Shortest Vector Research Articles

BITRU: Binary Version of the NTRU Public Key Cryptosystem via Binary Algebra

New terms such as closest vector problem (CVP) and the shortest vector problem (SVP), which have been illustrated as NP-hard problem, emerged, leading to a new hope for designing public key cryptosystem based on certain lattice hardness. A new cryptosystem called NTRU is proven computationally efficient and it can be implemented with low cost. With these characteristics, NTRU possesses advantage over others system that rely on number-theoretical problem in a finite field (e.g. integer factorization problem or discrete logarithm problem). These advantages make NTRU a good choice for many applications. After the adaptation of NTRU, many attempts to generalize its algebraic structure have appeared. In this study, a new variant of the NTRU public key cryptosystem called BITRU is proposed. BITRU is based on a new algebraic structure used as an alternative to NTRU-mathematical structure called binary algebra. This commutative and associative. Establishing two public keys in the proposed system has distinguished it from NTRU and those similar to NTRU cryptosystems. This new structure helps to increase the security and complexity of BITRU. The clauses of BITRU, which include key generation, encryption, decryption, and decryption failure, are explained in details. Its suitability of the proposed system is proven and its security is demonstrated by comparing it with NTRU.

On the distribution of lengths of short vectors in a random lattice

We use an idea from sieve theory to estimate the distribution of the lengths of $k$th shortest vectors in a random lattice of covolume 1 in dimension $n$. This is an improvement of the results of Rogers and S\"odergren in that it allows $k$ to increase with $n$.

Lattice reduction‐ordered successive interference cancellation detection algorithm for multiple‐input–multiple‐output system

Lattice reduction (LR) is a powerful technique for improving the performance of linear multiple-input–multiple-output detection methods. The efficient LR algorithms can largely improve the performance of the linear detectors (LDs). Note that the ordered successive interference cancellation (OSIC) system can decrease the interference between antennas and provide performance gain of the LDs. In this paper, a novel LR-aided algorithm called NLR-OSIC improving the performance of the OSIC system has been proposed. Most existing LR algorithms are designed to improve the orthogonality of channel matrices, which is not directly related to the error performance of the OSIC system. While the authors’ algorithm maximises the signal-to-interference-plus-noise ratio (SINR) of the detected symbol in each stage of the OSIC system, thus exhibiting improved error rate than the previous LR-aided LDs and their corresponding OSIC algorithms. In each stage, the authors verify that maximising the SINR of the detected symbol can be formulated as a shortest vector problem which is solved by a suboptimal algorithm in this study. In the end of this study, the error rate performance of the proposed algorithm as well as the required complexity has been demonstrated through extensive computer simulations.

Extreme value theory for random walks on homogeneous spaces

In this paper we study extreme events for random walks on homogeneous spaces. We consider the following three cases. On the torus we study closest returns of a random walk to a fixed point in the space. For a random walk on the space of unimodular lattices we study extreme values for lengths of the shortest vector in a lattice. For a random walk on a homogeneous space we study the maximal distance a random walk gets away from an arbitrary fixed point in the space. We prove an exact limiting distribution on the torus and upper and lower bounds for sparse subsequences of random walks in the two other cases. In all three settings we obtain a logarithm law.

Analysis of Genotype x Environment Interaction for Grain Yield in Maize Hybrids

Maize is an important staple crop for millions of people in Sub-Saharan Africa. The crop grows in a wide range of environmental conditions. The present study was performed to analyze the genotype-by-environment (G×E) interaction for grain yield of fourty two single cross hybrids grown in three environments located at different agro-ecological zones of Kenya. The maize genotypes were grown in randomized complete block design with three replicates in 2006 and 2007. Grain yield data of the single cross hybrids was analysed using the additive main effects and multiplicative interaction (AMMI) and GGE biplot methods. The additive main effects and multiplicative interaction model (AMMI) analysis of variance (additive main effects) showed significant effects for genotypes, environment and the genotype by environment interaction. The total proportion contributed by environment and G x E interaction accounted for 78% and 10% of the total variation. The results also showed that 64.5% of the total variation was contributed by the environment while the genotype and interaction was associated with 20.7% and 10.3% of the total variation, respectively. The most stable genotype in the high yielding group in this study was CML312/TZMI 711 (X20), followed by genotypes TZMI 102/CML384 (designated as X33), and CML312/TZMI 712 (X21). The genotype CML312/TZMI 711 (designated as X20) could be used for broad selection since it was found to be the most ideal genotypes with both high mean yield and high stability. In the biplot analysis, X20 was also close to the origin, had the shortest vector from the ATC and with large PC1 scores (high mean yield) and small (absolute) PC2 scores (high stability) implying it had the least contribution to genotype by environment interaction (GEI). Through the stability analysis, the superior genotypes identified could also be used as references for genotype evaluation and also included in further testing in both early and later stages of selection.

The Laplacian lattice of a graph under a simplicial distance function

We provide a complete description of important geometric invariants of the Laplacian lattice of a multigraph under the distance function induced by a regular simplex, namely Voronoi Diagram, Delaunay Triangulation, Delaunay Polytope and its combinatorial structure, Shortest Vectors, Covering Radius and Packing Radius. We use this information to obtain the following results: i. We provide an explicit description of the vertices, edges and facets of the Delaunay polytope of the origin under the simplicial distance function and using the description, we show that the Delaunay triangulation contains complete information of the multigraph up to isomorphism. ii. A weak generalization of the matrix-tree theorem: the number of multigraphs with a given Laplacian lattice is controlled, in particular upper bounded, by the number of different Delaunay triangulations. iii. We obtain formulae for the covering and packing densities of a Laplacian lattice and deduce that in the space of Laplacian lattices of undirected connected multigraphs, the Laplacian lattices of highly connected multigraphs such as Ramanujan multigraphs possess high covering and packing properties in the space of Laplacian lattices of connected graphs.

Speed and direction response profiles of neurons in macaque MT and MST show modest constraint line tuning

Several models of heading detection during smooth pursuit rely on the assumption of local constraint line tuning to exist in large scale motion detection templates. A motion detector that exhibits pure constraint line tuning responds maximally to any 2D-velocity in the set of vectors that can be decomposed into the central, or classic, preferred velocity (the shortest vector that still yields the maximum response) and any vector orthogonal to that. To test this assumption, we measured the firing rates of isolated middle temporal (MT) and medial superior temporal (MST) neurons to random dot stimuli moving in a range of directions and speeds. We found that as a function of 2D velocity, the pooled responses were best fit with a 2D Gaussian profile with a factor of elongation, orthogonal to the central preferred velocity, of roughly 1.5 for MST and 1.7 for MT. This means that MT and MST cells are more sharply tuned for speed than they are for direction; and that they indeed show some level of constraint line tuning. However, we argue that the observed elongation is insufficient to achieve behavioral heading discrimination accuracy on the order of 1–2 degrees as reported before.

On the Hardness of Subset Sum Problem from Different Intervals

SUMMARY The subset sum problem, which is often called as the knapsack problem, is known as an NP-hard problem, and there are several cryptosystems based on the problem. Assuming an oracle for shortest vector problem of lattice, the low-density attack algorithm by Lagarias and Odlyzko and its variants solve the subset sum problem efficiently, when the “density” of the given problem is smaller than some threshold. When we define the density in the context of knapsack-type cryptosystems, weights are usually assumed to be chosen uniformly at random from the same interval. In this paper, we focus on general subset sum problems, where this assumption may not hold. We assume that weights are chosen from different intervals, and make analysis of the effect on the success probability of above algorithms both theoretically and experimentally. Possible application of our result in the context of knapsack cryptosystems is the security analysis when we reduce the data size of public keys.

Grain Yield and Charcoal Rot Resistance Stability in Common Beans under Terminal Drought Conditions

AbstractCharcoal rot (Macrophomina phaseolina) is a major disease of beans (Phaseolus vulgaris L.) in Mexico. The use of germplasm combining high‐yield stability with resistance to drought and charcoal rot could reduce damage from this disease. In this study, we compared the Eberhart and Russell method and the Additive Main Effect and Multiplicative Interaction (AMMI) model plus biplot analysis for measuring grain yield (GY) and charcoal rot resistance (CHRR) stabilities in 98 F8 : 10 recombinant inbred lines (RILs) derived from a cross between bean adapted to the tropics (BAT) 477 (resistant) × Pinto UI‐114 (susceptible). Experiments were conducted from 2007 to 2009 in Isla, Cotaxtla, Río Bravo and Díaz Ordaz, México, under irrigated or terminal drought conditions. anova detected significant differences (P ≤ 0.05) in GY and CHRR among environments, genotypes and genotype × environment interactions (GEI). Most RILs showed good responses to unfavourable environments based on GY (48) and CHRR (40). AMMI anova s for both traits showed that all sources of variation in the model accounted for approximately 49% of the total squared sum. For the first principal component (PC1), we found 13 RILs that were stable for GY, and for the second (PC2), we found 9 that were stable for GI. For CHRR, we detected 14 stable RILs (PC1) and eight (PC2). Biplot analysis showed the largest vectors for Díaz Ordaz (irrigated and drought, 2008), where the highest and most variable GYs were detected. The shortest vectors were found in Isla (drought, 2007) and Río Bravo (irrigated and drought, 2008), where the lowest and least variable GY were found. We found differential responses of RILs to locations, years and soil humidity conditions as well as significant GEI based on GY and CHRR. The two methods were complementary, and both gave us information to select stable, high‐yield germplasm associated with resistance to charcoal rot disease.

On the distribution of angles between the N shortest vectors in a random lattice

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n tends to infinity. Moreover we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally we discuss the limit distribution of any finite number of successive minima of a random n-dimensional lattice as n tends to infinity.

An ideal lattice based IBE scheme in the standard model

Under the standard model, an IND-sID-CPA secure IBE (Identity Based Encryption) scheme is proposed, which is based on an improved trapdoor generation function over ring and decisional R-LWE (ring-learning with errors) assumption, thus the hardness of recovering system master key can be reduced to solve the Ideal-SIS (small integer solution) problem, and the attack to the ciphertext is equivalent to solving the SVP (the shortest vector problem) in the worst case on ideal lattice to within a fixed poly(n) factor. Moreover, the proposed scheme has advantages of high encryption and decryption speed and low encryption blowup factor compared with all known IBE scheme from the standard lattice under the standard model.

On the complexity of recognizing a set of vectors by a neuron

We consider the problems connected with the computational abilities of a neuron. The orderings of finite subsets of real vectors associated with neural computing are studied. We construct a lattice of such orderings and study some its properties. The interrelation between the orders on the sets and the neuron implementation of functions defined on these sets is derived. We prove the NP-hardness of “The Shortest Vector” problem and represent the relationship of the problem with neural computing.

The Contact Polytope of the Leech Lattice

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.

Optimisation-based proximity queries and penetration depth computation

The virtual reality (VR) was found to be a perfect technique that could be used as a training approach, since it shows many advantages despite its weakness. In the VR some major bottlenecks arises namely the proximity queries (PQ) and penetration depth computation. This paper shows a novel algorithm used to solve those problems. Problems of PQ are ubiquitous within many tasks in computer graphics, virtual environments, robotics, manufacturing, and mechanical design. Interactions in any virtual scene usually involve contact or close proximity between its objects. Determining which pairs of objects are in contact or at close proximity is a complex task in most of the virtual environments. The PQ is the shortest vector over which one object needs to be translated in order to bring the pair in contact.

On Nearly Orthogonal Lattice Bases and Random Lattices

We study lattice bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is at least $\frac{\pi}{3}$ radians; we denote such bases as “nearly orthogonal.” We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, we prove that if the basis vector lengths are “nearly equal,” then the basis is the unique nearly orthogonal lattice basis up to multiplication of basis vectors by $\pm 1$. We also study random lattices generated by the columns of random matrices with $n$ rows and $m \leq n$ columns. We show that if $m \leq c\,n$, with $c \approx 0.071$, then the random matrix forms a nearly orthogonal basis for the random lattice with high probability for large $n$ and almost surely as $n$ tends to infinity. Consequently, the columns of such a random matrix contain the shortest vector in the random lattice. Finally, we discuss an interesting JPEG image compression application where nearly orthogonal lattice bases play an important role.

Transmission electron microscopy of dislocations and grain boundaries in Bi[sbnd]Pb[sbnd]Sr[sbnd]Ca[sbnd]Cu[sbnd]O superconductors deformed at 800°C

Abstract Structures of dislocations and grain boundaries in Bi[sbnd]Pb[sbnd]Sr[sbnd]Ca[sbnd]Cu[sbnd]O superconductors deformed to 90% in compression at 800°C have been studied using transmission electron microscopy. It has been found that only 1/2(110) type dislocations (i.e. with Burgers vectors of the shortest fundamental translational vectors) are activated during deformation, although (100) type dislocations are also observed after a post-deformation annealing. Grain boundaries in an as-deformed specimen have been examined using high-resolution electron microscopy, and it has been found that the grain boundaries are very clean in the sense that neither grain-boundary phases such as amorphous phases nor pores are found. It has been concluded that local melting takes place preferentially at grain boundaries during deformation at 800°C; the melt freezes during subsequent cooling, leading to apparently clean grain boundaries.

How to Calculate Shortest Vectors in a Lattice

How to Calculate Shortest Vectors in a Lattice

A contribution to the indexing of transmission electron diffraction patterns

AbstractA method for the indexing of TED spot patterns is presented. The reduced cell of the TED pattern defined by the two shortest vectors, which enclose an acute angle, serves as reference cell for the indexing. The d‐values of both the cell defining vectors and of their sum and difference vectors are determined being the four largest d‐values, which can be determined from the pattern. These four d‐values are compared with suitable d‐value sets of the possible crystalline phases and probable phase identifications are made. Generally, there exist several indexing variants for the individual d‐values of the separate phase identification possibilities. It is possible to find the correct indexing by the determination of the indices of the sum and difference vectors, occasionally, for one pair of the indexing variant from the vectors defining the reference cell and by comparison of these indices with the indexing variants for the d‐values of the sum and difference vectors of the corresponding phase identification variant. The measuring and calculation of the intervectorial angles formerly necessarily hasn't to be made in this method. However, the determination of the sum and difference vectors involves it. The reliability of the indexing can be increased in a strong manner if two or three different reciprocal lattice planes of the specimen under investigation are recorded in the corresponding TED patterns and if, in the following, these patterns can be indexed by the mentioned method without any doubt.