ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

block recombination approach for subquadratic space complexity binary field multiplication based on toeplitz matrix vector product in this paper we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity the method is based on decomposing the building blocks of the fan hasan subquadratic toeplitz matrix vector multiplier we reduce the space complexity of their architecture by recombining the building blocks in comparison to other similar schemes available in the literature our proposal presents a better space complexity while having the same time complexity we also show that block recombination can be used for efficient implementation of the ghash function of galois counter mode gcm

a remark concerning m divisibility and the discrete logarithm in the divisor class group of curves the aim of this paper is to show that the computation of the discrete logarithm in the m torsion part of the divisor class group of a curve x over a finite field ko with char ko prime to m or over a local field k with residue field ko can be reduced to the computation of the discrete logarithm in k0 4m for this purpose we use a variant of the tame tate pairing for abelian varieties over local fields in the same way the problem to determine all linear combinations of a finite set of elements in the divisor class group of a curve over k or ko which are divisible by m is reduced to the computation of the discrete logarithm in ko cm

a survey of public key cryptosystems we give an overview of the most important public key cryptosystems and discuss the difficult task of evaluating the merit of such systems

on operator order and chaotic operator order for two operators abstract in this paper we characterize operator order a b o and chaotic operator order log a log b for positive and invertible operators a and b in terms of operator inequalities via the furuta inequality and operator equalities due to the douglas s majorization and factorization related results are obtained which include generalizations and characterizations of some well known results

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

a knapsack public key cryptosystem based on elliptic curve discrete logarithm there is no question that knapsack systems still warrant continuous researches as a result of the np completeness nature the faster speed and a desire to have a wide variety of available cryptosystems this paper presents a new knapsack type public key cryptosystem based on a novel application of the elliptic curve logarithm problem the difficulty can be made sufficient to frustrate attacks against the survey major knapsack public key cryptosystems developed in the past more than 20 years and hope fully shed some light on the future of the knapsack cryptosystems the proposed new cryptosystem can be found some theoretic support on confidentiality

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

bases for parametrized iterativity parametrized iterativity of an algebra means the existence of unique solutions of all finitary recursive systems of equations where recursion is allowed to use only some variables chosen as a parameter we show how such algebras can be introduced in an arbitrary category a by employing a base i e an operation interpreting objects of a as monads on a for every base we prove that free base algebras and free iterative base algebras exist the main result is a coalgebraic construction of the latter all equation morphisms form a diagram whose colimit is proved to be a free iterative base algebra

achieving ntru with montgomery multiplication we propose a new unified architecture that utilizes the montgomery multiplication algorithm to perform a modular multiplication for both integers and binary polynomials and ntru s polynomial multiplications the unified design is capable of supporting a majority of public key cryptosystems such as ntru rsa diffie hellman key exchange and elliptic curve schemes among others furthermore the architecture is highly efficient in terms of area and speed

short memory scalar multiplication for koblitz curves this paper presents a scalar multiplication method for koblitz curves koblitz curves are elliptic curves where the scalar multiplication can be computed in a much faster way than with other curves allowing designs and implementations without arithmetic coprocessor the new method is as fast as the fastest known techniques on koblitz curves but requires much less memory therefore it is of particular interest for environments with low resources our technique is well suited for both hardware and software implementations in hardware we show that a normal basis implementation reduces memory consumption by 85 percent compared to conventional methods but this still has exactly the same computational cost in software thanks to a mixed normal polynomial bases approach our technique allows memory savings up to 70 percent and depending on the instruction set of the cpu can be as fast as the fastest known scalar multiplication methods or can even beat them by a large margin therefore in software and in hardware our scalar multiplication technique offers high performance without sacrifice in view of memory

efficient subquadratic space complexity binary polynomial multipliers based on block recombination some applications like cryptography involve a large number of multiplications of binary polynomial in this paper we consider two three and four way methods for parallel implementation of binary polynomial multiplication we propose optimized three and four way split formulas which reduce the space and time complexity of the best known methods moreover we present a block recombination method which provides some further reduction in the space complexity of the considered two three and four way split multipliers

conversion algorithms and implementations for koblitz curve cryptography in this paper we discuss conversions between integers and tau adic expansions and we provide efficient algorithms and hardware architectures for these conversions the results have significance in elliptic curve cryptography using koblitz curves a family of elliptic curves offering faster computation than general elliptic curves however in order to enable these faster computations scalars need to be reduced and represented using a special base tau expansion hence efficient conversion algorithms and implementations are necessary existing conversion algorithms require several complicated operations such as multiprecision multiplications and computations with large rationals resulting in slow and large implementations in hardware and microcontrollers with limited instruction sets our algorithms are designed to utilize only simple operations such as additions and shifts which are easily implementable on practically all platforms we demonstrate the practicability of the new algorithms by implementing them on altera stratix ii fpgas the implementations considerably improve both computation speed and required area compared to the existing solutions

computing elliptic curve discrete logarithms with the negation map it is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the pollard rho method however the random walks defined on elliptic curve points equivalence class p used by pollard rho will always get trapped in fruitless cycles we propose an efficient alternative approach to resolve fruitless cycles besides the theoretical analysis we also examine the performance of the new algorithm in experiments with elliptic curve groups the experiment results show that we can achieve the speedup by a factor extremely close to 2 using the new algorithm with the negation map which is the best performance reported in the literature

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

extension of the furuta inequality and ando hiai log majorization abstract ando and hiai have obtained many excellent log majorization results for power means of positive semidefinite matrices one of these results ensures matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson inequality firstly we show an extension of the furuta inequality by using itself secondary by using techniques due to ando and hiai and also using this extension of the furuta inequality we extend some results on log majorization

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

low complexity multiplier architectures for single and hybrid double multiplications in gaussian normal bases the extensive rise in the number of resource constrained wireless devices and the needs for secure communications with the servers imply fast and efficient cryptographic computations for both parties efficient hardware implementation of arithmetic operations over finite field using gaussian normal basis is attractive for public key cryptography as it provides free squarings in this paper we first present two low complexity digit level multiplier architectures it is shown that the proposed multipliers outperform the existing gaussian normal basis gnb multiplier structures available in the literature then for the first time using these two architectures we propose a new digit level hybrid multiplier which performs two successive multiplications with the same latency as the one for one multiplication we have studied the efficiency of the proposed hybrid architecture in terms of area and time delay for different digit sizes the main advantage of this new hybrid architecture is to speed up exponentiation and point multiplication whenever double multiplication is required and the traditional schemes fail due to the data dependencies we have investigated the applicability of the proposed hybrid structure to reduce the latency of exponentiation based cryptosystems our analysis and timing results show that the expected acceleration in double exponentiation is considerable prototypes of the presented low complexity multiplier architectures and the proposed hybrid architecture are implemented and experimental results are presented

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

on the discrete logarithm in the divisor class group of curves let x be a curve which is defined over a finite field k of characteristic p we show that one can evaluate the discrete logarithm in pic 0 x p n by o n 2 log p operations in k this generalizes a result of semaev for elliptic curves to curves of arbitrary genus

linear and sublinear time algorithms for the basis of abelian groups it is well known that every finite abelian group g can be represented as a direct product of cyclic groups email protected g 1xg 2x xg t where each g i is a cyclic group of order p j for some prime p and integer j 1 if a i generates the cyclic group of g i i 1 2 t then the elements a 1 a 2 a t are called a basis of g we show a randomized algorithm such that given a set of generators m x 1 x k for an abelian group g and the prime factorization of order ord x i i 1 k it computes a basis of g in o m logn email protected i 1 tn ip i n i 2 time where n g has prime factorization p 1 n 1p 2 n 2 p t n t which is not a part of input this generalizes buchmann and schmidt s algorithm that takes o m g time in another model all elements in an abelian group are put into a list as a part of input we obtain an o n time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group

a hardware algorithm for modular multiplication division a mixed radix 4 2 algorithm for modular multiplication division suitable for vlsi implementation is proposed the algorithm is based on montgomery method for modular multiplication and on the extended binary gcd algorithm for modular division both algorithms are modified and combined into the proposed algorithm so that almost all the hardware components are shared the new algorithm carries out both calculations using simple operations such as shifts additions and subtractions the radix 2 signed digit representation is used to avoid carry propagation in all additions and subtractions a modular multiplier divider based on the algorithm performs an n bit modular multiplication division in o n clock cycles where the length of the clock cycle is constant and independent of n the modular multiplier divider has a linear array structure with a bit slice feature and can be implemented with much smaller hardware than that necessary to implement both multiplier and divider separately

on final coalgebras of continuous functors continuous endofunctors f of locally finitely presentable categories carry a natural metric on their final coalgebra whenever f 0 has an element this metric is proved to be a cauchy completion of the initial algebra of f this is illustrated on the poset of real numbers represented as a final coalgebra of an endofunctor of pos by pavlovic and pratt under additional assumptions on the locally finitely presentable category all finitary endofunctors are proved to have a final coalgebra constructed in steps of the natural iteration construction

on functors preserving coproducts and algebras with iterativity abstract an algebra for a functor h is called completely iterative cia for short if every flat recursive equation in it has a unique solution every cia is corecursive i e it admits a unique coalgebra to algebra morphism from every coalgebra if the converse also holds h is called a cia functor we prove that whenever the base category is hyper extensive i e countable coproducts are well behaved and h preserves countable coproducts then h is a cia functor surprisingly few cia functors exist among standard finitary set functors in fact the only ones are those preserving coproducts they are given by x w y for some sets w and y

low complexity bit parallel systolic montgomery multipliers for special classes of gf 2 sup m recently cryptographic applications based on finite fields have attracted much interest this paper presents a transformation method to implement low complexity montgomery multipliers for all one polynomials and trinomials using this method we propose a new bit parallel systolic architecture for computing multiplications over gf 2 sup m these new multipliers have a latency m 1 clock cycles and each cell incorporates at most one 2 input and gate two 2 input xor gates and four 1 bit latches moreover these new multipliers are shown to exhibit significantly lower latency and circuit complexity than the related systolic multipliers and are highly appropriate for vlsi systems because of their regular interconnection pattern modular structure and fully inherent parallelism

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

arithmetic operations in finite fields of medium prime characteristic using the lagrange representation in this paper we propose a complete set of algorithms for the arithmetic operations in finite fields of prime medium characteristic the elements of the fields ifpk are represented using the newly defined lagrange representation where polynomials are expressed using their values at sufficiently many points our multiplication algorithm which uses a montgomery approach can be implemented in o k multiplications and o k2 log k additions in the base field ifp for the inversion we propose a variant of the extended euclidean gcd algorithm where the inputs are given in the lagrange representation the lagrange representation scheme and the arithmetic algorithms presented in the present work represent an interesting alternative for elliptic curve cryptography

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

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

on the final sequence of a finitary set functor a standard construction of the final coalgebra of an endofunctor involves defining a chain of iterates starting at the final object of the underlying category and successively applying the functor in this paper we show that for a finitary set functor this construction always yields a final coalgebra in 2 steps

counterexample to a question on the operator equation t h1nt n k abstract let h and k be bounded positive operators on a hilbert space and assume that h is nonsingular our previous result shows that if there exists a positive operator t such that t h 1 n t n k for some natural number n then there exists a positive operator t1 such that t 1 h 1 m t 1 m k for any natural number m such that m n in each case there is at most one positive solution t or t1 respectively the purpose of this paper is to show that the reverse implication does not always hold that is although there exists the positive operator t1 such that t 1 h 1 m t 1 m k for some natural number m there need not exist a positive operator t such that t 1 h 1 n t n k for any natural number n such that m

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

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

the operator equation t h1nt n k abstract let h and k be bounded positive operators on a hilbert space and assume that h is nonsingular this paper shows that if there exists a positive operator t such that t h 1 n t n k for some natural number n then for any natural number m such that m n there exists a positive operator t 1 such that t 1 h 1 m t 1 m k in each case there is at most one positive solution t and t 1 respectively

terminal coalgebras in well founded set theory abstract this paper shows that in order to obtain the theorem of aczel and mendler on the existence of terminal coalgebras for an endofunctor on the category of sets it is entirely unnecessary to delve into such exotica as non well founded set theory in addition we discuss the canonical map from the initial algebra for an endofunctor on sets to the terminal coalgebra and show that in many cases it embeds the former as a dense subset of the latter in a certain natural topology by way of example we calculate the terminal coalgebra for various simple endofunctors

infinite trees and completely iterative theories a coalgebraic view infinite trees form a free completely iterative theory over any given signature this fact proved by elgot bloom and tindell turns out to be a special case of a much more general categorical result exhibited in the present paper we prove that whenever an endofunctor h of a category has final coalgebras for all functors h x then those coalgebras tx form a monad this monad is completely iterative i e every guarded system of recursive equations has a unique solution and it is a free completely iterative monad on h the special case of polynomial endofunctors of the category set is the above mentioned theory or monad of infinite trees this procedure can be generalized to monoidal categories satisfying a mild side condition if for an object h the endofunctor h i has a final coalgebra t then t is a monoid this specializes to the above case for the monoidal category of all endofunctors

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

low latency digit serial systolic double basis multiplier over mbi gf 2 m using subquadratic toeplitz matrix vector product approach recently in cryptography and security the multipliers with subquadratic space complexity for trinomials and some specific pentanomials have been proposed for such kind of multipliers alternatively we use double basis multiplication which combines the polynomial basis and the modified polynomial basis to develop a new efficient digit serial systolic multiplier the proposed multiplier depends on trinomials and almost equally space pentanomials aesps and utilizes the subquadratic toeplitz matrix vector product scheme to derive a low latency digit serial systolic architecture if the selected digit size is d bits the proposed digit serial multiplier for both polynomials i e trinomials and aesps requires the latency of 2 left lceil sqrt m over d right rceil while traditional ones take at least o left left lceil m over d right rceil right clock cycles analytical and application specific integrated circuit asic synthesis results indicate that both the area and the time times area complexities of our proposed architecture are significantly lower than the existing digit serial systolic multipliers

efficient multiplier architectures for galois fields gf 2 sup 4n this contribution introduces a new class of multipliers for finite fields gf 2 sup n sup 4 the architecture is based on a modified version of the karatsuba ofman algorithm koa by determining optimized field polynomials of degree four the last stage of the koa and the module reduction can be combined this saves computation and area in vlsi implementations the new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with koa based architectures with separate module reduction the new multipliers lead to highly modular architectures and are thus well suited for vlsi implementations three types of field polynomials are introduced and conditions for their existence are established for the small fields where n 2 3 8 which are of primary technical interest optimized field polynomials were determined by an exhaustive search for each field order exact space and time complexities are provided

the operator equation kp hδ2t12 t12hδ rt12 p δδ rt12hδ2 and its applications abstract let h and k be bounded positive operators on a hilbert space and assume that h is nonsingular based on pedersen and takesaki s research on the operator equation k t h t furuta and bach gave deep discussion on the equation k t 1 2 t 1 2 h 1 n t 1 2 n t 1 2 where n is a natural number as a continuation this paper is to consider the equation k p h 2 t 1 2 t 1 2 h r t 1 2 p r t 1 2 h 2 where p 0 r 0 and p r as applications we prove that the inclusion relations among class w a p r operators are strict and show a generalization of aluthge s result

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

on tree coalgebras and coalgebra presentations for deterministic systems expressed as coalgebras over polynomial functors every tree t an element of the final coalgebra turns out to represent a new coalgebra at the universal property of this family of coalgebras resembling freeness is that for every state s of every system s there exists a unique coalgebra homomorphism from a unique at which takes the root of t to s consequently the tree coalgebras are finitely presentable and form a strong generator thus these categories of coalgebras are locally finitely presentable in particular every system is a filtered colimit of finitely presentable systems in contrast for transition systems expressed as coalgebras over the finite power set functor we show that there are systems which fail to be filtered colimits of finitely presentable finite ones surprisingly if is an uncountable cardinal then presentation is always well behaved whenever an endofunctor f preserves filtered colimits i e is accessible then presentable coalgebras are precisely those whose underlying objects are presentable consequently every f coalgebra is a filtered colimit of presentable coalgebras thus coalg f is a locally presentable category this holds for all endofunctors of accessible categories with colimits of chains corollary a set functor is bounded at in the sense of kawahara and mori iff it is accessible

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

additions and corrections to terminal coalgebras in well founded set theory abstract this note is to correct certain mistaken impressions of the author s that were in the original paper terminal coalgebras in well founded set theory which appeared in theoretical computer science 114 1993 299 315

how iterative reflections of monads are constructed every ideal monad m on the category of sets is known to have a reflection m in the category of all iterative monads of elgot here we describe the iterative reflection m as the monad of free iterative eilenberg moore algebras for m this yields numerous concrete examples if m is the free semigroup monad then m is obtained by adding a single absorbing element if m is the monad of finite trees then m is the monad of rational trees etc

coequational logic for accessible functors covarieties of coalgebras are those classes of coalgebras for an endofunctor h on the category of sets that are closed under coproducts subcoalgebras and quotients equivalently covarieties are classes of h coalgebras that can be presented by coequations adamek introduced a logic of coequations and proved soundness and completeness for all polynomial functors on the category of sets here this result is extended to accessible functors given a presentation of an accessible functor h simple deduction systems for coequations are formulated and it is shown that regularity of the presentation implies soundness and completeness of these deduction systems the converse is true whenever h has a non trivial terminal coalgebra also a method is found to obtain concrete descriptions of cofree and thus terminal coalgebras of accessible functors and is applied to the finite and countable powerset functor as well as to the finite distribution functor

terminal coalgebras and free iterative theories every finitary endofunctor h of set can be represented via a finitary signature and a collection of equations called basic we describe a terminal coalgebra for h as the terminal coalgebra of all trees modulo the congruence of applying the basic equations potentially infinitely often as an application we describe a tree iterative theory on h in the sense of calvin elgot as the theory of all rational trees modulo the analogous congruence this yields a number of new examples of iterative theories e g the theory of all strongly extensional rational finitely branching trees free on the finite power set functor or the theory of all binary rational unordered trees free on one commutative binary operation

low weight polynomial form integers for efficient modular multiplication in 1999 solinas introduced families of moduli called the generalized mersenne numbers gmns which are expressed in low weight polynomial form p f t where t is limited to a power of 2 gmns are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a gmn requires only integer additions and subtractions however since there are not many gmns and each gmn requires a dedicated implementation gmns are hardly useful for other cryptosystems here we modify gmn by removing restriction on the choice of t and restricting the coefficients of f t to 0 and plusmn1 we call such families of moduli low weight polynomial form integers lwpfis we show an efficient modular multiplication method using lwpfi moduli lwpfis allow general implementation and there exist many lwpfi moduli one may consider lwpfis as a trade off between general integers and gmns

the tate pairing and the discrete logarithm applied to elliptic curve cryptosystems the tate pairing is used to reduce the discrete logarithm dl problem on certain elliptic curves to the dl in the multiplicative group of finite fields

provably sublinear point multiplication on koblitz curves and its hardware implementation we describe algorithms for point multiplication on koblitz curves using multiple base expansions of the form k sigmaplusmntaua tau 1 b and k sigmaplusmntaua tau mu b tau2 mutau 1 c we prove that the number of terms in the second type is sublinear in the bit length of k which leads to the first provably sublinear point multiplication algorithm on koblitz curves for the first type we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of tau adic nonadjacent form expansions we present details of an innovative fpga implementation of our algorithm and performance data demonstrating the efficiency of our method we also show that implementations with very low computation latency are possible with the proposed method because parallel processing can be exploited efficiently

on pattern structures of the n soliton solution of the discrete kp equation over a finite field the existence and properties of coherent pattern in the multisoliton solutions of the dkp equation over a finite field are investigated to that end starting with an algebro geometric construction over a finite field we derive a travelling wave formula for n soliton solutions in a finite field however despite it having a form similar to its analogue in the complex field case the finite field solutions produce patterns essentially different from those of classical interacting solitons

completely iterative algebras and completely iterative monads completely iterative theories of calvin elgot formalize potentially infinite computations as solutions of recursive equations one of the main results of elgot and his coauthors is that infinite trees form a free completely iterative theory their algebraic proof of this result is extremely complicated we present completely iterative algebras as a new approach to the description of free completely iterative theories examples of completely iterative algebras include algebras on complete metric spaces it is shown that a functor admits an initial completely iterative algebra iff it has a final coalgebra the monad given by free completely iterative algebras is proved to be the free completely iterative monad on the given endofunctor this simplifies substantially all previous descriptions of these monads moreover the new approach is much more general than the classical one of elgot et al a necessary and sufficient condition for the existence of a free completely iterative monad is proved

several security schemes constructed using ecc based self certified public key cryptosystems in this paper we develop an efficient ecc based self certified public key cryptosystem eccscpkc quite suitable for efficiently securing electronic transactions the proposed eccscpkc is constructed based on the elliptic curve cryptosystems ecc and the id based self certified public key cryptosystems the approaches proposed in this paper possess the following advantages 1 when verifying the validity of public key it does not need to spend extra much time to verify the signature in the digital certificate used in the certificate based public key cryptosystem 2 both distributing a session key and verifying the validity of public key can be concurrently achieved in a logically single step 3 verifying both a signature and the validity of public key can be concurrently accomplished in a logically single step 4 both decrypting a cipher correctly and verifying the validity of public key can be concurrently finished in a logically single step 5 since the proposed methods are combined with the id based and ecc public key cryptosystems they can reduce the computation cost greatly in summary based on the above characteristics the proposed eccscpkc and its related security schemes can gain much efficiency in saving both the communicational cost and the computational effort because they can simplify public key distribution also since the proposed eccscpkc does not need to manage the key directory the cost of system maintenance can be greatly reduced

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