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In this paper we provide a characterization of the complexity class PSPACE by using a purely continuous model defined with polynomial ordinary differential equations. [ABSTRACT FROM AUTHOR]

In 1941, Claude Shannon introduced the General Purpose Analog Computer (GPAC) as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later on electronic) machines of that time. Following Shannon’s arguments, functions generated by the GPAC must satisfy a polynomial differential algebraic equation (DAE). As it is known that some computable functions like Euler’s Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t or Riemann’s Zeta function ζ ( x ) = ∑ k = 0 ∞ 1 k x do not satisfy any polynomial DAE, this argument has often been used to demonstrate in the past that the GPAC is less powerful than digital computation. It was proved in Bournez et al. (2007), that if a more modern notion of computation is considered, i.e. in particular if computability is not restricted to real-time generation of functions, the GPAC is actually equivalent to Turing machines. Our purpose is first to discuss the robustness of the notion of computation involved in Bournez et al. (2007), by establishing that many natural variants of the notion of computation from this paper lead to the same computability result. Second, to go from these computability results towards considerations about (time) complexity: we explore several natural variants for measuring time/space complexity of a computation. Quite surprisingly, whereas defining a robust time complexity for general continuous time systems is a well known open problem, we prove that all variants are actually equivalent even at the complexity level. As a consequence, it seems that a robust and well defined notion of time complexity exists for the GPAC, or equivalently for computations by polynomial ordinary differential equations. Another side effect of our proof is also that we show in some way that polynomial ordinary differential equations can actually be used as a kind of programming model, and that there is a rather nice and robust notion of ordinary differential equation (ODE) programming. [ABSTRACT FROM AUTHOR]

Periodic sequences over finite fields have been used as key streams in private-key cryptosystems since the 1950s. Such periodic sequences should have a series of cryptographic properties in order to resist many attack methods. The binary generalized cyclotomic periodic sequences, constructed by the cyclotomic classes over finite fields, have good pseudo-random properties and correlation properties. In this paper, the linear complexity and minimal polynomials of some generalized cyclotomic sequences over GF( q ) have been determined where q = p m and p is an odd prime. Results show that these sequences have high linear complexity over GF( q ) for a large part of odd prime power q , which means they can resist the linear attack method. [ABSTRACT FROM AUTHOR]