Lenstra's Algorithm Research Articles

Substantiation of correctness and advantages of Lenstra factorization method on Edwards curves

The factorization problem, which is the basis for many classical asymmetric cryptosystems (RSA, Rabin, and others) and a cryptographically strong generator of pseudo-random sequences (PBS), has been investigated in this paper. The methods that served as prototypes for the Lenstra method were described, the method for factorization of numbers, which is analogous to the Lenstra method on Edwards curves, has been proposed. To substantiate the correctness of the method, an appropriate mathematical apparatus was developed. In addition, an analog of the Lenstra method on Edwards curves was constructed with the use of the presented apparatus; the appropriate algorithm for the factorization of numbers was designed. The correctness of the method and correctness of the algorithm operation were substantiated mathematically; the top analytical estimates of its performance speed, as well as the lower estimates of success probability, have been strictly proved. The advantages of the developed method in comparison with the classical Lenstra method, which applies elliptic curves in the Weierstrass form, were presented and strictly substantiated. A comparative analysis of the new and the classical algorithms was performed. Results of the research provided a strict proof that the new algorithm on full Edwards curves, in comparison with the classic one, has some advantages in terms of performance speed, by about 1.5 times. The presented experimental results show that the performance speed increases even larger (by up to 30 per cent) in case the twisted and quadratic curves are used instead of full Edwards curves. It was shown that the assessment of probability of success of the new method increases due to the emergence of new conditions that lead to success of the algorithm that are not satisfied for the classical Lenstra algorithm on Weierstrass curves. The obtained results make it possible to decrease the time required for solving the problem on factorization by approximately 1.5 times, and thus, enable the faster breaking of cryptosystems whose stability is based on this problem

ALGORITHMS FOR FACTORIZATION AND IRREDUCIBILITY TESTING OF POLYNOMIALS USING ELLIPTIC CURVES

In this article irreducibility criteria and factorization algorithms are formulated and proved, that are generalizations of correspondent criteria and algorithms for integers. Generalization of Lenstra’s theorem is of especial interest. It provides correct calculations of multiple points on elliptic curve that is used in Lenstra algorithm. Note that the generalization of this theorem in case of finite field of characteristic 2 is completely different from the classical case and is very non-trivial. Using the criteria obtained, we construct a series of algorithms for factoring and testing of irreducibility of polynomials over finite field. We should note that their performance isn’t their main advantage, but the main advantage is using some abelian group built on elliptic curve. So in case of failure of algorithm we can just chose the other elliptic curve over the corresponding field and repeat the algorithm. These algorithms have no large advantage in speed, like their preimages – analogical algorithms for integers, but they demonstrate new approaches to the solutions of important problem of irreducibility testing and also are very interesting from mathematical point of view.

Incorporating Condition Measures in the Context of Combinatorial Optimization

Integer programming algorithms have some kind of exponential complexity in the worst case. However, it is also observed that data instances of similar sizes might have very different practical complexity when solved by computer algorithms. This paper considers an alternative complexity analysis of some integer programming algorithms, which is based on measures of "intrinsic difficulty." The work extends to the setup of integer programming some notions of condition measures which have been developed for convex optimization. We present bounds on the so-called lattice width of polyhedra and address the impact on the complexity of integer programming algorithms like Lenstra's algorithm as well as branch and bound algorithms. The condition measures introduced here reflect shape and spatial orientation factors which are not fully captured by the traditional combinatorial analysis.

Complexity of integer quasiconvex polynomial optimization

We study a particular case of integer polynomial optimization: Minimize a polynomial F ^ on the set of integer points described by an inequality system F 1 ⩽ 0 , … , F s ⩽ 0 , where F ^ , F 1 , … , F s are quasiconvex polynomials in n variables with integer coefficients. We design an algorithm solving this problem that belongs to the time-complexity class O ( s ) · l O ( 1 ) · d O ( n ) · 2 O ( n 3 ) , where d ⩾ 2 is an upper bound for the total degree of the polynomials involved and l denotes the maximum binary length of all coefficients. The algorithm is polynomial for a fixed number n of variables and represents a direct generalization of Lenstra's algorithm [Math. Oper. Res. 8 (1983) 538–548] in integer linear optimization. In the considered case, our complexity-result improves the algorithm given by Khachiyan and Porkolab [Discrete Comput. Geom. 23 (2000) 207–224] for integer optimization on convex semialgebraic sets.

Non-standard approaches to integer programming

In this survey we address three of the principal algebraic approaches to integer programming. After introducing lattices and basis reduction, we first survey their use in integer programming, presenting among others Lenstra's algorithm that is polynomial in fixed dimension, and the solution of diophanine equations using basis reduction. The second topic concerns augmentation algorithms and test sets, including the role played by Hilbert and Gröbner bases in the development of a primal approach to solve a family of problems for all right-hand sides. Thirdly we survey the group approach of Gomory, showing the importance of subadditivity in integer programming and the generation of valid inequalities, as well the relation to the parametric problem cited above of solving for all right-hand sides.

An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming

In recent years many advances have been made in solution techniques for specially structured 0–1 integer programming problems. In contrast, very little progress has been made on solving general (mixed) integer problems. This, of course, is not true when viewed from the theoretical side: H. W. Lenstra (1983) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of L. Lovász and H. E. Scarf (1988). This method allows us to avoid the ellipsoidal approximations required in Lenstra's algorithm. We report on the solution of a number of small (but difficult) examples, with up to 100 integer variables. Our computer code uses the linear programming optimizer CPLEX as a subroutine to solve the linear programming problems that arise. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

The Generalized Basis Reduction Algorithm

Let F(x) be a convex function defined in Rn, which is symmetric about the origin and homogeneous of degree 1, and let L be the lattice of integers Zn. A definition of a reduced basis, b1, …, bn, of the lattice with respect to the distance function F is presented, and we describe an algorithm which yields a reduced basis in polynomial time, for fixed n. In the special case in which the bodies {x: F(x) ≤ t} are ellipsoids, the definition of a reduced basis is identical with that given by Lenstra, Lenstra and Lovász (1982) and the algorithm is the well-known basis reduction algorithm. We show that the basis vector b1, in a reduced basis, is an approximation to a shortest nonzero lattice point with respect to F and relate the basis vectors bi to Minkowski's successive minima. The results lead to an algorithm for integer programming which executes in polynomial time for fixed n, but which avoids the ellipsoidal approximations required by Lenstra's algorithm. We also discuss the properties of a Korkine-Zolotarev basis for the lattice.