The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Lower bounds for solving linear diophantine equations on random access machines
Journal of the ACM (JACM)
A knapsack type public key cryptosystem based on arithmetic in finite fields
Proceedings of CRYPTO 84 on Advances in cryptology
The cryptographic security of truncated linearly related variables
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
On parallel complexity of integer linear programming, GCD and the iterated mod function
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Random lattices, threshold phenomena and efficient reduction algorithms
Theoretical Computer Science
The Two Faces of Lattices in Cryptology
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
The optimal LLL algorithm is still polynomial in fixed dimension
Theoretical Computer Science - Latin American theoretical informatics
Hidden number problem with hidden multipliers, timed-release crypto, and noisy exponentiation
Mathematics of Computation
A Measure for the Non-Orthogonality of a Lattice Basis
Combinatorics, Probability and Computing
Journal of the ACM (JACM)
Hardness of approximating the shortest vector problem in lattices
Journal of the ACM (JACM)
ACM SIGSAM Bulletin
Hardness of Approximating the Closest Vector Problem with Pre-Processing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Finding short lattice vectors within mordell's inequality
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Cryptanalysis of General Lu-Lee Type Systems
Information Security and Cryptology
Efficient lattice-based signature scheme
International Journal of Applied Cryptography
Rigorous and Efficient Short Lattice Vectors Enumeration
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Parallelization of Sphere-Decoding Methods
High Performance Computing for Computational Science - VECPAR 2008
Low-dimensional lattice basis reduction revisited
ACM Transactions on Algorithms (TALG)
A list-based detection technique for long intersymbol interference channels
IEEE Transactions on Wireless Communications
A hybrid lattice-reduction and meet-in-the-middle attack against NTRU
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
Improved analysis of Kannan's shortest lattice vector algorithm
CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
A digital signature scheme based on CV P∞
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
Faster exponential time algorithms for the shortest vector problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Accelerating lattice reduction with FPGAs
LATINCRYPT'10 Proceedings of the First international conference on Progress in cryptology: cryptology and information security in Latin America
Parallel enumeration of shortest lattice vectors
Euro-Par'10 Proceedings of the 16th international Euro-Par conference on Parallel processing: Part II
SCN'10 Proceedings of the 7th international conference on Security and cryptography for networks
Solving Ellipsoid-Constrained Integer Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
Lattice reduction algorithms: theory and practice
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
Analyzing blockwise lattice algorithms using dynamical systems
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
Extreme enumeration on GPU and in clouds: how many dollars you need to break SVP challenges
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
Cryptanalysis of a quadratic compact knapsack public-key cryptosystem
Computers & Mathematics with Applications
Implicit factoring with shared most significant and middle bits
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Lattice enumeration using extreme pruning
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
Parallel shortest lattice vector enumeration on graphics cards
AFRICACRYPT'10 Proceedings of the Third international conference on Cryptology in Africa
BKZ 2.0: better lattice security estimates
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
Column basis reduction and decomposable knapsack problems
Discrete Optimization
A complexity analysis of a Jacobi method for lattice basis reduction
Proceedings of the Fifth International C* Conference on Computer Science and Software Engineering
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The integer programming problem is: Given m×n and m×l matrices A and b respectively of integers, find whether, there exists an all integer n×l vector x satisfying the m inequalities A×≤b. In settling an important open problem, Lenstra (1981) showed in an elegant way that when n, the number of dimensions is fixed, there is a polynomial-time algorithm to solve this problem. His algorithm achieves a running-time of 0(cn3•p(length of data)) where p is some polynomial and c a constant independent of n. Since such an algorithm has several important applications - cryptography (Shamir (1982)), diophantine approximations (Lagarias (1982)), coding theory (Conway and Sloane (1982), etc. it is important to improve the running time. We present an algorithm here that has a running time of 0(n9nL log L) where L is the length of the input. Whereas Lenstra's algorithm in the worst case reduces an n-dimensional problem to cn2−(n−) dimensional problems, our algorithm effectively reduces an n-dimensional problem to at most polynomially many (n−1) dimensional problems, thus achieving our time bound. The algorithm we propose, first finds a “more orthogonal” basis for a lattice (see the next section for the definition of a lattice) than those of Lenstra (1981) and Lenstra, Lenstra and Lovasz (1982), but in time 0(ndn poly (length of input)). It then uses an enumeration technique to solve integer programming and related problems. While this paper presents mainly the theoretical improvements that can be made in the algorithms, we discuss in section 6 why in practice our estimates of running time may be overly pessimistic. The last part of the paper discusses some complexity issues. It is an interesting open problem as to whether finding the Euclidean shortest non-zero vector of a given lattice is NP-hard. (See Lenstra (1981), Van Emde Boas (1981) and Lagarias (1982)).