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

Algorithms ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 362
Author(s):  
Priyanka Mukhopadhyay
Keyword(s):  
Approximation Algorithms ◽  
Time Complexity ◽  
Main Idea ◽  
Running Time ◽  
Closest Vector Problem ◽  
Vector Problem ◽  
Shortest Vector Problem ◽  
Quadratic Sieve ◽  
Better Than

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).

On the string matching with k differences in DNA databases

2021 ◽  
Vol 14 (6) ◽  
pp. 903-915
Author(s):  
Yangjun Chen ◽  
Hoang Hai Nguyen
Keyword(s):  
Time Complexity ◽  
String Matching ◽  
Main Idea ◽  
Second Step ◽  
Working Process ◽  
Typical Application ◽  
Average Value ◽  
Dna Database ◽  
A Genome ◽  
Better Than

In this paper, we discuss an efficient and effective index mechanism for the string matching with k differences, by which we will find all the substrings of a target string y of length n that align with a pattern string x of length m with not more than k insertions, deletions, and mismatches. A typical application is the searching of a DNA database, where the size of a genome sequence in the database is much larger than that of a pattern. For example, n is often on the order of millions or billions while m is just a hundred or a thousand. The main idea of our method is to transform y to a BWT-array as an index, denoted as BWT ( y ), and search x against it. The time complexity of our method is bounded by O( k · | T |), where T is a tree structure dynamically generated during a search of BWT ( y ). The average value of | T | is bounded by O(|Σ| 2 k ), where Σ is an alphabet from which we take symbols to make up target and pattern strings. This time complexity is better than previous strategies when k ≤ O(log |Σ| n ). The general working process consists of two steps. In the first step, x is decomposed into a series of l small subpatterns, and BWT ( y ) is utilized to speedup the process to figure out all the occurrences of such subpatterns with ⌊ k/l ⌋ differences. In the second step, all the found occurrences in the first step will be rechecked to see whether they really match x , but with k differences. Extensive experiments have been conducted, which show that our method for this problem is promising.

scholarly journals A Survey of Solving SVP Algorithms and Recent Strategies for Solving the SVP Challenge

2020 ◽  
pp. 189-207
Author(s):  
Masaya Yasuda
Keyword(s):  
Quantum Cryptography ◽  
Point Of View ◽  
Classical Lattice ◽  
Mathematical Point ◽  
Closest Vector Problem ◽  
Vector Problem ◽  
Shortest Vector Problem ◽  
Post Quantum Cryptography ◽  
Lattice Based Cryptography

Abstract Recently, lattice-based cryptography has received attention as a candidate of post-quantum cryptography (PQC). The essential security of lattice-based cryptography is based on the hardness of classical lattice problems such as the shortest vector problem (SVP) and the closest vector problem (CVP). A number of algorithms have been proposed for solving SVP exactly or approximately, and most of them are useful also for solving CVP. In this paper, we give a survey of typical algorithms for solving SVP from a mathematical point of view. We also present recent strategies for solving the Darmstadt SVP challenge in dimensions higher than 150.

scholarly journals Approximate Voronoi cells for lattices, revisited

2020 ◽  
Vol 15 (1) ◽  
pp. 60-71
Author(s):  
Thijs Laarhoven
Keyword(s):  
Open Problem ◽  
Time Complexity ◽  
Worst Case ◽  
Average Case ◽  
Voronoi Cells ◽  
Closest Vector Problem ◽  
Vector Problem ◽  
Trade Offs ◽  
Enumeration Algorithms ◽  
Asymptotic Scaling

AbstractWe revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis–Laarhoven–De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than $2^{0.076d + o(d)}$ memory, and we show how to obtain time–memory trade-offs even when using less than $2^{0.048d + o(d)}$ memory. We also settle the open problem of obtaining a continuous trade-off between the size of the advice and the query time complexity, as the time complexity with subexponential advice in our approach scales as $d^{d/2 + o(d)}$ matching worst-case enumeration bounds, and achieving the same asymptotic scaling as average-case enumeration algorithms for the closest vector problem.

scholarly journals Cryptanalysis of Loiss Stream Cipher-Revisited

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Lin Ding ◽  
Chenhui Jin ◽  
Jie Guan ◽  
Qiuyan Wang
Keyword(s):  
Time Complexity ◽  
Stream Cipher ◽  
Linear Equations ◽  
Data Complexity ◽  
Secret Key ◽  
Key Recovery ◽  
Systems Of Linear Equations ◽  
Key Recovery Attack ◽  
Better Than

Loiss is a novel byte-oriented stream cipher proposed in 2011. In this paper, based on solving systems of linear equations, we propose an improved Guess and Determine attack on Loiss with a time complexity of 2231and a data complexity of 268, which reduces the time complexity of the Guess and Determine attack proposed by the designers by a factor of 216. Furthermore, a related key chosenIVattack on a scaled-down version of Loiss is presented. The attack recovers the 128-bit secret key of the scaled-down Loiss with a time complexity of 280, requiring 264chosenIVs. The related key attack is minimal in the sense that it only requires one related key. The result shows that our key recovery attack on the scaled-down Loiss is much better than an exhaustive key search in the related key setting.

scholarly journals An Efficient Algorithm for the Shortest Vector Problem

IEEE Access ◽  
2018 ◽  
Vol 6 ◽  
pp. 61478-61487 ◽  
Author(s):  
Yu-Lun Chuang ◽  
Chun-I Fan ◽  
Yi-Fan Tseng
Keyword(s):  
Efficient Algorithm ◽  
Vector Problem ◽  
Shortest Vector Problem
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