Journal ArticleDOI
Reconstructing truncated integer variables satisfying linear congruences
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TLDR
A general polynomial time algorithm is proposed to find small integer solutions to systems of linear congruences and will solve most problems when twice as much information as that necessary to uniquely determine the variables is available.Abstract:
We propose a general polynomial time algorithm to find small integer solutions to systems of linear congruences. We use this algorithm to obtain two polynomial time algorithms for reconstructing the values of variables $x_1 , \cdots ,x_k $ when we are given some linear congruences relating them together with some bits obtained by truncating the binary expansions of the variables. The first algorithm reconstructs the variables when either the high order bits or the low order bits of the $x_i $ are known. It is essentially optimal in its use of information in the sense that it will solve most problems almost as soon as the variables become uniquely determined by their constraints. The second algorithm reconstructs the variables when an arbitrary window of consecutive bits of the variables is known. This algorithm will solve most problems when twice as much information as that necessary to uniquely determine the variables is available. Two cryptanalytic applications of the algorithms are given: predicting li...read more
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References
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Proceedings ArticleDOI
Improved algorithms for integer programming and related lattice problems
TL;DR: The proposed algorithm first finds a “more orthogonal” basis for a lattice than those of Lenstra (1981) and Lenstra, Lenstra and Lovasz (1982), but in time 0(ndn poly (length of input)).
Journal ArticleDOI
Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix
Ravi Kannan,Achim Bachem +1 more
TL;DR: Recently, Frumkin pointed out that none of the well-known algorithms that transform an integer matrix into Smith or Hermite normal form is known to be polynomially bounded in its runn...
Book ChapterDOI
Cracking a multiplicative congruential encryption algorithm
TL;DR: This paper shows how the computationally efficient solution of a mathematics problem leads to the successful cracking of a cryptographic system.