Ring learning with errors key exchange

 Background

Since the 1980's the security of cryptographic key exchanges and digital signatures over the internet has been based on a small number of public key algorithms. The security of these algorithms is based on a small number of computationally hard problems in classical computing. These problems are the difficulty of 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 but are rather easily solved by a relatively small quantum computer. If a quantum computer of sufficient size were build, all of the public key algorithms based on these classically hard problems would become extremely insecure.

Cryptography that is not susceptible to attack by a quantum computer is referred to as Quantum Resistant, 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 in 2005^[1]. A specialized form of Learning with errors operates within the Ring of Polynomials over a Finite Field. This specialised form is called Ring Learning with Errors or Ring-LWE.

There are a variety of cryptographic algorithms which work using the Ring-LWE paradigm. There are public key encryption algorithms, homomorphic encryption algorithms, and digital signatures in addition to the public key, key exchange algorithm presented in this article

A key exchange algorithm is a type of public key algorithm which establishes a shared secret key between to communicants on a communications link. The classic example of a key exchange is the Diffie-Hellman key exchange. The exchange consists of one transmission from one end of the line and one transmission from the other end of the link. Diffie-Hellman and Elliptic Curve Diffie-Hellman are the two most popular key exchange algorithms.

The Ring-LWE Key Exchange is designed to be a "quantum safe" replacement for the widely used Diffie-Hellman and Elliptic Curve Diffie-Hellman key exchanges that are used to secure the establishment of secret keys over untrusted communications channels.

Introduction

For an integer, n, the Ring-LWE key exchange works in the ring of polynomials modulo a polynomial g(x) with coefficients in the field of integers mod q, F_q. (i.e. the ring F_q[x]/g(x) ). Multiplication and addition of polynomials will work in the usual fashion with results of a multiplication reduced mod g(x). This article will closely follow the Ring-LWE work of Peikert in "Lattice Cryptography for the Internet" as further explained by Singh ^[2][3]. For this presentation a typical polynomial is expressed as:

a(x) = a₀ + a₁x + a₂x² + ... + a_n-3x^n-3 + a_n-2x^n-2 + a_n-1x^n-1

The coefficients of this polynomial, the a_i's, are integers mod q where q is a prime number.

The Ring-LWE key exchange 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 vice Z/qZ). The algorithm's security will depend 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:

1. Using 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+1, -b+2. ... -2, -1, 0, 1, 2, ... , b-2, b-1, b} the polynomial will be small with respect to the bound (b).
2. Using Discrete Gaussian Sampling - For an odd value for 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 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.^[4]

For the rest of this article, the random small polynomials will be sampled according do a distribution which is simply be specified as D. Further q will be an odd prime such that q is congruent to 1 mod 4. The maximum degree of the polynomials (n) will be a power of 2. This follows the work of Singh.^[3] The other cases for q and n are thoroughly discussed in the referenced paper by Peikert. A fixed public polynomial, a(x), shared by all users of the network. It is deterministically generated from a cryptographically secure source.

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 responder designated as (R). Both I and R know, q, n, a(x), and have the ability to generate small polynomials according to the distribution D. 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 further understanding the reader should look at the referenced works by Peikert and Singh.^[2][3]

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

Initiator's First Steps:

1. Generate two small polynomials s_I(x) and e_I(x) by sampling from the distribution D.
2. Compute t_I(x) = a(x)·s_I(x) + e_I(x).
3. The initiator sends the polynomial t_I(x) to the Responder.

Responder's Steps:

1. Generate two small polynomials s_R(x) and e_R(x) by sampling from the distribution D.
2. Compute v(x) = t_I(x)·s_R(x) + e_R(x) Note that this equals: a(x)s_I(x)s_R(x) + e_I(x)s_R(x) + e_R(x) and that e_I(x)s_R(x) will also be small relative to the infinity norm on q.
3. The distribution of the coefficients of v(x) are smoothed by looping through the coefficients and randomly adjusting certain values. For j = 0 to n-1:
   1. If v_j = 0, draw a random bit (b). If b = 0 then v_j = 0 otherwise v_j = q-1
   2. If v_j = (q-1)/4, draw a random bit (b). If b = 0 then v_j = (q-1)/4 otherwise v_j = (q+3)/4
4. Two n-long bit streams, uj, and cj, are formed from the coefficients of v(x), (v_n-1, ... , v₀ ), via "Modular Rounding" and "Cross Rounding" respectively. For j = 0 to n-1:
   1. Find m_j and r_j such that 2v_j = m_jq + r_j
   2. Find s_j and y_j such that 4v_j = s_jq + y_j
   3. If r_j > (q-1)/2 (in Z) then set u_j = m_j + 1 (mod 2) otherwise u_j = m_j (mod 2)
   4. If y_j > (q-1)/2 (in Z) then set c_j = s_j + 1 (mod 2) otherwise c_j = s_j (mod 2)
5. Form the key (k) as the concatenation of u_n-1, ..., u₀.
6. Form an n-long "reconciliation" bit string (c) as the concatenation of c_n-1, ..., c₀.
7. Compute t_R(x) = a(x)·s_R(x) + e_R(x).
8. The Responder sends t_R(x) and c to the Initiator.

Initiators Final Steps:

1. Receive t_R(x) and c from the Responder
2. Compute w(x) = t_R(x)·s_I(x) + e_I(x) = a(x)s_I(x)s_R(x) + e_R(x)s_I(x) + e_I(x) Note that this does not equal v(x) (above) but is "close."
3. An n-long bit stream (uj) is formed by looping through the coefficients of w(x) and performing "Key Reconciliation." For j = 0 to n-1:
   1. If c_j = 0 and -q/8 ≤ w_j < 3q/8 then u_j = 0 otherwise u_j = 1
   2. If c_j = 1 and q/8 ≤ w_j < 5q/8 then u_j = 0 otherwise u_j = 1
4. Form the key (k) as the concatenation of u_n-1, ..., u₀

If the key exchange worked properly, the initiator's string: u_n-1, ..., u₀ and the responder's string: u_n-1, ..., u₀ will be the same.

Depending on the specifics of the parameters chosen n, q, σ, or b, there is an extremely small probability that this key exchange will fail to produce the same key. Implementors of the scheme might want to introduce a key validation step before ciphertext is produced.

Parameter Choices

The Ring-LWE Key exchange presented above worked in the Ring of Polynomials of degree n-1 or less mod a polynomial F(x). The presentation assumed that n was a power of 2 and that q was a prime which was congruent to 1 (mod 4). Following the guidance given in Peikert's paper, Singh suggested two sets of paramters for the Ring-LWE Key Exchange.

For 128 bits of security, n = 512, q = 25601, and F(x) = x⁵¹² + 1

For 256 bits of security, n = 1024, q = 40961, and F(x) = x¹⁰²⁴ + 1

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. This is generally referred to as the soundness of the If we assume that the gaussian parameter σ is 8/sqrt(2π) and the uniform sampling bound (b) = 5(see Singh)^[3], then the probability of key agreement failure is less than 2^-71 for the 128-bit secure parameters and 2^-91 for the 256-bit secure parameters.

Key Exchange Security

Based on the BKZ 2.0 lattice reduction algorithm, these choices of parameters will provide greater than 128 or 256 bits of security, respectively. ^[5] It is also worth noting that Peikert demonstrates that there is a provable security equivalence between breaking this key exchange and the Shortest Vector Problem (SVP) in an integer lattice.

References

1. Regev, Oded (2005). "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography". Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing. STOC '05. New York, NY, USA: ACM: 84–93. doi:10.1145/1060590.1060603. ISBN 1-58113-960-8.
2. Peikert, Chris (2014). Mosca, Michele (ed.). Lattice Cryptography for the Internet. Lecture Notes in Computer Science. Springer International Publishing. pp. 197–219. ISBN 978-3-319-11658-7.
3. Singh, Vikram (2015). "A Practical Key Exchange for the Internet using Lattice Cryptography". {{cite journal}}: Cite journal requires |journal= (help)
4. "http://www.cc.gatech.edu/~cpeikert/pubs/slides-pargauss.pdf" (PDF). www.cc.gatech.edu. Retrieved 2015-05-29. {{cite web}}: External link in |title= (help)
5. Chen, Yuanmi; Nguyen, Phong Q. (2011). Lee, Dong Hoon; Wang, Xiaoyun (eds.). BKZ 2.0: Better Lattice Security Estimates. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 1–20. ISBN 978-3-642-25384-3.