We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the complexity of solving arbitrary ordinary polynomial differential equations in terms of Puiseux series. Introduction LetK = Q(T1, . . . , Tl)[η] be a finite extens...

We consider the problem of solving a linear system Ax = b over a cyclotomic field. What makes cyclotomic fields of special interest is that we can easily find a prime p that splits the minimal polynomial m(z) for the field into linear factors. This makes it possible to develop very fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese re...

This paper presents a general framework for Shanks transformations of sequences of elements in a vector space. It is shown that the Minimal Polynomial Extrapolation (MPE), the Modified Minimal Polynomial Extrapolation (MMPE), the Reduced Rank Extrapolation (RRE), the Vector Epsilon Algorithm (VEA), the Topological Epsilon Algorithm (TEA), and Anderson Acceleration (AA), which are standard gener...

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and 〈B〉 denote the subring of the full matrix ring generated by B. Then 〈B〉 is a free Z-module of finite rank, which guarantees that there are only finitely many ideals of 〈B〉 with given finite index. Thus, the formal Diric...

Zhang Shi-Yi, Wang Yong-juan, Gao Yang, Wang Tao Corresponding author: Wang Yong-juan, E-mail: [email protected] Abstract: The 4 4  MDS matrix over 2 F is widely used in the design of block cipher's linear diffusion layers. However, considering the cost of a lightweight cipher's implementation, the sum of XOR operations of a MDS matrix usually plays the role of measure. During the research on t...

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices nk of the k-iterated line digraph L (G), for k ≥ 0, where L(G) = G. We obtain this result by using the minimal polynomial of a quotient digraph π(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our ...

We present new, efficient algorithms for computations on separable matrix algebras over infinite fields. We provide a probabilistic method of the Monte Carlo type to find a generator for the centre of a given algebra A ⊆ Fm×m over an infinite field F. The number of operations used is within a logarithmic factor of the cost of solving m×m systems of linear equations. A Las Vegas algorithm is als...

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a—possibly randomized— verification algorithm that proves the correctness of each output. Wiedemann’s algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divid...

Many lightweight block ciphers apply a very simple key schedule in which the round keys only differ by addition of a roundspecific constant. Generally, there is not much theory on how to choose appropriate constants. In fact, several of those schemes were recently broken using invariant attacks, i.e., invariant subspace or nonlinear invariant attacks. This work analyzes the resistance of such c...

We present and analyse a Monte-Carlo algorithm to compute the minimal polynomial of an n× n matrix over a finite field that requires O(n) field operations and O(n) random vectors, and is well suited for successful practical implementation. The algorithm, and its complexity analysis, use standard algorithms for polynomial and matrix operations. We compare features of the algorithm with several o...