On the Hardness of Learning With Errors with Binary Secrets

We give a simple proof that the decisional Learning With Errors (LWE) problem with binary secrets (and an arbitrary polynomial number of samples) is at least as hard as the standard LWE problem (with unrestricted, uniformly random secrets, and a bounded, quasi-linear number of samples). This proves that the binary-secret LWE distribution is pseudorandom, under standard worst-case complexity assumptions on lattice problems. Our results are similar to those proved by (Brakerski, Langlois, Peikert, Regev and Stehle, STOC 2013), but provide a shorter, more direct proof, and a small improvement in the noise growth of the reduction

On the hardness of the shortest vector problem

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 77-84).An n-dimensional lattice is the set of all integral linear combinations of n linearly independent vectors in Rm. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest non-zero vector in it. We prove that the shortest vector problem is NP-hard (for randomized reductions) to approximate within some constant factor greater than 1 in any 1, norm (p >\=1). In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2. The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an n-dimensional ball of radius greater than 1 + [square root of] 2 grows exponentially in n, and a new constructive version of Sauer's lemma (a combinatorial result somehow related to the notion of VC-dimension) is presented, considerably simplifying all previously known constructions.by Daniele Micciancio.Ph.D

Embedded Lattice and Properties of Gram Matrix

In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [16] and cryptographic systems with lattice [17].Futa Yuichi - Tokyo University of Technology, Tokyo, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Yuichi Futa and Yasunari Shidama. Lattice of Z-module. Formalized Mathematics, 24 (1):49-68, 2016. doi: 10.1515/forma-2016-0005.Yuichi Futa and Yasunari Shidama. Divisible Z-modules. Formalized Mathematics, 24 (1):37-47, 2016. doi: 10.1515/forma-2016-0004.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathe matics, 20(1):47-59, 2012. doi: 10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205-214, 2012.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29-49, 2015.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.A. K. Lenstra, H. W. Lenstra Jr., and L. Lov´asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515-534, 1982.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Michał Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):3-11, 1991.Karol Pak. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199-211, 2007.Karol Pak and Andrzej Trybulec. Laplace expansion. Formalized Mathematics, 15(3): 143-150, 2007.Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127-136, 2007.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990.Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877-882, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Lattice of ℤ-module

In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].Futa Yuichi - Japan Advanced Institute of Science and Technology Ishikawa, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537-541, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of ℤ-module. Formalized Mathematics, 20(3):205-214, 2012. doi:10.2478/v10037-012-0024-y.Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, and Yasunari Shidama. Torsion ℤ-module and torsion-free ℤ-module. Formalized Mathematics, 22(4):277-289, 2014. doi:10.2478/forma-2014-0028.Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29-49, 2015. doi:10.2478/forma-2015-0003.A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990

Asymptotically Efficient Lattice-Based Digital Signatures

We present a general framework that converts certain types of linear collision-resistant hash functions into one-time signatures. Our generic construction can be instantiated based on both general and ideal (e.g. cyclic) lattices, and the resulting signature schemes are provably secure based on the worst-case hardness of approximating the shortest vector (and other standard lattice problems) in the corresponding class of lattices to within a polynomial factor. When instantiated with ideal lattices, the time complexity of the signing and verification algorithms, as well as key and signature size is almost linear (up to poly-logarithmic factors) in the dimension n of the underlying lattice. Since no sub-exponential (in n) time algorithm is known to solve lattice problems in the worst case, even when restricted to ideal lattices, our construction gives a digital signature scheme with an essentially optimal performance/security trade-off

Error Correction and Ciphertext Quantization in Lattice Cryptography

Recent work in the design of rate $1 - o(1)$ lattice-based cryptosystems have used two distinct design paradigms, namely replacing the noise-tolerant encoding $m \mapsto (q/2)m$ present in many lattice-based cryptosystems with a more efficient encoding, and post-processing traditional lattice-based ciphertexts with a lossy compression algorithm, using a technique very similar to the technique of ``vector quantization\u27\u27 within coding theory. We introduce a framework for the design of lattice-based encryption that captures both of these paradigms, and prove information-theoretic rate bounds within this framework. These bounds separate the settings of trivial and non-trivial quantization, and show the impossibility of rate $1 - o(1)$ encryption using both trivial quantization and polynomial modulus. They furthermore put strong limits on the rate of constructions that utilize lattices built by tensoring a lattice of small dimension with $\mathbb{Z}^k$, which is ubiquitous in the literature. We additionally introduce a new cryptosystem, that matches the rate of the highest-rate currently known scheme, while encoding messages with a ``gadget\u27\u27, which may be useful for constructions of Fully Homomorphic Encryption

Equational Security Proofs of Oblivious Transfer Protocols

We exemplify and evaluate the use of the equational framework of Micciancio and Tessaro (ITCS 2013) by analyzeing a number of concrete Oblivious Transfer protocols: a classic OT transformation to increase the message size, and the recent (so called ``simplest\u27\u27) OT protocol in the random oracle model of Chou and Orlandi (Latincrypt 2015), together with some simple variants. Our analysis uncovers subtle timing bugs or shortcomings in both protocols, or the OT definition typically employed when using them. In the case of the OT length extension transformation, we show that the protocol can be formally proved secure using a revised OT definition and a simple protocol modification. In the case of the ``simplest\u27\u27 OT protocol, we show that it cannot be proved secure according to either the original or revised OT definition, in the sense that for any candidate simulator (expressible in the equational framework) there is an environment that distinguishes the real from the ideal system

Hardness of SIS and LWE with Small Parameters

The Short Integer Solution (SIS) and Learning With Errors (LWE) problems are the foundations for countless applications in lattice-based cryptography, and are provably as hard as approximate lattice problems in the worst case. A important question from both a practical and theoretical perspective is how small their parameters can be made, while preserving their hardness. We prove two main results on SIS and LWE with small parameters. For SIS, we show that the problem retains its hardness for moduli $q \geq \beta \cdot n^{\delta}$ for any constant $\delta > 0$, where $\beta$ is the bound on the Euclidean norm of the solution. This improves upon prior results which required $q \geq \beta \cdot \sqrt{n \log n}$, and is essentially optimal since the problem is trivially easy for $q \leq \beta$. For LWE, we show that it remains hard even when the errors are small (e.g., uniformly random from $\{0,1\}$), provided that the number of samples is small enough (e.g., linear in the dimension $n$ of the LWE secret). Prior results required the errors to have magnitude at least $\sqrt{n}$ and to come from a Gaussian-like distribution

Symbolic security of garbled circuits

We present the first computationally sound symbolic analysis of Yao\u27s garbled circuit construction for secure two party computation. Our results include an extension of the symbolic language for cryptographic expressions from previous work on computationally sound symbolic analysis, and a soundness theorem for this extended language. We then demonstrate how the extended language can be used to formally specify not only the garbled circuit construction, but also the formal (symbolic) simulator required by the definition of security. The correctness of the simulation is proved in a purely syntactical way, within the symbolic model of cryptography, and then translated into a concrete computational indistinguishability statement via our general computational soundness theorem. We also implement our symbolic security framework and the garbling scheme in Haskell, and our experiment shows that the symbolic analysis performs well and can be done within several seconds even for large circuits that are useful for real world applications