Error linear complexity measures for multisequences

Complexity measures for sequences over finite fields, such as the linear complexity and the k-error linear complexity, play an important role in cryptology. Recent developments in stream ciphers point towards an interest in word-based stream ciphers, which require the study of the complexity of multisequences. We introduce various options for error linear complexity measures for multisequences. For finite multisequences as well as for periodic multisequences with prime period, we present formulas for the number of multisequences with given error linear complexity for several cases, and we present lower bounds for the expected error linear complexity.     

On the joint linear complexity profile of explicit inversive multisequences

We prove lower bounds on the joint linear complexity profile of multisequences obtained by explicit inversive methods and show that for some suitable choices of parameters these joint linear complexity profiles are close to be perfect.     

The expectation and variance of the joint linear complexity of random periodic multisequences

The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences has been investigated. By using the generalized discrete Fourier transform for multisequences, Meidl and Niederreiter determined the expectation of the joint linear complexity of random N-periodic multisequences explicitly. In this paper, we study the expectation and variance of the joint linear complexity of random periodic multisequences. Several new lower bounds on the expectation of the joint linear complexity of random periodic multisequences are given. These new lower bounds improve on the previously known lower bounds on the expectation of the joint linear complexity of random periodic multisequences. By further developing the method of Meidl and Niederreiter, we derive a general formula and a general upper bound for the variance of the joint linear complexity of random N-periodic multisequences. These results generalize the formula and upper bound of Dai and Yang for the variance of the linear complexity of random periodic sequences. Moreover, we determine the variance of the joint linear complexity of random periodic multisequences with certain periods.     

Successive minima profile,lattice profile,and joint linear complexity profile of pseudorandom multisequences

In this paper we use the successive minima profile to measure structural properties of pseudorandom multisequences. We show that both the lattice profile and the joint linear complexity profile of a multisequence can be expressed in terms of the successive minima profile.     

A family of keystream generators with large linear complexity

In this work, a new class of keystream generators with a large linear complexity has been derived. The design criteria are easily compatible with those given in the literature to prevent correlation attacks.     

基于代数曲线上具有高广义联合线性复杂度的多重序列

多重序列的联合线性复杂度是衡量基于字的流密码体系安全的一个重要指标. 由元素取自F_q上的m 重序列和元素取自F_q^m 上的单个序列之间的一一对应, Meidl 和Özbudak 定义多重序列的广义联合线性复杂度为对应的单个序列的线性复杂度. 在本文中, 我们利用代数曲线的常数域扩张, 研究两类多重序列的广义联合线性复杂度. 更进一步, 我们指出这两类多重序列同时具有高联合线性复杂度和高广义联合线性复杂度.     

Round-off error analysis of iterations for large linear systems

Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA _x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A ^*>0 andA has PropertyA. This means that the computed resultx _k approximates the exact solution with relative error of order A·A ^–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax _k –b is of order A² A ^–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55     

On the complexity of scheduling with large communication delays

Given a directed acyclic graph (dag) with unit execution time tasks and constant communication delays c ⩾ 2, we are interested in deciding if there is a schedule for the dag of length at most L. We prove that the problem is polynomial when L is equal to (c + 1), or (c + 2) for the special case of c = 2, and that it is NP-complete for (c + 3) for any value of c, even in the case of a bipartite dag of depth one.     

The proof complexity of linear algebra

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley–Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB=I→BA=I.     

Computational complexity of parametric linear programming

We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.This research is partially supported by Air Force Office of Scientific Research, Air Force Number AFOSR-78-3646.     

Periodic schedules for linear precedence constraints

We consider the computation of periodic cyclic schedules for linear precedence constraints graphs: a linear precedence constraint is defined between two tasks and induces an infinite set of usual precedence constraints between their executions such that the difference of iterations is a linear function. The objective function is the minimization of the maximal period of a task.We recall first that this problem may be modelled using linear programming. A polynomial algorithm is then developed to solve it for a particular class of linear precedence graphs called unitary graphs. We also show that a periodic schedule may not exist for unitary graphs. In the general case, a decomposition of the linear precedence graph into unitary components is computed and we assume that a periodic schedule exists for each of these components. Lower bounds on the periods are exhibited and we show that an optimal periodic schedule may not achieve them. The notion of quasi-periodic schedule is then introduced and we prove that this new class of schedules always reaches these bounds.     

On the complexity of linear boolean operators with thin matrices

Under consideration is the problem of constructing a square Booleanmatrix A of order n without “rectangles” (it is a matrix whose every submatrix of the elements that are in any two rows and two columns does not consist of 1s). A linear transformation modulo two defined by A has complexity o(ν(A) − n) in the base {⊕}, where ν(A) is the weight of A, i.e., the number of 1s (the matrices without rectangles are called thin). Two constructions for solving this problem are given. In the first construction, n = p ² where p is an odd prime. The corresponding matrix H _p has weight p ³ and generates the linear transformation of complexity O(p ² log p log log p). In the second construction, the matrix has weight nk where k is the cardinality of a Sidon set in ℤ _n . We may assume that

The complexity of recognizing linear systems with certain integrality properties

Let A be a 0 − 1 matrix with precisely two 1’s in each column and let 1 be the all-one vector. We show that the problems of deciding whether the linear system (1) defines an integral polyhedron, (2) is totally dual integral (TDI), and (3) box-totally dual integral (box-TDI) are all co-NP-complete, thereby confirming the conjecture on NP-hardness of recognizing TDI systems made by Edmonds and Giles in 1984. Supported in part by NSA grant H98230-05-1-0081 and NSF grants DMS-0556091 and ITR-0326387. Supported in part by the Research Grants Council of Hong Kong and Seed Funding for Basic Research of HKU.     

Periodic points of a linear transformation

We discuss the structure of periodic points of a linear transformation and find the possible set of the primitive periods of periodic points of a linear transformation.