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The main drawback of following this approach is that as the Gaussian elimination algorithm proceeds, the matrix being processed becomes more and more dense, which implies that its sparseness will get completely destroyed after some few iterations. (1) can be solved using standard Gaussian elimination at a computational complexity cost of O ( N 3 ). In the examples analyzed in this work this non-zero row density, which in the reminder of this paper will be denoted as λ, will always be less than one thousand non-zero entries, where an overwhelming majority of these entries take a small integer value.Įq. Moreover, the case of interest in this paper are matrices with a dimension N of tens, and even hundreds of thousands columns and rows.Īt the same time, the average per-row density of non-zero elements in these matrices will be very low. ![]() This means that there is a unique vector w such that the matrix-vector product, ℬ ⋅ w, is equal to the zero vector. The families of matrices studied in this paper will always have a kernel of dimension one, which basically means that for a given matrix ℬ, there is only one non-trivial vector solution w that annihilates ℬ. The space of solutions of Eq.( 1) is sometimes referred as the kernel or null-space of the matrix ℬ. Equivalently, Eq.( 1) has non-zero solutions if and only if ℬ has not full rank. We stress that Eq.( 1) has non-zero solutions if and only if ℬ is a singular matrix, i.e., the determinant of ℬ is zero. Were both, w and 0 in Equation (1) are considered to be N × 1 vectors with integer entries in the range. The linear algebra problem addressed in this work consists of finding a non-trivial vector w ∈ F p (with w ≠ 0) such that: Let ℬ be an N × N square matrix defined over a finite field F p, where p is a large odd prime. In the context of cryptographic applications the aforementioned linear algebra problem can be formally stated as follows. ![]() įor example, solving the DLP problem over small characteristic finite fields, can lead to thousands of linear algebra problems some of them containing up to 266,086 equations and variables and up to 289 million ( ≈ 2 28.1) nonzero entries in the corresponding sparse matrix.įurther, the sought solution for this system of equations must be exact because the matrices arising from these cryptanalysis applications are defined over the integers modulo a prime number. Indeed, when one wants to factorize extremely large integers, or to solve the Discrete Logarithm Problem (DLP) using state-of-the-art index-calculus methods, one needs to solve high dimension linear algebra problems. Ĭryptography is another important application where the problem of solving large sparse linear systems shows up. Besides its famous Google application, the PageRank algorithm has found its way for solving problems in the areas of biology and social network analysis, among others. An emblematic application of sparse linear systems occurs in the process of finding the PageRank vector of the so-called Google matrix, which is a sparse Markov matrix with dimension of about ten billions ( ≈ 2 33) of rows and columns. ![]() ![]() There is a natural question to ask here: is it possible to write the solution to a homogeneous matrix equation using fewer vectors than the one given in the above recipe? We will see in Example 2.5.3 in Section 2.5 that the answer is no: the vectors from the recipe are always linearly independent, which means that there is no way to write the solution with fewer vectors.Īnother natural question is: are the solution sets for inhomogeneuous equations also spans? As we will see shortly, they are never spans, but they are closely related to spans.The computational problem of solving large sparse linear systems of equations shows up in several computer science subdisciplines. Since two of the variables were free, the solution set is a plane.
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