Modular Linear Equations Research Articles

On modular inverse matrices a computation approach

This paper describes the proposal of a numerical method and its extension, to compute Modular Inverse Matrices and so, Modular Linear Equations Systems (with one, infinite or no-solution set), with no theoretical limit, inZ_n; considering polynomial and logarithmic time computational complexity. The geometric interpretation of this, implies that elements, such as planes of these vector spaces, interact in the n-dimensional grid. The interaction and ‘movement’ inside the Grid, can only be possible in a discrete way; from one point to another, like digital states. On the other hand, this work also considers applied mathematics in fields such as cryptography. Based on research, it was observed that this method is an algorithm, because it is precise, defined and finite, so it can be programmed in any computer language. This work constitutes a new approach in numerical analysis for modular inverse matrix computation, plotted in 3-axis linearly. Uses and applications of this proposal are diverse.

Anti Quasi-Cyclic LDPC Codes

Low-density parity-check (LDPC) codes based on affine permutation matrices, APM-LDPC codes, have been attracted recently, because of some advantages rather than QC-LDPC codes in minimum-distance, cycle distribution and error-rate performance. In this letter, circulant and anti-circulant permutation matrices are used to define a class of LDPC codes, called AQC-LDPC codes, which can be considered as an special case of APM-LDPC codes. In fact, each AQC-LDPC code can be verified by a sign matrix and a slope matrix which are helpful to show each cycle in the Tanner graph by a modular linear equation. For the normal sign matrix $A$ , if −1 $\in A$ , it is shown that the corresponding AQC-LDPC code has maximum-girth 8. Finally, two explicit constructions for AQC-LDPC codes with girths 6, 8 are presented which have some benefits rather than the explicitly constructed QC and APM LDPC codes in minimum-distance, cycle distributions and bit-error-rate performances.

Cryptanalysis of two cryptosystems based on multiple intractability assumptions

Two public key cryptosystems based on the two intractable number-theoretic problems, integer factorisation and simultaneous Diophantine approximation, were proposed in 2005 and 2009, respectively. In this study, the authors break these two cryptosystems for the recommended minimum parameters by solving the corresponding modular linear equations with small unknowns. For the first scheme, the public modulus is factorised and the secret key is recovered with the Gauss algorithm. By using the LLL basis reduction algorithm for a seven-dimensional lattice, the public modulus in the second scheme is also factorised and the plaintext is recovered from a ciphertext. The author's attacks are efficient and verified by experiments which were done within 5s.

Cryptanalysis of a Public Key Cryptosystem Based on the Matrix Combinatorial Problem

In this paper, we present a cryptanalysis of a public key cryptosystem based on the matrix combinatorial problem proposed by Wang and Hu (2010). Using lattice-based methods finding small integer solutions of modular linear equations, we recover the secret key of this cryptosystem for a certain range of parameters. In experiments, for the suggested parameters by Wang and Hu, the secret key can be recovered in seconds.

An exhaustive search method to find all small solutions of a multivariate modular linear equation

An exhaustive search method to find all small solutions of a multivariate modular linear equation

Cryptanalysis of a quadratic compact knapsack public-key cryptosystem

Recently, Wang and Hu have proposed a high-density quadratic compact knapsack public-key cryptosystem using the Chinese remainder theorem to disguise two secret cargo vectors. The system is claimed to be secure against certain known attacks; however, it has not been demonstrated to fulfill any provable security goals. In this work, we show that this system is not secure. Exploiting the special structure of system parameters, we first show that a candidate list for the secret modulus can be obtained by solving linear equations with small solutions. Next, we show that with this candidate list, all other secrets can be recovered in succession with lattice-based methods by solving certain modular linear equations with small solutions. As a result, recovering a private key can be done in about 11 h for the proposed system parameter n = 100 . We also discuss a method to thwart the proposed attack.

Probability method for cryptanalysis of general multivariate modular linear equation

Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. However, one problem arises here that if the equation has a short solution in given bounded range, the results outputted by earlier algorithms are often not the ones we are interested in. In this paper, we present a probability method based on lattice basis reduction to solve the problem. For a general multivariate modular linear equation with short solution in the given bounded range, the new method outputs this short solution in polynomial time, with a high probability. When the number of unknowns is not too large (smaller than 68), the probability is approximating 1. Experimental results show that Knapsack systems and Lu-Lee type systems are easily broken in polynomial time with this new method.

Worst‐Case to Average‐Case Reductions Based on Gaussian Measures

We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well‐known closest vector problem). The approximation factor we obtain is $n \log^{O(1)} n$ for all four problems. This greatly improves on all previous work on the subject starting from Ajtai’s seminal paper [Generating hard instances of lattice problems, in Complexity of Computations and Proofs, Quad. Mat. 13, Dept. Math., Seconda Univ. Napoli, Caserta, Italy, 2004, pp. 1–32] up to the strongest previously known results by Micciancio [SIAM J. Comput., 34 (2004), pp. 118–169]. Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the high‐dimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its moments remain essentially unchanged.