use today will be easily broken by a quantum computer if and when such computers are implemented. [[Ring learning with errors|RLWE]]-KEX is one of a set of [[post-quantum cryptography|post-quantum cryptographic]] algorithms which are based on the difficulty of solving certain mathematical problems involving [[lattice-based cryptography|lattices]]. Unlike older lattice based cryptographic algorithms, the [[ring learning with errors|RLWE]]-KEX is provably reducible to a known hard problem in lattices.

== Background ==

Since the 1980s the security of cryptographic [[key exchange]]s and [[digital signature]]s over the Internet has been primarily based on a small number of [[public key]] algorithms. The security of these algorithms is based on a similarly small number of computationally hard problems in classical computing. These problems are the difficulty of [[Integer factorization|factoring the product of two carefully chosen prime numbers]], the difficulty to compute [[discrete logarithms]] in a carefully chosen finite field, and the difficulty of computing discrete logarithms in a carefully chosen [[elliptic curve]] group. These problems are very difficult to solve on a classical computer (the type of computer the world has known since the 1940s through today) but are rather easily solved by a relatively small [[Quantum computing|quantum computer]] using only 5 to 10 thousand of bits of memory. There is optimism in the computer industry that larger scale quantum computers will be available around 2030. If a [[quantum computer]] of sufficient size were built, all of the public key algorithms based on these three classically hard problems would be insecure. This public key cryptography is used today to secure Internet websites, protect computer login information, and prevent our computers from accepting malicious software.

Cryptography that is not susceptible to attack by a quantum computer is referred to as [[post-quantum cryptography|quantum safe]], or [[post-quantum cryptography]]. One class of quantum resistant cryptographic algorithms is based on a concept called "[[learning with errors]]" introduced by [[Oded Regev (computer scientist)|Oded Regev]] in 2005.<ref name=":4">{{Cite book|title = On Lattices, Learning with Errors, Random Linear Codes, and Cryptography|publisher = ACM|journal = Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing|date = 2005|___location = New York, NY, USA|isbn = 978-1-58113-960-0|pages = 84–93|series = STOC '05|doi = 10.1145/1060590.1060603|first = Oded|last = Regev| title=Proceedings of the thirty-seventh annual ACM symposium on Theory of computing | chapter=On lattices, learning with errors, random linear codes, and cryptography |citeseerx = 10.1.1.110.4776|s2cid = 53223958}}</ref> A specialized form of Learning with errors operates within the [[polynomial ring|ring of polynomials]] over a [[finite field]]. This specialized form is called [[ring learning with errors]] or [[ideal lattice cryptography|RLWE]].

There are a variety of cryptographic algorithms which work using the RLWE paradigm. There are [[Public-key cryptography|public-key encryption]] algorithms, [[homomorphic encryption]] algorithms, and [[Ring learning with errors signature|RLWE digital signature]] algorithms in addition to the public key, key exchange algorithm presented in this article

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== Introduction ==

Starting with a [[Prime number|prime]] integer q, the [[fingRing learning with errors|Ring-LWE]] key exchange works in the [[ring of polynomials]] modulo a polynomial <math>\Phi(x)</math> with coefficients in the field of integers mod q (i.e. the ring <math>R_q := Z_q[x] / \Phi(x)</math>). Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod <math>\Phi(x)</math>.

InThe 2014,idea Peikert<ref>{{Citeof journal|last=Peikert|first=Chris|date=2014-01-01|title=Latticeusing CryptographyLWE forand theRing Internet|url=http://eprint.iacr.org/2014/070|journal=|volume=|issue=|doi=|pmid=|access-date=|via=}}</ref> presented a key transport scheme based on Ring-LWE. For somewhat greater than 128 [[bits of security]], Singh presents a set of parameters which have 6956-bit public keys for thekey Peikert'sexchange scheme.<refwas name=":1">{{Cite journal|last=Singh|first=Vikram|date=2015|title=A Practicalproposed Keyand Exchangefiled forat the Internet using Lattice Cryptography|url=http://eprint.iacr.org/2015/138}}</ref> The corresponding private key would be roughly 14,000 bits. An RLWE versionUniversity of theCincinnati classicin MQV variant of a Diffie–Hellman key exchange was later published2011 by ZhangJintai et al. in 2014Ding. The securityidea ofcomes both key exchanges is directly related tofrom the problemassociativity of findingmatrix approximatemultiplications, short vectors in an ideal lattice. This article will closely followand the RLWEerrors workare ofused Dingto in "A Simple Provably Secure Key Exchange Scheme Based onprovide the Learningsecurity. withThe Errors Problem".paper<ref name=":0">{{Cite book|url=https://eprint.iacr.org/2012/688.pdf|title=A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem|lastlast1=Ding|firstfirst1=Jintai|last2=Xie|first2=Xiang|last3=Lin|first3=Xiaodong|publisher=|year=2012|isbn=|___location=|pages=|via=}}</ref> appeared Forin this2012 presentationafter a typicalprovisional polynomialpatent application was filed in 2012. The security of the protocol is expressedproven as:based on the hardness of solving the LWE problem.

In 2014, Peikert presented a key-transport scheme<ref>{{Cite journal|last=Peikert|first=Chris|date=2014-01-01|title=Lattice Cryptography for the Internet|journal=Cryptology ePrint Archive |url=https://eprint.iacr.org/2014/070}}</ref> following the same basic idea of Ding's, where the new idea of sending an additional 1-bit signal for rounding in Ding's construction is also used.

The "New Hope" implementation<ref>{{Cite journal|last1=Alkim|first1=Erdem|last2=Ducas|first2=Léo|last3=Pöppelmann|first3=Thomas|last4=Schwabe|first4=Peter|date=2015-01-01|title=Post-quantum key exchange - a new hope|journal=Cryptology ePrint Archive |url=https://eprint.iacr.org/2015/1092}}</ref> selected for Google's post-quantum experiment,<ref>{{Cite news|url=https://security.googleblog.com/2016/07/experimenting-with-post-quantum.html|title=Experimenting with Post-Quantum Cryptography|newspaper=Google Online Security Blog|access-date=2017-02-08|language=en-US}}</ref> uses Peikert's scheme with variation in the error distribution.

For somewhat greater than 128 [[bits of security]], Singh presents a set of parameters which have 6956-bit public keys for the Peikert's scheme.<ref name=":1">{{Cite journal|last=Singh|first=Vikram|date=2015|title=A Practical Key Exchange for the Internet using Lattice Cryptography|journal=Cryptology ePrint Archive |url=http://eprint.iacr.org/2015/138}}</ref> The corresponding private key would be roughly 14,000 bits. An RLWE version of the classic MQV variant of a Diffie–Hellman key exchange was later published by Zhang et al. in 2014. The security of both key exchanges is directly related to the problem of finding approximate short vectors in an ideal lattice. This article will closely follow the RLWE work of Ding in "A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem".<ref name=":0">{{Cite book|url=https://eprint.iacr.org/2012/688.pdf|title=A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem|last1=Ding|first1=Jintai|last2=Xie|first2=Xiang|last3=Lin|first3=Xiaodong|year=2012}}</ref> For this presentation a typical polynomial is expressed as:

: <math> a(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n-3} x^{n-3} + a_{n-2} x^{n-2} + a_{n-1} x^{n-1} </math>

The coefficients <math>a_i</math> of this polynomial, the ''a''_''i''s, are integers&nbsp;mod&nbsp;''q''. The polynomial <math>\Phi(x)</math> will be the [[cyclotomic polynomial]]. When ''n'' is a power of 2 then <math>\Phi(x) = x^n +1.</math><ref name=":1" /><ref>{{Cite web|title = Cryptology ePrint Archive: Report 2015/1120|url = https://eprint.iacr.org/2015/1120|website = eprint.iacr.org|accessdateaccess-date = 2015-12-23}}</ref>

The RLWE-KEX uses polynomials which are considered "small" with respect to a measure called the "[[infinity norm]]." The infinity norm for a polynomial is simply the value of the largest coefficient of the polynomial when the coefficients are considered as integers in '''Z''' rather than <math>Zq</math> (i.e.from the set {−(''q''&nbsp;−&nbsp;1)/2,..., 0, ... (''q''&nbsp;−&nbsp;1)/2} ). The algorithm's security depends on an ability to generate random polynomials which are small with respect to the infinity norm. This is done simply by randomly generating the coefficients for a polynomial (s_n-1, ..., s₀) which are guaranteed or very likely to be small. There are two common ways to do this:

# 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 coefficients from the set: { −''b'', −b&nbsp;+&nbsp;1, −b&nbsp;+&nbsp;2. ... −2, −1, 0, 1, 2, ... , ''b''&nbsp;−&nbsp;2, ''b''&nbsp;−&nbsp;1, ''b''} the polynomial will be small with respect to the bound (b). Singh suggest using b = 5.<ref name=":1" /> Thus coefficients would be chosen from the set {''q''&nbsp;−&nbsp;5, ''q''&nbsp;−&nbsp;4, ''q''&nbsp;−&nbsp;3, ''q''&nbsp;−&nbsp;2, ''q''&nbsp;−&nbsp;1, 0, 1, 2, 3, 4, 5 }.

# Using [[Gaussian distribution|Discrete Gaussian]] Sampling – For an odd value for q, the coefficients are randomly chosen by sampling from the set { −(q&nbsp;−&nbsp;1)/2 to (''q''&nbsp;−&nbsp;1)/2 } according to a discrete Gaussian distribution with mean 0 and distribution parameter&nbsp;''σ''. 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. An overview of Gaussian sampling is found in a presentation by Peikert.<ref>{{Cite web|title = An Efficient and Parallel Gaussian Sampler for Lattices|url = https://web.eecs.umich.edu/~cpeikert/pubs/slides-pargauss.pdf|website = www.cc.gatech.edu|accessdateaccess-date = 2015-05-29}}</ref>

For the rest of this article, the random small polynomials will be sampled according to a distribution which is simply specified as '''D'''. Further q will be an odd prime such that q is congruent to 1 mod 4 and 1 mod 2n. Other cases for q and n are thoroughly discussed in "A Toolkit for Ring-LWE Cryptography" and in Singh's "Even More Practical Key Exchange for the Internet using Lattice Cryptography."<ref name=":2">{{Cite journal|lastlast1=Lyubashevsky|firstfirst1=Vadim|last2=Peikert|first2=Chris|last3=Regev|first3=Oded|date=2013|title=A Toolkit for Ring-LWE Cryptography|journal=Cryptology ePrint Archive |url=http://eprint.iacr.org/2013/293}}</ref><ref>{{Cite web|title = Cryptology ePrint Archive: Report 2015/1120|url = http://eprint.iacr.org/2015/1120|website = eprint.iacr.org|access-date = 2016-01-17}}</ref> and another paper by Singh. A fixed public polynomial, a(x), shared by all users of the network. It is deterministically generated from a cryptographically secure source.

Given ''a''(''x'') as stated, we can randomly choose small polynomials ''s''(''x'') and ''e''(''x'') to be the "private key" in a public key exchange. The corresponding public key will be the polynomial ''p''(''x'') = ''a''(''x'')''s''(''x'') + 2''e''(''x'').

== The key exchange ==

The key exchange will take place between two devices. There will be an initiator for the key exchange designated as (I) and a respondent designated as (R). Both I and R know ''q'', ''n'', ''a''(''x''), and have the ability to generate small polynomials according to the distribution <math>\chi_\alpha</math> with parameter <math>\alpha</math>. The distribution <math>\chi_\alpha</math> is usually the discrete Gaussian distribution on the ring <math> R_q = ZqZ_q[x]/\Phi(x)</math>. The description which follows does not contain any explanation of why the key exchange results in the same key at both ends of a link. Rather, it succinctly specifies the steps to be taken. For a thorough understanding of why the key exchange results in the initiator and responder having the same key, the reader should look at the referenced work by Ding et al.<ref name=":0" />

The key exchange begins with the initiator (I) doing the following:

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# Receive <math>p_R</math> and <math>w</math> from the Responder.

# Sample <math>e'_I</math> from <math>\chi_\alpha</math> and Compute <math>k_I = p_Rs_I + 2e'_I = as_Is_R + 2e_Rs_I + 2e'_I</math>.

# Initiator side's key stream is produced as <math>sk_I = \operatorname{Mod}_2(k_I,w)</math> from the reconciliation information <math>w</math> and polynomial <math>k_I</math>.

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== Parameter choices ==

The RWLERLWE-KEX exchange presented above worked in the Ring of Polynomials of degree ''n''&nbsp;−&nbsp;1 or less mod a polynomial <math>\Phi(x)</math>. The presentation assumed that n was a power of 2 and that q was a prime which was congruent to 1 (mod 2n). Following the guidance given in Peikert's paper, Singh suggested two sets of parameters for the RLWE-KEX.

For 128 bits of security, ''n'' = 512, ''q'' = 25601, and <math>\Phi(x) = x^{512} + 1</math>

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For 256 bits of security, ''n'' = 1024, ''q'' = 40961, and <math>\Phi(x) = x^{1024} + 1</math>

Because the key exchange uses random sampling and fixed bounds there is a small probability that the key exchange will fail to produce the same key for the initiator and responder. If we assume that the Gaussian parameter ''σ'' is <math display=inline>\frac{8/&radic;(2}{\sqrt{2\pi}})</math> and the uniform sampling bound (''b'') = 5 (see Singh),<ref name=":1" /> then the probability of key agreement failure is <u>less than</u> 2⁻⁷¹ for the 128-bit secure parameters and <u>less than</u> 2⁻⁹¹ for the 256-bit secure parameters.

In their November 2015 paper, Alkim, Ducas, PopplemannPöppelmann, and Schwabe recommend the following parameters n = 1024, q =12289, and <math>\Phi(x)</math> = x¹⁰²⁴ + 1.<ref name=":3" /> This represents a 70% reduction in public key size over the n = 1024 parameters of Singh, and was submitted to NIST's [[Post-Quantum Cryptography Standardization]] project under the name [[NewHope]]. A listing of a number of different parameter choices for key exchanges using the Ring Learning with Errors problem are given at this link ([http://www.ringlwe.info/parameters-for-rlwe.html click here]).<ref>{{Cite web

| url = http://www.ringlwe.info/parameters-for-rlwe.html

| title = Parameters for RLWE

| website = Ring Learning with Errors

| access-date = 2016-02-28

}}</ref>

Also in their November 2015 paper, Alkim, Ducas, PopplemannPöppelmann and Schwabe recommend that the choice of the base polynomial for the key exchange ( a(x) above ) be either generated randomly from a secure random number generator for each exchange or created in a verifiable fashion using a "nothing up my sleeve" or NUMS technique.<ref name=":3" /> An example of parameters generated in this way are the prime numbers for the Internet Key Exchange (<nowiki>RFC 2409</nowiki>) which embed the digits of the mathematical constant pi in the digital representation of the prime number.<ref>{{Cite webjournal|url=https://tools.ietf.org/html/rfc2409|title=The Internet Key Exchange (IKE)|lastlast1=D.|firstfirst1=Carrel|last2=D.|first2=Harkins|website=tools.ietf.org|date=November 1998 |language=en|access-date=2017-03-16}}</ref> Their first method prevents amortization of attack costs across many key exchanges at the risk of leaving open the possibility of a hidden attack like that described by Dan Bernstein against the NIST elliptic curves.<ref>{{Cite web|url=https://crypto.stackexchange.com/q/35488 |title=Is the "New Hope" Lattice Key Exchange vulnerable to a lattice analog of the Bernstein BADA55 Attack?|website=crypto.stackexchange.com|access-date=2017-03-16}}</ref> The NUMS approach is open to amortization but generally avoids the Bernstein attack if only common mathematical constants such as pi and e are used.

== Key exchange security ==

The security of this key exchange is based on the underlying hardness of [[ring learning with errors]] problem that has been proven to be as hard as the worst case solution to the [[shortest vector problem]] (SVP) in an [[ideal lattice cryptography|ideal lattice]].<ref name=":4" /><ref name=":0" /> The best method to gauge the practical security of a given set of lattice parameters is the BKZ 2.0 lattice reduction algorithm.<ref>{{Cite book|title = BKZ 2.0: Better Lattice Security Estimates|publisher = Springer Berlin Heidelberg|date = 2011|isbn = 978-3-642-25384-3|pages = 1–20|series = Lecture Notes in Computer Science|firstfirst1 = Yuanmi|lastlast1 = Chen|first2 = Phong Q.|last2 = Nguyen| title=Advances in Cryptology – ASIACRYPT 2011 | chapter=BKZ 2.0: Better Lattice Security Estimates | volume=7073 |editor-first = Dong Hoon|editor-last = Lee|editor-first2 = Xiaoyun|editor-last2 = Wang|doi = 10.1007/978-3-642-25385-0_1}}</ref> According to the BKZ 2.0 algorithm the key exchange parameters listed above will provide greater than 128 or 256 bits of security, respectively.

==Implementations==

In 2014 Douglas Stebila made [http://www.douglas.stebila.ca/research/papers/bcns15 a patch] for OpenSSL 1.0.1f. based on his work and others published in "Post-quantum key exchange for the TLS protocol from the ring learning with errors problem."<ref>{{Cite journal|title = Post-quantum key exchange for the TLS protocol from the ring learning with errors problem|url = http://eprint.iacr.org/2014/599|date = 2014-01-01|firstfirst1 = Joppe W.|lastlast1 = Bos|first2 = Craig|last2 = Costello|first3 = Michael|last3 = Naehrig|first4 = Douglas|last4 = Stebila| journal=Cryptology ePrint Archive }}</ref> Software implementing the work of Singh is found on GitHub at [https://github.com/vscrypto/ringlwe https://github.com/vscrypto/ringlwe.]<ref name=":1" />

== Other approaches ==

A variant of the approach described above is an authenticated version in the work of Zhang, Zhang, Ding, Snook and Dagdelen in their paper, "Post Quantum Authenticated Key Exchange from Ideal Lattices."<ref>{{Cite webjournal|title = Workshop on Cybersecurity in a Post-Quantum World|url = https://www.nist.gov/itl/csd/ct/post-quantum-crypto-workshop-2015.cfm|websitejournal = www.nist.govNIST|accessdateaccess-date = 2015-06-06|date = 2015-04-02}}</ref> The concept of creating what has been called a Diffie–Hellman-like Key Exchange using lattices with a reconciliation function appears to have first been presented by French researchers Aguilar, Gaborit, Lacharme, Schrek, and Zemor at PQCrypto 2010 in their talk, "Noisy Diffie–Hellman Protocols."<ref>{{Cite web|title = Noisy Diffie–Hellman protocols|url = https://pqc2010.cased.de/rr/03.pdf|website = pqc2010.cased.de|accessdateaccess-date = 2015-06-06|archive-url=https://web.archive.org/web/20150614110435/https://pqc2010.cased.de/rr/03.pdf |archive-date=2015-06-14 |url-status=dead}}</ref>

In November 2015, Alkim, Ducas, PopplemannPöppelmann, and Schwabe built on the prior work of Peikert and used what they believe is a more conservative costing of lattice attacks to recommend parameters.<ref name=":3">{{Cite web|title = Cryptology ePrint Archive: Report 2015/1092|url = https://eprint.iacr.org/2015/1092|website = eprint.iacr.org|accessdateaccess-date = 2015-11-11}}</ref> Software based on the work of Alkim, Ducas, PopplemannPöppelmann, and Schwabe is found on GitHub at https://github.com/tpoeppelmann/newhope<ref name=":3" />

== See also ==

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* [[Ring learning with errors signature]]

* [[Ring learning with errors]]

== References ==

{{reflist}}

==External links==

{{ Cryptography navbox | public-key }}

[[Category:Cryptographic algorithms]]

[[Category:Cryptography]]

[[Category:Post-quantum cryptography]]

[[Category:Lattice-based cryptography]]