Shortest Vector Research Articles

On the distribution of angles between increasingly many short lattice vectors

Following Södergren, we consider a collection of random variables on the space Xn of unimodular lattices in dimension n: normalizations of the angles between the N=N(n) shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions (whose support depends on a parameter K=K(n)) evaluated at these random variables in the regime where K and N are allowed to tend to infinity with n at the rate KN=o(n1/6). Our main result is that as n⟶∞, these random variables exhibit a joint Poissonian and Gaussian behavior.

Tef (<i>Eragrostis tef</i>) Variety Development for High Potential Areas of Ethiopia

The aims of the experiment are to determine and understand the effect of genotype, environment, and their interaction on grain yield of tef, and to identify and release stable and high yielding tef genotype for high potential areas of country. Twelve tef genotypes including two checks were laid out in randomized complete block design using four replications for two years (2016 and 2017) at twelve representative locations of high potential areas of the country. The trial was conducted on the plot size of 2m*2m with 10 rows per plot throughout all trial sites and 1.5m between replication, 1m between plot, and 20cm between rows. Agronomic and yield data were collected and subjected to statistical analysis in order to identify the best genotypes of the evaluated genotypes. Data from individual environments and combined over twelve locations were analyzed by using R 3.5 software version. The combined data analysis over locations and years indicated that the candidate variety Kaye Murri X 3774-1 (RIL18) performed better than the two checks and the other test genotypes. It gave 7.5% and 10.65% grain yield advantages over the standard check (Quncho) and local check, respectively. The significant genotypes x environments interaction effects indicated the inconsistent performance of genotypes across the tested environments. The candidate variety DZ-Cr-458 (RIL18) is the shortest vector from the AEC axis that identified as the most stable genotype. The National Variety Release Committee in Ethiopia investigated the two-year performance of Kaye Murri X 3774-1 (RIL18) and visited several locations where the new candidate variety was grown for evaluation under variety verification trial. Based on the critiques, the National Variety Release Committee the candidate variety was approved for release Kaye Murri X 3774-1 (RIL18), for high potential areas of the country with the vernacular name of “Ebba” as a standing witnessed for the earliest known tef scientist, Dr. Tadesse Ebba.

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in ℓp norm (1≤p≤∞). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all ℓp norm (1≤p≤∞). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of 22.751n+o(n), which is much less than the 23.849n+o(n) complexity of the previous best algorithm. We also introduce a mixed sieving procedure, where a point is mapped to a hypercube within a ball and then a quadratic sieve is performed within each hypercube. This improves the running time, especially in the ℓ2 norm, where we achieve a time complexity of 22.25n+o(n), while the List Sieve Birthday algorithm has a running time of 22.465n+o(n). We adopt our sieving techniques to approximation algorithms for SVP and CVP in ℓp norm (1≤p≤∞) and show that our algorithm has a running time of 22.001n+o(n), while previous algorithms have a time complexity of 23.169n+o(n).

Efficient Implementations of Sieving and Enumeration Algorithms for Lattice-Based Cryptography

The security of lattice-based cryptosystems is based on solving hard lattice problems such as the shortest vector problem (SVP) and the closest vector problem (CVP). Various cryptanalysis algorithms such as (Pro)GaussSieve, HashSieve, ENUM, and BKZ have been proposed to solve these hard problems. Several implementations of these algorithms have been developed. On the other hand, the implementations of these algorithms are expected to be efficient in terms of run time and memory space. In this paper, a modular software package/library containing efficient implementations of GaussSieve, ProGaussSieve, HashSieve, and BKZ algorithms is developed. These implementations are considered efficient in terms of run time. While constructing this software library, some modifications to the algorithms are made to increase the performance. Then, the run times of these implementations are compared with the others. According to the experimental results, the proposed GaussSieve, ProGaussSieve, and HashSieve implementations are at least 70%, 75%, and 49% more efficient than previous ones, respectively.

Three Strategies for Improving Shortest Vector Enumeration Using GPUs

Hard Lattice problems are assumed to be one of the most promising problems for generating cryptosystems that are secure in quantum computing. The shortest vector problem (SVP) is one of the most famous lattice problems. In this paper, we present three improvements on GPU-based parallel algorithms for solving SVP using the classical enumeration and pruned enumeration. There are two improvements for preprocessing: we use a combination of randomization and the Gaussian heuristic to expect a better basis that leads rapidly to a shortest vector and we expect the level on which the exchanging data between CPU and GPU is optimized. In the third improvement, we improve GPU-based implementation by generating some points in GPU rather than in CPU. We used NVIDIA GeForce GPUs of type GTX 1060 6G. We achieved a significant improvement upon Hermans’s improvement. The improvements speed up the pruned enumeration by a factor of almost 2.5 using a single GPU. Additionally, we provided an implementation for multi-GPUs by using two GPUs. The results showed that our algorithm of enumeration is scalable since the speedups achieved using two GPUs are almost faster than Hermans’s improvement by a factor of almost 5. The improvements also provided a high speedup for the classical enumeration. The speedup achieved using our improvements and two GPUs on a challenge of dimension 60 is almost faster by factor 2 than Correia’s parallel implementation using a dual-socket machine with 16 physical cores and simultaneous multithreading technology.

Extreme value distributions for one-parameter actions on homogeneous spaces**This research was supported by ERC Grant 239606 and DFF/Marie Curie COFUND Grant 5051-00184.

In this paper we study extreme value distributions for one-parameter actions on homogeneous spaces of Lie groups. We study both shortest vectors in unimodular lattices, maximal distance excursions and closest distance returns of a one-parameter action. For certain sparse subsequences of the one-parameter action and by taking the maximum over a moving interval of indices we prove non-trivial estimates for the limiting distribution in all cases. We also consider the kth largest element, , as opposed to just the largest element and obtain analogue estimates for the limiting distribution of this quantity.

A new algorithm on the minimal rational fraction representation of feedback with carry shift registers

In 1994, Klapper and Goresky (Proceedings of the 1993 Cambridge Security Workshop, Lecture Notes in Computer Science, vol 809, Cambridge, pp 174–178, 1994) proposed a new device called feedback with carry shift register to generate pseudo-random sequences instead of using the traditional device linear feedback shift register. They raised an algorithm called as rational approximation algorithm to recover the device for a given sequence (Klapper and Goresky, Advances in Cryptology, Crypto’95, Lecture Notes in Computer Science, vol 963, Springer, Berlin, pp 262–274, 1995). In this paper, we propose a new algorithm by introducing a new parameter and get the best rational approximation of the sequence much more quickly, especially when the size of the sequence increases dramatically. Unlike most of known algorithms, we can solve the minimal lattice basis instead of one shortest vector. Besides, we can prove that the solution of each step is optimal regardless of the length of the input sequence theoretically.

Explicit Relation between Low-Dimensional LLL-Reduced Bases and Shortest Vectors

Explicit Relation between Low-Dimensional LLL-Reduced Bases and Shortest Vectors

Novel Properties of Successive Minima and Their Applications to 5G Tactile Internet

The lattice $\mathcal {L}(\boldsymbol{A})$ of a full-column rank matrix $\boldsymbol{A}\in \mathbb {R}^{m\times n}$ is defined as the set of all the integer linear combinations of the column vectors of $\boldsymbol{A}$ . The successive minima $\lambda _i(\boldsymbol{A}),\,1\leq i\leq n,$ of lattice $\mathcal {L}(\boldsymbol{A})$ are important quantities since they have close relationships with the following problems: shortest vector problem, shortest independent vector problem, and successive minima problem. These problems arise from many practical applications, such as communications and cryptography. This paper first investigates some properties of $\lambda _i(\boldsymbol{A})$ . Specifically, we develop lower and upper bounds on $\lambda _i(\boldsymbol{A})$ , where $\boldsymbol{A}$ are, respectively, the Cholesky factor of $\boldsymbol{G}_1+\boldsymbol{G}_2$ and $(\boldsymbol{G}_1+\boldsymbol{G}_2)^{-1}$ for two given symmetric positive definitive matrices $\boldsymbol{G}_1$ and $\boldsymbol{G}_2$ . The bounds are, respectively, expressed as the successive minima of $\mathcal {L}(\boldsymbol{A}_1)$ and $\mathcal {L}(\boldsymbol{A}_2)$ , and $\mathcal {L}(\hat{\boldsymbol{A}}_1)$ and $\mathcal {L}(\hat{\boldsymbol{A}}_2)$ , where $\boldsymbol{A}_1, \boldsymbol{A}_2, \hat{\boldsymbol{A}}_1$ and $\hat{\boldsymbol{A}}_2$ are, respectively, the Cholesky factors of $\boldsymbol{G}_1, \boldsymbol{G}_2, \boldsymbol{G}_1^{-1}$ , and $\boldsymbol{G}_2^{-1}$ . Then, we show how some properties of $\lambda _i(\boldsymbol{A})$ are used to design a suboptimal integer-forcing strategy for cloud radio access network. Our approach provides much higher time efficiency while keeping the same achievable rate as the algorithm reported by Bakoury and Nazer (I. E. Bakoury and B. Nazer, “Integer-forcing architectures for uplink cloud radio access networks,” in Proc. 55th Annu. Allerton Conf. Commun. Control Comput. , Oct. 2007, pp. 67–75). Simulation tests are performed to illustrate our main results.

Preprocessing optimisation: revisiting recursive‐BKZ lattice reduction algorithm

Preprocessing is applied to certain lattice reduction algorithms such as block Korkine–Zolotarev (BKZ) variants to reduce the search time in the enumeration tree for a shortest vector. The most classical form of preprocessing the authors observe is with polynomial time Lenstra–Lenstra–Lovasz algorithm to work with a slow enumeration-based algorithm like BKZ. The trade-off between the preprocessing and the enumeration stages in the context of time complexity of the whole algorithm is not well studied and explored. The main goal of this study is to re-investigate the preprocessing approach presented by Chen and Nguyen and improve its performance through optimisation. They extend the numeral results published by Haque et al. in IET Inf. Secur. for larger block sizes and report a comparison.

Deterministic lattice reduction on knapsacks with collision‐free properties

The knapsack problem is an important problem in computer science and had been used to design public key cryptosystems. Low-density subset sum algorithms are powerful tools to reduce the security of trapdoor knapsacks to the shortest vector problem (SVP) over lattices. Several knapsack ciphers Chor–Rivest, Okamoto–Tanaka–Uchiyama, and Kate–Goldberg were proposed to defend low-density attacks by utilising low-weight knapsack problems. Some evidence was also found on the vulnerabilities of the above three knapsack ciphers to lattice attacks. However, previous lattice-based cryptanalytic results have been established via a probabilistic approach. The authors investigate some collision-free properties and derive from the properties a deterministic reduction from the knapsack problems in the Chor–Rivest, Okamoto–Tanaka–Uchiyama, and Kate–Goldberg knapsack ciphers to SVP without imposing any restriction and assumption. To the best of the authors' knowledge, the proposed reduction is the first deterministic reduction from public key cryptographic knapsacks to SVP.

Application of multivariate analysis to study genotype × environment interaction in sesame (Sesamum indicum) under stress conditions

The present research aimed to study seed yield stability of 12 sesame (Sesamum indicum L.) genotypes evaluated across six environments (two growing years × three irrigation intervals). Data on seed yield were subjected to additive main effects and multiplicative interaction (AMMI) and genotype main effect and genotype × environment interaction (GGE) analyses using GenStat software. Results showed that environments main effect explained 73.9% of the total variation in seed yield compared with genotypes (12.3%) and genotypes × environment interactions (GEI) (10.9%). IPCAs showed significant effects and accounted for a total of 65.9% of GEI. IPCA1 had higher significant contribution to GEI (44.0%) than IPCA2 (21.9%). Based on AMMI stability value (ASV), genotypes 2, 3 followed by genotypes 1 and 11 were identified as the most stable and high yielding genotypes across tested environments. Furthermore, Environment 5, which is close to the biplot origin, is expected to be candidate environment for stable and repeatable genotype testing. Using GGE biplot model, considered G1, G7, G3 and G2 to be the most stable genotypes as they had the shortest vectors, moreover the most yielding genotypes were G7 and G2 as they were the most distance from the right side of the biplot and their yield surpassed the environmental mean. Introduced breeding lines are effective genetic source than local cultivated cultivars for high seed yield, such genotypes would be potential parental lines in sesame breeding programs.

The linear transformation that relates the canonical and coefficient embeddings of ideals in cyclotomic integer rings

The geometric embedding of an ideal in the algebraic integer ring of some number field is called an ideal lattice. Ideal lattices and the shortest vector problem (SVP) are at the core of many recent developments in lattice-based cryptography. We utilize the matrix of the linear transformation that relates two commonly used geometric embeddings to provide novel results concerning the equivalence of the SVP in these ideal lattices arising from rings of cyclotomic integers.

A polynomial time algorithm for a variant of shortest vector problem with applications in compute-and-forward design

In wireless communication networks, one of the most significant problems in Compute-and-Forward (CPF) strategy based on Integer-Forcing (IF) technique is to search the optimal integer coefficient vector for a relay to decode the received messages. While this is a variant of shortest vector problem (SVP), which is prohibitively complex in its general form. In this paper, we mainly focus on the optimal integer solution with non-zero entries in multi-source one-relay model. For comparison we extend the number (or dimension) of communication nodes from 2 to any larger integer n based on an existing 2-dimensional search algorithm with non-zero entries. Then we add this non-zero-entries constraint to an existing search algorithm and make some improvements to search the optimal integer vector by solving an optimization problem over only one variable instead of searching all vector components, which has a much lower time complexity. We prove that our new search algorithm is more efficient with polynomial complexity compared with the extended n-dimensional algorithm, meanwhile the same optimal integer solution is guaranteed.

Analysing recursive preprocessing of BKZ lattice reduction algorithm

Lattice problems are considered as the key elements in many areas of computer science as well as in cryptography; the most important of which is the shortest vector problem and its approximate variants. Algorithms for this problem are known as lattice reduction algorithms. Currently, the most practical lattice reduction algorithm for such problems is the block Korkine–Zolotarev (BKZ) algorithm and its variants. The authors optimise both the pruning and the preprocessing parameters of the recursive (aborted, extreme pruned) preprocessing of the BKZ lattice reduction algorithm and improve the results from Asiacrypt'11 by Chen and Nguyen. The authors derive approximate closed-form complexity formulas (based on the sandpile model assumption model by Hanrot et al.) for the enumeration time which allow a simple estimation of complexity without running the simulation algorithm (by Chen and Nguyen) and asymptotically suggests a modified extreme pruning bounding profiles with different parameters. Hence, the authors’ contributions are in optimising and improving the analysis of the complexity upper bound estimates presented by Chen and Nguyen, based on the same recursive-BKZ preprocessing model.

B 7 as a supergroup of crystal and quasicrystal symmetries

In sharp contrast to the generation of a finite group that includes all the 14 types of Bravais lattices as its subgroups [Hosoya (2000). Acta Cryst. A56, 259-263; Hosoya (2002). Acta Cryst. A58, 208], it was proved that a signed permutation group Bk may be interpreted as the supergroup of both crystal and quasicrystal symmetries. Minimal dimension k = 6 is adequate for lattices referred to their three non-coplanar shortest vectors, or for symmetry groups of most quasicrystal types. If one prefers complete, well defined semi-reduced lattice descriptions or needs a dodecagonal group, the B7 supergroup is necessary. All considered matrix groups correspond to isometric transformations in extended k-bases and may be easily derived from B7 and projected onto three-dimensional crystallographic space. Three models of extended bases are proposed: semi-reduced, cyclic and axial. In all cases additional basis vectors are strictly (functionally) related to three original basis vectors.

Efficient Chosen-Ciphertext Secure Encryption from R-LWE

In order to construct efficient public-key encryption scheme that is secure against adaptive chosen-ciphertext attacks (CCA), an efficient signature scheme and an identity-based encryption (IBE) scheme from the learning with errors over rings are presented firstly in this paper, whose security are reducible to the hardness of the shortest vector problem in the worst case on ideal lattices. Secondly, a CCA-secure public key cryptosystem is constructed on the basis of the IBE and signature proposed above. The efficiency analysis indicates the proposed signature and encryption schemes are much more efficient than correlative cryptosystems. The security analysis shows that the IBE scheme is secure against chosen-plaintext attacks, and the public-key encryption scheme is CCA-secure in the random oracle model.

An Efficient Algorithm for Optimally Solving a Shortest Vector Problem in Compute-and-Forward Design

We consider the problem of finding the optimal coefficient vector that maximizes the computation rate at a relay in the compute-and-forward scheme. Based on the idea of sphere decoding, we propose a highly efficient algorithm that finds the optimal coefficient vector. First, we derive a novel algorithm to transform the original quadratic form optimization problem into a shortest vector problem (SVP) using the Cholesky factorization. Instead of computing the Cholesky factor explicitly, the proposed algorithm realizes the Cholesky factorization with only $ {\mathcal {O}}(n)$ flops by taking advantage of the structure of the Gram matrix in the quadratic form. Then, we propose some conditions that can be checked with $ {\mathcal {O}}(n)$ flops, under which a unit vector is the optimal coefficient vector. Finally, by considering some useful properties of the optimal coefficient vector, we modify the Schnorr-Euchner search algorithm to solve the SVP. We show that the estimated average complexity of our new algorithm is $ {\mathcal {O}}(n^{1.5}P^{0.5})$ flops for independent identically distributed (i.i.d.) Gaussian channel entries with SNR $P$ based on the Gaussian heuristic. Simulations show that our algorithm is not only much more efficient than the existing ones that give the optimal solution, but also faster than some best known suboptimal methods. Besides, we show that our algorithm can be readily adapted to output a list of $L$ best candidate vectors for use in the compute-and-forward design. The estimated average complexity of the resultant list-output algorithm is $ {\mathcal {O}}(n^{2.5}P^{0.5} + n^{1.5}P^{0.5}\log (L) + nL)$ flops for i.i.d. Gaussian channel entries.

Simultaneous approximation problems of p-adic numbers and p-adic knapsack cryptosystems - Alice in p-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l ∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.

Bounds for the traveling salesman paths of two-dimensional modular lattices

We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi---Perron type algorithms.