on the density of irreducible nfsrs let n be a positive integer an nfsr of n stages is called irreducible if the family of output sequences of any nfsr of stages less than n is not included in that of the nfsr in this paper we prove that the density of the irreducible nfsrs of n stages is larger than 0 39 this implies that it is expected to find an irreducible nfsr of n stages among three randomly chosen nfsrs of n stages

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

observability of boolean networks a graph theoretic approach boolean networks bns are discrete time dynamical systems with boolean state variables and outputs bns are recently attracting considerable interest as computational models for genetic and cellular networks we consider the observability of bns that is the possibility of uniquely determining the initial state given a time sequence of outputs our main result is that determining whether a bn is observable is np hard this holds for both synchronous and asynchronous bns thus unless p np there does not exist an algorithm with polynomial time complexity that solves the observability problem we also give two simple algorithms with exponential complexity that solve this problem our results are based on combining the algebraic representation of bns derived by d cheng with a graph theoretic approach some of the theoretical results are applied to study the observability of a bn model of the mammalian cell cycle

controllability and observability of boolean control networks the controllability and observability of boolean control networks are investigated after a brief review on converting a logic dynamics to a discrete time linear dynamics with a transition matrix some formulas are obtained for retrieving network and its logical dynamic equations from this network transition matrix based on the discrete time dynamics the controllability via two kinds of inputs is revealed by providing the corresponding reachable sets precisely then the problem of observability is also solved by giving necessary and sufficient conditions

a p adic algorithm to compute the hilbert class polynomial classically the hilbert class polynomial p z x of an imaginary quadratic discriminant a is computed using complex analytic techniques in 2002 couveignes and henocq suggested a p adic algorithm to compute p unlike the complex analytic method it does not suffer from problems caused by rounding errors in this paper we give a detailed description of the algorithm in the paper by couveignes and henocq and our careful study of the complexity shows that if the generalized riemann hypothesis holds true the expected runtime of the p adic algorithm is o log 8 e instead of o 1 e we illustrate the algorithm by computing the polynomial p 639 using a 643 adic algorithm

a survey of full length nonlinear shift register cycle algorithms shift registers have been used to generate sequences of 0 s and 1 s for over thirty years a wide variety of applications has been made of these sequences principally communications have made use of the sequences generated one particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences these sequences are periodic and of length 2 n and all 2 n different binary n tuples appear exactly one time in a periodic portion of the sequence in this paper we discuss various algorithms which have been suggested for generating these sequences

optimal control of boolean control networks in this paper we address the optimal control problem for boolean control networks bcns we first consider the problem of finding the input sequences that minimize a given cost function over a finite time horizon the problem solution is obtained by means of a recursive algorithm that represents the analogue for bcns of the difference riccati equation for linear systems we prove that a significant number of optimal control problems for bcns can be easily reframed into the present setup in particular the cost function can be adjusted so as to include penalties on the switchings provided that we augment the size of the bcn state variable in the second part of the paper we address the infinite horizon optimal control problem and we provide necessary and sufficient conditions for the problem solvability the solution is obtained as the limit of the solution over the finite horizon 0 t and it is always achieved in a finite number of steps finally the average cost problem over the infinite horizon investigated in optimal control of logical control networks y zhao ieee trans autom control vol 56 no 8 pp 1766 1776 aug 2011 is addressed by making use of the results obtained in the previous sections

computing modular polynomials in quasi linear time we analyse and compare the complexity of several algorithms for computing modular polynomials we show that an algorithm relying on floating point evaluation of modular functions and on interpolation which has received little attention in the literature has a complexity that is essentially up to logarithmic factors linear in the size of the computed polynomials in particular it obtains the classical modular polynomials phi ell of prime level ell in time o ell 3 log 4 ell log log ell besides treating modular polynomials for gamma 0 ell which are an important ingredient in many algorithms dealing with isogenies of elliptic curves the algorithm is easily adapted to more general situations composite levels are handled just as easily as prime levels as well as polynomials between a modular function and its transform of prime level such as the schlafli polynomials and their generalisations our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors

boolean dynamics of networks with scale free topology the dynamics of boolean networks with scale free topology are studied the existence of a phase transition from ordered to chaotic dynamics governed by the value of the scale free exponent of the network is shown analytically by analyzing the overlap between two distinct trajectories the phase diagram shows that the phase transition occurs for values of the scale free exponent in the open interval 2 2 5 since the boolean networks under study are directed graphs the scale free topology of the input connections and that of the output connections are studied separately ultimately these two topologies are shown to be equivalent a numerical study of the attractor structure of the configuration space reveals that this structure is similar in both networks with scale free topologies and networks with homogeneous random topologies however an important result of this work is that the fine tuning usually required to achieve stability in the dynamics of networks with homogeneous random topologies is no longer necessary when the network topology is scale free finally based on the results presented in this work it is hypothesized that the scale free topology favors the evolution and adaptation of network functioning from a biological perspective

output feedback stabilization of boolean control networks in the paper output feedback control of boolean control networks bcns is investigated first necessary and sufficient conditions for the existence of a time invariant output feedback tiof law stabilizing the bcn to some equilibrium point are given and constructive algorithms to test the existence of such a feedback law are proposed two sufficient conditions for the existence of a stabilizing time varying output feedback tvof are then given finally an example concerning the lac operon in the bacterium escherichia coli is presented to illustrate the effectiveness of the proposed techniques

dirichlet problems for plurisubharmonic functions on compact sets in this paper we solve the dirichlet problems for different classes of plurisubharmonic functions on compact sets in mathbb c n including continuous pluriharmonic and maximal functions

ramanujan s class invariants and their use in elliptic curve cryptography the complex multiplication cm method is a method frequently used for the generation of elliptic curves ecs over a prime field f p the most demanding and complex step of this method is the computation of the roots of a special type of class polynomials called hilbert polynomials however there are several polynomials called class polynomials which can be used in the cm method having much smaller coefficients and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding hilbert polynomials in this paper we propose the use of a new class of polynomials which are derived from ramanujan s class invariants t n we explicitly describe the algorithm for the construction of the new polynomials and give the necessary transformation of their roots to the roots of the corresponding hilbert polynomials we provide a theoretical asymptotic bound for the bit precision requirements of all class polynomials and also using extensive experimental assessments we compare the efficiency of using the new polynomials against the use of the other class polynomials our comparison shows that the new class of polynomials clearly surpass all of the previously used polynomials when they are used in the generation of prime order elliptic curves

a scalable method for constructing galois nlfsrs with period 2 n 1 using cross join pairs a method for constructing n stage galois nlfsrs with period 2n 1 from n stage maximum length lfsrs is presented nonlinearity is introduced into state cycles by adding a nonlinear boolean function to the feedback polynomial of the lfsr each assignment of variables for which this function evaluates to 1 acts as a crossing point for the lfsr state cycle the effect of nonlinearity is cancelled and state cycles are joined back by adding a copy of the same function to a later stage of the register the presented method requires no extra time steps and it has a smaller area overhead compared to the previous approaches based on cross join pairs it is feasible for large n

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

kalman decomposition for boolean control networks the kalman decomposition problem for boolean control networks bcns is solved completely a necessary and sufficient condition described by vertex partitions of digraphs is proposed for the kalman decomposition and an algorithm is designed to compute the logical coordinate transformation that results in the kalman decomposition finally two examples are presented one example illustrates the procedure of realizing the kalman decomposition the other one reveals that not all the bcns can be transformed into the kalman decomposition form this is an essential property different from that of the traditional linear control systems

proof of a conjecture of zahariuta concerning a problem of kolmogorov on the e entropy we prove a conjecture of zahariuta which itself solves a problem of kolmogorov on the e entropy of some classes of analytic functions for a given holomorphically convex compact subset k in a pseudoconvex domain d in c n zahariuta s conjecture consists in approximating the relative extremal function u k d uniformly on any compact subset of d k by pluricomplex green functions on d with logarithmic poles in the compact subset k

further results on the decomposition of an nfsr into the cascade connection of an nfsr into an lfsr nonlinear feedback shift registers nfsrs are widely used in stream cipher design as building blocks in this paper we study the problem of decomposing an nfsr into the cascade connection of an nfsr into a linear feedback shift register lfsr which is a kind of concatenation of an nfsr and lfsr a necessary and sufficient condition for such decomposition is provided and other algebraic properties about such decomposition are also studied based on these theoretical results a binary decision diagram bdd based algorithm for such decomposition is proposed compared with the previous algorithm proposed by ma et al our algorithm can find more accurate candidate lfsr and the algebraic properties presented in this paper guarantee that the memory requirement during our verification is linear in the size of the bdd of the nfsrs characteristic function

on minimum period of nonlinear feedback shift registers in grain like structure grain is one of three hardware oriented finalists of the estream project a nonlinear feedback shift register nfsr in grain like structure is a cascade connection of a linear feedback shift register lfsr into an nfsr in which the characteristic polynomial of the lfsr is primitive and the feedback function of the nfsr is nonsingular in 2011 hu and gong pointed out that the period of the sequence generated by an nfsr in grain like structure is a multiple of the period of the sequence generated by its lfsr if the initial state of the lfsr is nonzero meanwhile they proposed an open problem for fixed feedback functions of an nfsr and an lfsr determine whether the sequences generated by the nfsr in grain like structure can achieve the minimum period i e the period of the lfsr when the initial state of the lfsr is nonzero and if they can achieve provide at least one pair of the initial states of the nfsr and lfsr clearly from a security point of view it is not preferable if the sequences generated by an nfsr in grain like structure achieve the minimum period this paper converts the open problem into a problem of solving an integer equation with respect to two unknown integers that uniquely correspond to the initial states of the nfsr and lfsr by viewing the nfsr as a boolean control network based on the integer equation this paper shows that for any given initial state of an n stage nfsr and any given nonzero initial state of an m stage lfsr the probability that the sequence generated by the nfsr in grain like structure achieves the minimum period 2 m 1 is at most 2 n this implies that the probability of the cascade connection used in grain achieving the minimum period is very small

approximation of plurifinely plurisubharmonic functions abstract in this study we consider the approximation of a negative plurifinely plurisubharmonic function defined on a plurifinely domain by an increasing sequence of plurisubharmonic functions defined in euclidean domains

choosing the correct elliptic curve in the cm method we give an elementary way to distinguish between the twists of an ordinary elliptic curve e over f p in order to identify the one with p 1 2u points when p u 2 dv 2 with 2u 2v and e is constructed using the cm method for finding elliptic curves with a prescribed number of points our algorithms consist in most cases of reading off simple congruence conditions on u and v modulo 4

on boolean control networks with maximal topological entropy boolean control networks bcns are discrete time dynamical systems with boolean state variables and inputs that are interconnected via boolean functions bcns are recently attracting considerable interest as computational models for genetic and cellular networks with exogenous inputs the topological entropy of a bcn with m inputs is a nonnegative real number in the interval 0 m log 2 roughly speaking a larger topological entropy means that asymptotically the control is more powerful we derive a necessary and sufficient condition for a bcn to have the maximal possible topological entropy our condition is stated in the framework of cheng s algebraic state space representation of bcns this means that verifying this condition incurs an exponential time complexity we also show that the problem of determining whether a bcn with n state variables and m n inputs has a maximum topological entropy is np hard suggesting that this problem cannot be solved in general using a polynomial time algorithm

plurisubharmonic approximation and boundary values of plurisubharmonic functions we study the problem of approximating plurisubharmonic functions on a bounded domain omega by continuous plurisubharmonic functions defined on neighborhoods of omega over bar it turns out that t

approximation of plurisubharmonic functions by multipole green functions for a strongly hyperconvex domain d c c n we prove that multipole pluricomplex green functions are dense in the cone in l 1 d of negative plurisubharmonic functions with zero boundary values

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method