the laplacian spectral radius of some graphs the laplacian spectral radius of a graph is the largest eigenvalue of the associated laplacian matrix in this paper we determine those graphs which maximize the laplacian spectral radius among all bipartite graphs with edge connectivity at most k we also characterize graphs of order n with k cut edges having laplacian spectral radius equal to n

the double base number system and its application to elliptic curve cryptography we describe an algorithm for point multiplication on generic elliptic curves based on a representation of the scalar as a sum of mixed powers of 2 and 3 the sparseness of this so called double base number system combined with some efficient point tripling formulae lead to efficient point multiplication algorithms for curves defined over both prime and binary fields side channel resistance is provided thanks to side channel atomicity

the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices in this paper we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices respectively

fast elliptic scalar multiplication using new double base chain and point halving the fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication based on the double base chain representation of scalar using powers of 2 and 3 we propose a new representation with powers of 1 2 and 3 instead thus the efficient point halving operation can be incorporated in the new double base chain to achieve fast scalar multiplication experimental results show that our approach leads to a lower complexity which contributes to the efficient implementation of elliptic curve cryptosystems

extended elliptic curve montgomery ladder algorithm over binary fields with resistance to simple power analysis in this paper we propose a scalar multiplication algorithm on elliptic curves over gf 2 m the proposed algorithm is an extended version of the montgomery ladder algorithm with the quaternary representation of the scalar in addition in order to improve performance we have developed new composite operation formulas and apply them to the proposed scalar multiplication algorithm the proposed composite formulas are 2p 1 2p 2 3p 1 p 2 and 4p 1 where p 1 and p 2 are points on an elliptic curve they can be computed using only the x coordinate of a point p x y in the affine coordinate system however the proposed scalar multiplication algorithm is vulnerable to simple power analysis attacks because different operations are performed depending on the bits of the scalar unlike the original montgomery ladder algorithm therefore we combine the concept of the side channel atomicity with the proposed composite operation formulas to prevent simple power analysis furthermore to optimize the computational cost we use the montgomery trick which can reduce the number of finite field inversion operations used in the affine coordinate system as the result the proposed scalar multiplication algorithm saves at least 26 of running time with small storage compared to the previous algorithms such as window based methods and comb based methods

a nonlinear elliptic curve cryptosystem based on matrices we propose a new mathematical problem that is applicable to public key cryptography based on the discrete logarithm problem dlp it uses certain elements formed by two matrices with elements in a finite field and a matrix whose elements are points of an elliptic curve with this system we get a larger key space without increasing the underlying elliptic curve and consequently without the computational requirements inherent to the set up of elliptic curves at random also we expose the diffie hellman key agreement protocol with this system acting as the underlying mathematical problem

evaluation of discrete logarithms in a group of p torsion points of an elliptic curve in characteristic p we show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs o ln p operations in this field

construction of cm picard curves in this article we generalize the cm method for elliptic and hyperelliptic curves to picard curves we describe the algorithm in detail and discuss the results of our implementation

on the spectral radius of graphs with cut edges we study the spectral radius of graphs with n vertices and k cut edges in this paper we show that of all the connected graphs with n vertices and k cut edges the maximal spectral radius is obtained uniquely at k k where k k n is a graph obtained by joining k independent vertices to one vertex of kn k we also discuss the limit point of the maximal spectral radius

on the laplacian spectral radii of bipartite graphs the laplacian spectral radius of a graph is the largest eigenvalue of the associated laplacian matrix in this paper we provide structural and behavioral details of graphs with maximum laplacian spectral radius among all bipartite connected graphs of given order and size using these results we provide a unified approach to determine the graphs with maximum laplacian spectral radii among all trees and all bipartite unicyclic bicyclic tricyclic and quasi tree graphs respectively

uncertainty measure on fuzzy partitions in this paper we face the problem of constructing the uncertainty or perhaps expected information of a fuzzy partition in terms of the crisp measures of its cuts we solve the problem in the cases in which the measure is either compositive or branching

id based threshold digital signature schemes on the elliptic curve discrete logarithm problem we present three identification group oriented t n threshold digital signature schemes based on the elliptic curve discrete logarithm problem with a view to optimizing these techniques for implementation within a network by employing these schemes any t out of n users in a group can represent this group to sign the group signature this paper has the following five notable advantages 1 this work demonstrates the feasibility of constructing a fast and extremely secure user identification system 2 any group signature can be mutually generated by at least t group member 3 the signature verification process is simplified because there is only one group public key required 4 the group signature can be verified by any outsider 5 these schemes offer higher speeds lower power consumption smaller bandwidth requirements and smaller certificate sizes than other existing public key schemes

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

the montgomery modular inverse revisited we modify an algorithm given by kaliski to compute the montgomery inverse of an integer modulo a prime number we also give a new definition of the montgomery inverse and introduce efficient algorithms for computing the classical modular inverse the kaliski montgomery inverse and the new montgomery inverse the proposed algorithms are suitable for software implementations on general purpose microprocessors

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

reducing elliptic curve logarithms to logarithms in a finite field elliptic curve cryptosystems have the potential to provide relatively small block size high security public key schemes that can be efficiently implemented as with other known public key schemes such as rsa and discrete exponentiation in a finite field some care must be exercised when selecting the parameters involved in this case the elliptic curve and the underlying field specific classes of curves that give little or no advantage over previously known schemes are discussed the main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field for the class of supersingular elliptic curves the reduction takes probabilistic polynomial time thus providing a probabilistic subexponential time algorithm for the former problem

a reformulation of entropy in the presence of indistinguishability operators this paper deals with the measurement of entropy when an indistinguishability relation on the set of events has been defined our approach states that entropy could be measured in terms of the observed distinguishability of the set of events in this sense the observer paradigm is introduced and definitions for joint and conditional entropy under this paradigm are given we also present some interesting properties and relationships with shannon s entropy measure

multicore curve based cryptoprocessor with reconfigurable modular arithmetic logic units over gf 2 n this paper presents a reconfigurable curve based cryptoprocessor that accelerates scalar multiplication of elliptic curve cryptography ecc and hyperelliptic curve cryptography hecc of genus 2 over gf 2n by allocating a copies of processing cores that embed reconfigurable modular arithmetic logic units malus over gf 2n the scalar multiplication of ecc hecc can be accelerated by exploiting instruction level parallelism ilp the supported field size can be arbitrary up to a n 1 1 the superscaling feature is facilitated by defining a single instruction that can be used for all field operations and point divisor operations in addition the cryptoprocessor is fully programmable and it can handle various curve parameters and arbitrary irreducible polynomials the cost performance and security trade offs are thoroughly discussed for different hardware configurations and software programs the synthesis results with a 0 13 mum cmos technology show that the proposed reconfigurable cryptoprocessor runs at 292 mhz whereas the field sizes can be supported up to 587 bits the compact and fastest configuration of our design is also synthesized with a fixed field size and irreducible polynomial the results show that the scalar multiplication of ecc over gf 2163 and hecc over gf 283 can be performed in 29 and 63 mus respectively

improved computation of square roots in specific finite fields in this paper we study exponentiation in the specific finite fields f with very special exponents such as those that occur in algorithms for computing square roots here q is a prime power q pk where k 1 and k is odd our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent to the best of our knowledge it is the first major improvement to the tonelli shanks algorithm for example the number of multiplications can be reduced to at least 60 percent on the average when p 1 mod 16 several numerical examples are given that show the speedup of the proposed methods

the information content of fuzzy relations and fuzzy rules based on different emphases some general information content measures of fuzzy relations are given distinguished from the measures related to fuzzy relations proposed before these measures try to estimate the information conveyed by multiple domain fuzzy relations without preassigned probability distribution since fuzzy rules can be fully captured by fuzzy relations from an input universe to an output universe the information content of fuzzy rules can be easily measured by the information content of fuzzy relations proposed by us however there also exists a difference between fuzzy relations and fuzzy rules rules especially classification rules always have a direction from the antecedent and consequent while relations do not have direction based on this difference some general measures for the information content of fuzzy rules are proposed in practice these measures can do well in the evaluation and selection of fuzzy rules finally the measures of the information content of fuzzy rules are used to evaluate the stability and sensitivity of fuzzy implication operators in fuzzy control

efficient elliptic curve scalar multiplication algorithms resistant to power analysis this paper presents four algorithms for securing elliptic curve scalar multiplication against power analysis the highest weight binary form hbf of scalars and randomization are applied to resist power analysis by using a special method to recode the scalars the proposed algorithms do not suffer from simple power analysis spa with the randomization of the secret scalar or base point three of the four algorithms are secure against differential power analysis dpa refined power analysis rpa and zero value point attacks zpa the countermeasures are also immune to the doubling attack fast shamir s method is used in order to improve the efficiency of parallel scalar multiplication compared with previous countermeasures the new countermeasures achieve higher security and do not impact overall performance

measures of entropy and fuzziness related to aggregation operators we suggest some new measures for the calculation of the entropy of a probability distribution one motivation for this is to provide tools for the application of the maximum entropy principle the first class of entropy measures is based on the ordered weighted averaging operator another class of these measures is based on the distance of the probability distribution from the uniform distribution a third class of entropy measures is based on the t conorms we then suggest some measures of fuzziness associated with a fuzzy set

optimal left to right binary signed digit recoding this paper describes new methods for producing optimal binary signed digit representations this can be useful in the fast computation of exponentiations contrary to existing algorithms the digits are scanned from left to right i e from the most significant position to the least significant position this may lead to better performances in both hardware and software

interlacing eigenvalues on some operations of graphs in this paper we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix standard laplacian and normalized laplacian also we explore some applications of this interlacing

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

the laplacian spectral radius of some bipartite graphs in this paper we study the largest laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs respectively identifying the center of a star k1 k and one vertex of degree n of km n we denote by km nk the resulting graph we show that the graph k2 n k 2k 1 k n 4 is the unique graph with the largest laplacian spectral radius among the bipartite graphs with n vertices and k cut edges and k2 3n 5 n 7 is the unique graph with the largest laplacian spectral radius among all the bicyclic bipartite graphs

elliptic curve scalar multiplication algorithm using complementary recoding elliptic curve based cryptosystem is an efficient public key cryptosystem which is more suitable for limited environments the very expensive operation in elliptic curve based cryptographic protocol is the scalar multiplication in this paper scalar multiplication using complementary recoding method is proposed the intermediate operations carried out in conversion are less than the other existing scalar multiplication algorithm the efficiency of our proposed method is tested and found that our method is an efficient than the other methods

a w nnaf method for the efficient computation of scalar multiplication in elliptic curve cryptography based on the non adjacent form naf expression here we present a new method to express a large positive integer k with the base of any positive integer w other than 2 we call this method the w nnaf method as the expression is near to the naf one and with base w this expression leads to the minimal hamming weight of all the base w expressions based on the proposed expression and some existing methods an algorithm is developed to compute kp efficiently

fast simultaneous scalar multiplication scalar multiplication is a very expensive operation in elliptic curve based cryptographic protocol in this paper we propose a modified simultaneous elliptic curve scalar multiplication algorithm using complementary recoding the signed binary representation of an integer is obtained by simple arithmetic operations such as complement and bitwise subtraction this is an efficient method to obtain the signed binary representation of scalar when compared to other standard methods such as naf mof and jsf

fuzzy measures and the entropy of fuzzy partitions abstract using an additive fuzzy measure a notion of the entropy of a finite fuzzy partition has been defined d dumitrescu on fuzzy partitions in itinerant seminar on functional equations approximation and convexity cluj napoca 1983 pp 57 60 d dumitrescu and m barbu fuzzy entropy and processes in itinerant seminar on functional equations approximation and convexity cluj napoca 1985 pp 71 74 this notion contains as a particular case a well known concept of entropy m a gil et al fuzzy sets and systems15 1985 65 78 h tanaka et al kybernetes5 1976 25 30 i j taneja bivariate entropies of finite partitions of fuzzy sets in preprints of second ifsa congress tokyo 1987 vol 2 pp 696 699 the aim of this paper is to review some basic properties of the considered entropy

an efficient undeniable group oriented signature scheme three main methods with favorable security and efficiency can be considered in the design of cryptosystems these methods are integer factorization systems of which rsa is the best known example discrete logarithm systems such as dsa and elliptic curve cryptosystems ecc math comput 48 1987 203 advances in cryptology crypto 85 springer verlag 1986 p 417 the elliptic curve cryptosystem has been applied in this study because of its low computational cost and short key size which reduces the overheads in a limited hardware environment thus the elliptic curve constitutes a suitable environment for the cryptosystems this study improves the undeniable signature scheme proceedings of the eleventh national conference on information security 2001 p 331 by reducing its time complexity such an improvement makes the proposed scheme more efficient and secure

security analysis of the public key algorithm based on chebyshev polynomials over the integer ring zn recently kocarev and tasev 20 proposed to use chebyshev polynomials over real numbers to design a public key algorithm by employing the semigroup property bergamo et al 4 pointed out that the public key algorithm based on chebyshev polynomials working on real numbers is not secure and devised an attack which permits to recover the corresponding plaintext from a given ciphertext later kocarev et al 19 generalized the chebyshev polynomials from real number fields to finite fields and finite rings to make the public key algorithm more secure and practical however we analyzed the period distribution of the sequences generated by the chebyshev polynomials over finite fields 21 when the modulus n is prime we found this algorithm was also not secure and proposed an attack on this algorithm over finite fields we then proposed some schemes to improve the security in this paper we further analyze in detail the period distribution of the sequences generated by chebyshev polynomials over the integer ring z n when n is composite it turns out that the period distribution is poor if n is not chosen properly and there are many small periods which are not secure in the sense of cryptology based on these findings we devise an attack on the public key algorithm based on chebyshev polynomials over the integer ring z n we also propose some suggestions to avoid this attack