analysis of the variable length nonzero window method for exponentiation abstract the variable length nonzero window method is a way of computing exponentiation and modular exponentiation this method has been analyzed but there is a substantial difference between the analysis and experimental results the difference results from some logical errors in the analysis we analyze the variable length nonzero window method by modeling it as a markov chain we consider all details and our analysis predicts the behavior of the variable length nonzero window method exactly

high speed double precision computation of reciprocal division square root and inverse square root a new method for the high speed computation of double precision floating point reciprocal division square root and inverse square root operations is presented in this paper this method employs a second degree minimax polynomial approximation to obtain an accurate initial estimate of the reciprocal and the inverse square root values and then performs a modified goldschmidt iteration the high accuracy of the initial approximation allows us to obtain double precision results by computing a single goldschmidt iteration significantly reducing the latency of the algorithm two unfolded architectures are proposed the first one computing only reciprocal and division operations and the second one also including the computation of square root and inverse square root the execution times and area costs for both architectures are estimated and a comparison with other multiplicative based methods is presented the results of this comparison show the achievement of a lower latency than these methods with similar hardware requirements

existence of minimum upcrossing controllers an optimal stochastic control problem that minimizes the probability that a signal upcrosses a level is solved by rewriting it as a one parametric optimization problem over a set of lqg control problem solutions finding the optimal controller can be interpreted as finding an optimal costing transfer function the existence of the optimal controller is investigated in a constructive way and it is shown that it is equivalent to the existence of a controller with sufficiently small closed loop variance of the controlled signal

an efficient montgomery exponentiation algorithm by using signed digit recoding and folding techniques abstract the motivation for designing fast modular exponentiation algorithms comes from their applications in computer science in this paper a new csd ef montgomery binary exponentiation algorithm is proposed it is based on the montgomery algorithm using the canonical signed digit csd technique and the exponent folding ef binary exponentiation technique by using the exponent folding technique of computing the common parts in the folded substrings the same common part in the folding substrings can be simply computed once we can thus improve the efficiency of the binary exponentiation algorithm by decreasing the number of modular multiplications moreover the signed digit representation has less occurrence probability of the nonzero digit than binary number representation taking this advantage we can further effectively decrease the amount of modular multiplications and we can therefore decrease the computational complexity of modular exponentiation as compared with the ha moon s algorithm 1 261718m multiplications and the lou chang s algorithm 1 375m multiplications the proposed csd ef montgomery algorithm on average only takes 0 5m multiplications to evaluate modular exponentiation where m is the bit length of the exponent

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the full classification of orthogonal easy quantum groups we study easy quantum groups a combinatorial class of orthogonal quantum groups introduced by banica speicher in 2009 we show that there is a countable descending chain of easy quantum groups interpolating between bichon s free wreath product with the permutation group s n and a semi direct product of a permutation action of s n on a free product this reveals a series of new commutation relations interpolating between a free product construction and the tensor product furthermore we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi direct product quantum groups recently studied by the authors this completes the classification of easy quantum groups we also study combinatorial and operator algebraic aspects of the new interpolating series

left to right optimal signed binary representation of a pair of integers the common computation in elliptic curve cryptography ecc ap bq is performed by extending shamir s method for the computation of the product of powers of two elements in a group the complexity of computing ap bq is dependent on the joint weight of the binary expansion of positive integers a and b we give a method of finding a minimum joint weight signed binary representation of a pair of integers our method examines the integers a and b from left to right thereby making the conversion to signed binary form compatible with shamir s method this reduces the memory required to perform the computation of ap bq

low latency floating point division and square root unit digit recurrence algorithms are widely used in actual microprocessors to compute floating point division and square root these iterative algorithms present a good trade off in terms of performance area and power we present a floating point division and square root unit which implements a radix 64 floating point division and a radix 16 floating point square root to have an affordable implementation each radix 64 division iteration and radix 16 square root iteration are made of simpler radix 4 iterations 3 radix 4 iterations in division and 2 in square root speculation is used between consecutive radix 4 iterations to get a reduced timing there are three different parts in digit recurrence implementations initialization digit iterations and rounding the digit iteration is the iterative part and it uses the same logic for several cycles division and square root share partially the initialization and rounding stages whereas each one has different logic for the digit iterations the result is a low latency floating point divider and square root requiring 11 6 and 4 cycles for double single and half precision division with normalized operands and result and 15 8 and 5 cycles for square root one or two additional cycles are needed in case of subnormal operand s or result

hybrid binary ternary number system for elliptic curve cryptosystems single and double scalar multiplications are the most computational intensive operations in elliptic curve based cryptosystems improving the performance of these operations is generally achieved by means of integer recoding techniques which aim at minimizing the scalars density of nonzero digits the hybrid binary ternary number system provides both short representations and small density in this paper we present three novel algorithms for both single and double scalar multiplication we present a detailed theoretical analysis together with timings and fair comparisons over both tripling oriented doche ichart kohel curves and generic weierstrass curves our experiments show that our algorithms are almost always faster than their widely used counterparts

elliptic curve cryptography with efficiently computable endomorphisms and its hardware implementations for the internet of things verification of an ecdsa signature requires a double scalar multiplication on an elliptic curve in this work we study the computation of this operation on a twisted edwards curve with an efficiently computable endomorphism which allows reducing the number of point doublings by approximately 50 percent compared to a conventional implementation in particular we focus on a curve defined over the 207 bit prime field mathbb f p with p 2 207 5 131 we develop several optimizations to the operation and we describe two hardware architectures for computing the operation the first architecture is a small processor implemented in 0 13 mu m cmos asic and is useful in resource constrained devices for the internet of things iot applications the second architecture is designed for fast signature verifications by using fpga acceleration and can be used in the server side of these applications our designs offer various trade offs and optimizations between performance and resource requirements and they are valuable for iot applications

bipartite modular multiplication method this paper proposes a new modular multiplication method that uses montgomery residues defined by a modulus m and a montgomery radix r whose value is less than the modulus m this condition enables the operand multiplier to be split into two parts that can be processed separately in parallel increasing the calculation speed the upper part of the split multiplier can be processed by calculating a product modulo m of the multiplicand and this part of the split multiplier the lower part of the split multiplier can be processed by calculating a product modulo m of the multiplicand this part of the split multiplier and the inverse of a constant r two different implementations based on this method are proposed one uses a classical modular multiplier and a montgomery multiplier and the other generates partial products for each part of the split multiplier separately which are added and accumulated in a single pipelined unit a radix 4 version of a multiplier based on a radix 4 classical modular multiplier and a radix 4 montgomery multiplier has been designed and simulated the proposed method is also suitable for software implementation in a multiprocessor environment

practical cellular dividers a discussion is presented of parallel division algorithms that can be classified among modified higher radix nonrestoring online division methods where redundant representations are extensively utilized to speed up the operation the network realizations of these algorithms are cellular or even systolic with exclusively local control they have both size area and time of o n where n is the length of the dividend representation the same structures can also be used as a signed digit serial multiplier when suitably equipped with some control and a few registers the divider multiplier brings remarkable performance to large modular arithmetic rsa cryptography and greatest common divisor computations they are also of interest for the design of floating point units and signal processing applications

analysis of sliding window techniques for exponentiation abstract the m ary method for computing x e partitions the bits of the integer e into words of constant length and then performs as many multiplications as there are nonzero words variable length partitioning strategies have been suggested to reduce the number of nonzero words and thus the total number of multiplications algorithms for exponentiation using such partitioning strategies are termed sliding window techniques in this paper we give algorithmic descriptions of two recently proposed sliding window techniques and calculate the average number of multiplications by modeling the partitioning process as a markov chain we tabulate the optimal values of the partitioning parameters and show that the sliding window algorithms require up to 8 fewer multiplications than the m ary method

a new symbolic substitution based addition algorithm symbolic substitution a parallel processing technique has been proposed in computing literature to perform fast carry free addition of numbers several algorithms and corresponding symbolic substitution tables have been developed for addition of two numbers represented in binary modified signed digit msd and canonical modified signed digit cmsd number systems in this paper however we present a new symbolic substitution based algorithm and corresponding substitution table for the addition of two numbers represented in canonical modified signed digit cmsd notation in contrast with the existing algorithm in this regard that derives the addition result in msd system our algorithm derives the result in cmsd notation and thus it could be employed to perform symbolic substitution based associative addition of a set of cmsd numbers

woronowicz construction of compact quantum groups for functions on permutations classification result for n 3 abstract we provide a classification of compact quantum groups which can be obtained by the woronowicz construction when the arrays used in the twisted determinant condition are extensions of functions on permutations general properties of such quantum groups are revealed with the aid of operators intertwining tensor powers of the fundamental corepresentation two new families of quantum groups appear three versions of the quantum group su q 3 with complex q and a non commutative analogue of the semi direct product of two dimensional torus with the alternating group a 3

sampled data and discrete time h_2 optimal control this paper deals with the sampled data h 2 optimal control problem given a linear time invariant continuous time system the problem of minimizing the h 2 performance over all sampled data controllers with a fixed sampling period can be reduced to a pure discrete time h 2 optimal control problem this discrete time h 2 problem is always singular motivated by this in this paper we give a treatment of the discrete time h 2 optimal control problem in its full generality the results we obtain are then applied to the singular discrete time h 2 problem arising from the sampled data h 2 problem in particular we give conditions for the existence of optimal sampled data controllers we also show that the h 2 performance of a continuous time controller can always be recovered asymptotically by choosing the sampling period sufficiently small finally we show that the optimal sampled data h 2 performance converges to the continuous time optimal h 2 performance as the sampling period converges to zero

identification of arx systems with non stationary inputs asymptotic analysis with application to adaptive input design a key problem in optimal input design is that the solution depends on system parameters to be identified in this contribution we provide formal results for convergence and asymptotic optimality of an adaptive input design method based on the certainty equivalence principle i e for each time step an optimal input design problem is solved exactly using the present parameter estimate and one sample of this input is applied to the system the results apply to stable arx systems with the input restricted to be generated by white noise filtered through a finite impulse response filter or a binary signal obtained from the latter by a static nonlinearity

analysis of fractional window recoding methods and their application to elliptic curve cryptosystems elliptic curve cryptosystems ecc are suitable for memory constraint devices like smart cards due to their small key size a standard way of computing elliptic curve scalar multiplication the most frequent operation in ecc is window methods which enhance the efficiency of the binary method at the expense of some precomputation the most established window methods are sliding window on naf naf sw wnaf and wmof where naf and mof are acronyms for nonadjacent form and mutually opposite form respectively a common drawback of these schemes is that only a small portion of the numbers is possible sizes for precomputation tables therefore in practice it is often necessary to waste memory because there is no table fitting exactly the available storage in the case of wnaf there exists a variant that allows arbitrary table sizes the so called fractional wnaf frac wnaf in this paper we give a comprehensive proof using markov theory for the estimation of the average nonzero density of the frac wnaf representation then we propose the fractional wmof frac wmof which is a left to right analogue of frac wnaf we prove that frac wmof inherits the outstanding properties of frac waf however because of its left to right nature frac wmof is preferable as it reduces the memory consumption of the scalar multiplication finally we show that the properties of all discussed previous schemes can be achieved as special instances of the frac wmof method to demonstrate the practicability of frac wmof we develop an on the fly algorithm for computing elliptic curve scalar multiplication with a flexibly chosen amount of memory

on calculating multiplicative inverses modulo 2 m this paper presents a procedure for calculating multiplicative inverses modulo 2m based on a novel mathematical approach the procedure is suitable for software implementation on a general purpose processor when counting the total number of word level processor multiplications the computational effort involved in calculating a multiplicative inverse is 2 3 that of a single multiplication of m bit values in addition to a few word level multiplications for standard processor word sizes the number of these additional multiplications does not exceed 12 this introduces a clear advantage of the proposed method when compared to other known methods presented in the literature

high radix multiplier dividers theory design and hardware this paper describes the theory and design of digital high radix multiplier dividers patent pending the theory of high radix division is extended to high radix multiplier dividers that can perform fused multiplication and division operations using a single recurrence relation with the fused implementation of multiplication and division the two operations can be executed using a single instruction implying only a single rounding operation the recurrence relation is described the quotient digit selection function derived and important design parameters together with their optimal values and relations are defined efficient design procedure and implementation hardware are described and important system parameter values for various radix systems computed compared to pure dividers the multiplier divider requires a slightly more complex data path and quotient digit selection function

hybrid signed digit number systems a unified framework for redundant number representations with bounded carry propagation chains a novel hybrid number representation is proposed it includes the two s complement representation and the signed digit representation as special cases the hybrid number representations proposed are capable of bounding the maximum length of carry propagation chains during addition to any desired value between 1 and the entire word length the framework reveals a continuum of number representations between the two extremes of two s complement and signed digit number systems and allows a unified performance analysis of the entire spectrum of implementations of adders multipliers and alike we present several static cmos implementations of a two operand adder which employ the proposed representations we then derive quantitative estimates of area in terms of the required number of transistors and the maximum carry propagation delay for such an adder the analysis clearly illustrates the trade offs between area and execution time associated with each of the possible representations we also discuss adder trees for parallel multipliers and show that the proposed representations lead to compact adder trees with fast execution times in practice the area available to a designer is often limited in such cases the designer can select the particular hybrid representation that yields the most suitable implementation fastest lowest power consumption etc while satisfying the area constraint similarly if the worst case delay is predetermined the designer can select a hybrid representation that minimizes area or power under the delay constraint

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

fast modular multi exponentiation using modified complex arithmetic abstract modular multi exponentiation i 1 n m i e i mod n is a very important but time consuming operation in many modern cryptosystems in this paper a fast modular multi exponentiation is proposed utilizing the binary like complex arithmetic method complement representation method and canonical signed digit recoding technique by performing complements and canonical signed digit recoding technique the hamming weight number of 1 s in the binary representation or number of non zero digits in the binary signed digit representations of the exponents can be reduced based on these techniques an algorithm with efficient modular multi exponentiation is proposed for modular multi exponentiation in average case the proposed algorithm can reduce the number of modular multiplications mms from 1 503k to 1 306k where k is the bit length of the exponent we can therefore efficiently speed up the overall performance of the modular multi exponentiation for cryptographic applications

a classification of unitary highest weight modules of the quantum analogue of the symmetric pair an an 1 abstract we study the irreducibility and unitarity of highest weight modules of the quantized enveloping algebra associated to s u n 1 we obtain a classification of these modules analogous to that of ehw in the classical context we also compute all commutation relations between root vectors in u q s u n 1

families of linear quadratic problems continuity properties in this paper we investigate for given one parameter families of linear time invariant finite dimensional systems the parameter dependence of the linear quadratic optimal cost optimal control inputs optimal state trajectories and optimal outputs it is shown that results that have been obtained in the past in the context of the problem of cheap control can in fact be generalized to a much broader class of parameter dependent cost functionals including cost functionals in which for every parameter value the weighting matrix of the control inputs is singular essentially only two assumptions on the parameter dependence of the cost functionals are required in order to have continuity of the optimal cost and optimal control inputs with respect to the underlying parameter one assumption is concerned with the continuity of the weighting matrices with respect to this parameter the other with the monotonicy of the weighting matrices with respect to the parameter instrumental in our development is a characterization of the linear quadratic optimal cost in terms of the so called dissipation inequality the results obtained are applied to the problem of cheap control and to a problem of priority control the latter provides an example of a family of quadratic cosl functionals with a polynomial parameter dependence

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

improving goldschmidt division square root and square root reciprocal the aim of this paper is to accelerate division square root and square root reciprocal computations when the goldschmidt method is used on a pipelined multiplier this is done by replacing the last iteration by the addition of a correcting term that can be looked up during the early iterations we describe several variants of the goldschmidt algorithm assuming 4 cycle pipelined multiplier and discuss obtained number of cycles and error achieved extensions to other than 4 cycle multipliers are given if we call g sub m the goldschmidt algorithm with m iterations our variants allow us to reach an accuracy that is between that of g sub 3 and that of g sub 4 with a number of cycle equal to that of g sub 3

tannaka duals in semisimple tensor categories abstract tannaka duals of finite dimensional hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories which are shown to include the tannaka dual of the dual hopf algebras the second orbifolds are then monoidally equivalent to the initial tensor categories in a canonical fashion

how to compute modular exponentiation with large operators based on the right to left binary algorithm abstract when the lengths of the operators are at least 1024 binary or 300 decimal digits modular exponentiation can be time consuming and is often the dominant part of the computation in many computer algebra systems the prime approach on this computational problem is known as the square and multiply method which includes two versions i e the left to right binary algorithm and the right to left binary algorithm for the past years too many attentions have been paid to propose the fast modular exponentiation methods based on the left to right binary algorithm however extremely few attentions have been paid on developing the fast modular exponentiation methods based on the right to left binary algorithm in this paper we propose a t fold exponent method based on the right to left binary algorithm from the performance view our t fold exponent method is similar to the m ary method based on the left to right binary algorithm from the structure view our t fold exponent method offers a framework for the fast modular exponentiation methods based on the right to left binary algorithm more important it is the first but steady step to develop the fast modular exponentiation methods based on the right to left binary algorithm

space time trade offs for higher radix modular multiplication using repeated addition the value of using a higher radix for modular multiplication in the context of rsa is investigated the main conclusion is that for algorithms which perform the multiplication via repeated addition there is broadly speaking a direct trade off between space and time provided by change of radix thus chip area utilized is roughly proportional to speed however initially as the radix is increased from 2 there is a short lived increase in speed greater than the extra area used

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

the alternating greedy expansion and applications to computing digit expansions from left to right in cryptography the central topic of this paper is the alternating greedy expansion of integers which is defined to be a binary expansion with digits 0 1 with the property that the nonzero digits have alternating signs we collect known results about this alternating greedy expansion and complement it with other useful properties and algorithms in the second part we apply it to give an algorithm for computing a joint expansion of d integers of minimal joint hamming weight from left to right i e from the column with the most significant bits towards the column with the least significant bits furthermore we also compute an expansion equivalent to the so caled w naf from left to right using the alternating greedy expansion

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

a characterization of right coideals of quotient type and its application to classification of poisson boundaries let mathbb g be a co amenable compact quantum group we show that a right coideal of mathbb g is of quotient type if and only if it is the range of a conditional expectation preserving the haar state and is globally invariant under the left action of the dual discrete quantum group we apply this result to the theory of poisson boundaries introduced by izumi for discrete quantum groups and generalize a work of izumi neshveyev tuset on suq n for co amenable compact quantum groups with the commutative fusion rules more precisely we prove that the poisson integral is an isomorphism between the poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of kac type in particular the poisson boundary and the quantum flag manifold are isomorphic for any q deformed classical compact lie group

measuring the accuracy of rom reciprocal tables we prove that a convenient rom reciprocal table construction algorithm generates tables that minimize the relative error the worst case relative errors realized for such optimally computed k bits in m bits out rom reciprocal tables are then determined for all table sizes 3 spl les k m spl les 12 we next prove the table construction algorithm always generates a k bits in k bits out table with relative errors never any greater than 3 4 2 sup k and more generally with g guard bits that for k g bits out the relative error is never any greater than 2 k 1 1 1 2g 1 to provide for determining extreme case test data and to compute the precision of a reciprocal table without prior construction of the full rom reciprocal table we describe a procedure that requires generation and inspection of only a small portion of such a table to identify input values guaranteed to include the worst case relative errors in the table

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

signed digit representations of minimal hamming weight the authors give an online algorithm for computing a canonical signed digit representation of minimal hamming weight for any integer n using combinatorial techniques the probability distributions pr k sub r h where k sub r is taken to be a random variable on the uniform probability space of k digit integers is computed also using a markov chain analysis it is shown that e k sub r approximately r 1 k r 1 as k to infinity

discrete wavelet transform and data expansion reduction in homomorphic encrypted domain signal processing in the encrypted domain is a new technology with the goal of protecting valuable signals from insecure signal processing in this paper we propose a method for implementing discrete wavelet transform dwt and multiresolution analysis mra in homomorphic encrypted domain we first suggest a framework for performing dwt and inverse dwt idwt in the encrypted domain then conduct an analysis of data expansion and quantization errors under the framework to solve the problem of data expansion which may be very important in practical applications we present a method for reducing data expansion in the case that both dwt and idwt are performed with the proposed method multilevel dwt idwt can be performed with less data expansion in homomorphic encrypted domain we propose a new signal processing procedure where the multiplicative inverse method is employed as the last step to limit the data expansion taking a 2 d haar wavelet transform as an example we conduct a few experiments to demonstrate the advantages of our method in secure image processing we also provide computational complexity analyses and comparisons to the best of our knowledge there has been no report on the implementation of dwt and mra in the encrypted domain

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

generalized signed digit number systems a unifying framework for redundant number representations signed digit sd number representation systems have been defined for any radix r or 3 with digit values ranging over the set alpha 1 0 1 alpha where alpha is an arbitrary integer in the range 1 2r

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

discrete quantum groups letgbe any discrete group consider the algebraaof all complex functions with finite support ongwith pointwise operations the multiplication onginduces a comultiplication onaby f p q f pq wheneverf aandp q g ifgis finite one can identify the algebra of complex functions ong gwitha aso that a a a then a is a hopf algebra ifgis infinite we still have f g 1 and f 1 g ina afor allfandg in this case a is a multiplier hopf algebra in fact it is a multiplier hopf algebra whenais given the natural involution defined by formula for allf aandp g in this paper we call a multiplier hopf algebra a adiscrete quantum group if the underlying algebraais a direct sum of full matrix algebras we study these discrete quantum groups and we give a simple proof of the existence and uniqueness of a left and a right invariant haar measure

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

exponentiation using canonical recoding abstract the canonical bit recoding technique can be used to reduce the average number of multiplications required to compute xe provided that x 1 is supplied along with x we model the generation of the digits of the canonical recoding d of an n bit long exponent e as a markov chain and show that binary quaternary and octal methods applied to d require 4 3 n 4 3 n and 23 18 n multiplications compared to 3 2 n 11 8 n and 31 24 n required by these methods applied to e we show that in general the canonically recoded m ary method for constant m requires fewer multiplications than the standard m ary method however when m is picked optimally for each method for a given n then the average number of multiplications required by the standard method is fewer than those required by the recoded version

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

modular theory for bimodules abstract modular theory is developed for bimodules and its basic properties are established among them frobenius reciprocity is obtained in a fairly general setting

quantization of lie group and algebra of g2 type in the faddeev reshetikhin takhtajan approach based on the quantized universal enveloping que algebras a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries a quantum group of g2 type included in this scheme is studied in detail in the faddeev reshetikhin takhtajan frt approach also in the formulation employing noncommuting matrix entries the que algebra of g2 type is reconstructed through the pairing induced by the r matrix between the quantum group and the que algebra

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

signed digit addition and related operations with threshold logic assuming signed digit number representations we investigate the implementation of some addition related operations assuming linear threshold networks we measure the depth and size of the networks in terms of linear threshold gates we show first that a depth 2 network with o n size weight and fan in complexities can perform signed digit symmetric functions consequently assuming radix 2 signed digit representation we show that the two operand addition can be performed by a threshold network of depth 2 having o n size complexity and o 1 weight and fan in complexities furthermore we show that assuming radix 2n 1 signed digit representations the multioperand addition can be computed by a depth 2 network with o n sup 3 size with the weight and fan in complexities being polynomially bounded finally we show that multiplication can be performed by a linear threshold network of depth 3 with the size of o n sup 3 requiring o n sup 3 weights and o n sup 2 log n fan in

a rounding method to reduce the required multiplier precision for goldschmidt division a new rounding method to reduce the required precision of the multiplier for goldschmidt division is presented it applies special truncation methods at the final iteration step this requires a minor modification to the rounding constants of the multiplier it allows twice the error tolerance of conventional methods and inclusive error bounds the proposed method further reduces the required precision of the multiplier by considering the asymmetric error bounds of goldschmidt dividers where the factors are computed using a one s complement operation as a result the proposed rounding method allows the multiplier of a three iteration goldschmidt divider to be implemented using only three extra bits the proposed method has been verified using a systemc hardware model of the divider supporting variable precision the validity of the error analysis is also checked via simulation the final rounding results are checked with both 1010 random double precision floating point significands and an exhaustive suite of 17 bit test vectors

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

the h control problem with zeros on the boundary of the stability domain in this paper we study the discrete and continuous time h control problems without any assumptions on the system parameters our approach yields necessary and sufficient conditions for the existence of a suitable controller

the montgomery inverse and its applications the montgomery inverse of b module a is b sup 1 2 sup n mod a where n is the number of bits in a the right shifting binary algorithm for modular inversion is shown naturally to compute the new inverse in fewer operations than the ordinary modular inverse the new inverse facilitates recent work by koc on modular exponentiation and has other applications in cryptography

on unitary representation theories of compact quantum groups categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite jones index

high speed vlsi multiplication algorithm with a redundant binary addition tree a high speed vlsi multiplication algorithm internally using redundant binary representation is proposed in n bit binary integer multiplication n partial products are first generated and then added up pairwise by means of a binary tree of redundant binary adders since parallel addition of two n digit redundant binary numbers can be performed in a constant time independent of n without carry propagation n bit multiplication can be performed in a time proportional to log2 n the computation time is almost the same as that by a multiplier with a wallace tree in which three partial products will be converted into two in contrast to our two to one conversion and is much shorter than that by an array multiplier for longer operands the number of computation elements of an n bit multiplier based on the algorithm is proportional to n2 it is almost the same as those of conventional ones furthermore since the multiplier has a regular cellular array structure similar to an array multiplier it is suitable for vlsi implementation thus the multiplier is excellent in both computation speed and regularity in layout it can be implemented on a vlsi chip with an area proportional to n2 log2 n the algorithm can be directly applied to both unsigned and 2 s complement binary integer multiplication

continuity properties of solutions to h 2 and h riccati equations in h2 and h optimal control semi stabilizing solutions of algebraic riccati equations play an essential role it is well known that these solutions might have discontinuities as a function of the system parameters the paper shows that these discontinuities are directly linked to non left invertibility and in contrast to what one might think unrelated to zeros on the imaginary axis

algorithm of asynchronous binary signed digit recoding on fast multiexponentiation multiexponentiation for the 2 dimension one especially is a very important but time consuming operation in many elgamal like public key cryptosystems an efficient multiexponentiation method is firstly proposed by shamir in elgamal s paper shamir s method requires 1 75n modular multiplications in average where n is the bit length of the exponents using minimal weight binary signed digit bsd representations in shamir s method can reduce the number of multiplications to 1 556n due to there are many minimal weight bsd representations for an integer dimitrov et al proposed a new 2 dimension multiexponentiation algorithm to recode the canonical binary signed digit representations the average number of the number 1 556n is reduced to 1 534n after the publish of the dimitrov jullien miller algorithm there are many research on receding bsd representation of a pair of integers the most interesting one is the joint sparse form the number of multiplication can be reduced to 1 500n in joint sparse form when the exponent length is near infinite in this paper we propose a new asynchronous recoding algorithm to reduce the factor to 1 509n for comparison to all the previous algorithm the property of asynchronous receding is especially suitable for multi dimension multiexponentiation

efficient exponentiation of a primitive root in gf 2 sup m in this paper exponentiation of a primitive root in gf 2 sup m is considered signed digit sd number representation is used to efficiently represent the exponent and the corresponding algorithms and structures for exponentiation are developed for primitive multiplications required in exponentiations extended bidirectional linear feedback shift registers are proposed and used for the cases where the exponent is represented as a binary or a radix 4 sd number comparisons are made with other methods on the bases of space time and possible power consumption since the proposed structures can effectively reduce power and area when implemented in vlsi they are especially suitable for battery powered portable devices

a unified architecture for the accurate and high throughput implementation of six key elementary functions this paper presents a unified architecture for the compact implementation of several key elementary functions including reciprocal square root and logarithm in single precision floating point arithmetic the proposed high throughput design is based on uniform domain segmentation and curve fitting techniques numerically accurate least squares regression is utilized to calculate the polynomial coefficients the architecture is optimized by analyzing the trade off between the size of the required memory and the precision of intermediate variables to achieve the minimum 23 bit accuracy required for single precision floating point representation the efficiency of the proposed unified data path is demonstrated on a common field programmable gate array

a new minimal average weight representation for left to right point multiplication methods this paper introduces a new radix 2 representation with the same average weight as the width w nonadjacent form w naf in both w naf and the proposed representations each nonzero digit is an odd integer with absolute value less than m however for w naf m is of the form 2 sup w 1 while for the proposed representation it can be any positive integer therefore using the proposed integer representation we can use the available memory efficiently which is attractive for devices with limited memory another advantage of the proposed representation over w naf is that it can be obtained by scanning the bits from left to right this property is also useful for memory constrained devices because it can reduce both the time and space complexity of fast point multiplication techniques

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

control of level crossings in stationary gaussian random processes a new optimal stochastic control problem that minimizes the intensity for a signal to upcross a level is solved by rewriting it as a one parametric optimization problem over a set of lqg problem solutions the solution can sometimes be thought of as finding optimal weightings in an lqg problem

a carry free architecture for montgomery inversion a new carry free montgomery inversion algorithm which is suitable for hardware implementation is presented the algorithm utilizes a new redundant sign digit rsd representation and arithmetic to avoid carry propagation in addition and subtraction which are the atomic operations in the montgomery inversion algorithm the proposed algorithm is described in such a way that its hardware realization is straightforward the algorithm enables very fast computation of multiplicative inversion in gf p which is the most time consuming operation in elliptic and hyperelliptic curve cryptography complexity analysis and a gate level implementation of the algorithm reveal that the proposed algorithm provides a speedup of at least 1 95 over the original montgomery inversion algorithm

control of mean time between failures a new optimal stochastic control problem is posed the criterion is to maximize the mean time between failures criterion given a certain reference value it is shown that this control problem is closely related both to the problem of minimizing the variance of the signal minimum variance control and to the problem of minimizing the so called upcrossing probability minimum upcrossing control it is made plausible that the upcrossing probability is a better approximating criterion to minimize than the variance criterion the problem of minimizing the upcrossing probability can be thought of as finding optimal weighting matrices in an lqg problem the new controller is compared with the minimum variance controller for a first order process it is seen that the new controller causes a lower upcrossing probability and a larger mean time between failures the improvement in the example is up to about 25 this makes it possible to choose the reference value closer to the critical level without causing smaller

a low complexity lut based squaring algorithm the computation of large modular multi exponentiation is a time consuming arithmetic operation used in cryptography the standard squaring algorithm is well known and the guajardo paar algorithm fixes the improper carry handling bug produced by the standard squaring algorithm but produces error indexing bug in this paper a novel squaring algorithm is proposed which stores base products in the look up table before the squaring computation and base size comparison method the proposed algorithm can not only evaluate squaring efficiently but also avoid bugs produced in other proposed algorithms the guajardo paar algorithm and the yang heih laih algorithm the performance of the proposed algorithm is 1 615 times faster than the standard squaring algorithm and much faster than other algorithms

computing sequences with addition chains given a sequence n 1 cdots n m of positive integers what is the smallest number of additions needed to compute all m integers starting with 1 this generalization of the addition chain m 1 problem will be called the addition sequence problem we show that the sequence 2 0 2 1 cdots 2 n 1 2 n 1 can be computed with n 2 13 sqrt n log n additions and that n sqrt n 2 is a lower bound this lower bound result is applied to show that the addition sequence problem is np complete

a radix 4 modular multiplication hardware algorithm for modular exponentiation a fast radix 4 modular multiplication hardware algorithm is proposed it is efficient for modular exponentiation with a large modulus used in public key cryptosystems such as the rsa cryptosystem the operands and the result of multiplication which are intermediate results in modular exponentiation are represented in a redundant representation the computation proceeds in serial parallel fashion each subtraction for the division for residue calculation is embedded in the repeated multiply add each intermediate result is represented in a more redundant representation than that for the operands and the result so that the number of the required addition subtractions is reduced all addition subtraction are carried out without carry propagation a serial parallel modular multiplier based on the algorithm has a regular cellular array structure with a bit slice feature and is suitable for vlsi implementation

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

montgomery multiplier and squarer for a class of finite fields montgomery multiplication in gf 2 sup m is defined by a x b x r sup 1 x mod f x where the field is generated by a root of the irreducible polynomial f x a x and b x are two field elements in gf 2 sup m and r x is a fixed field element in gf 2 sup m in this paper first a slightly generalized montgomery multiplication algorithm in gf 2 sup m is presented then by choosing r x according to f x we show that efficient architectures of bit parallel montgomery multiplier and squarer can be obtained for the fields generated with an irreducible trinomial complexities of the montgomery multiplier and squarer in terms of gate counts and time delay of the circuits are investigated and found to be as good as or better than that of previous proposals for the same class of fields

improvement to montgomery modular inverse algorithm after a comprehensive study on the montgomery modular inverse algorithm and its revised versions two modified high radix algorithms are proposed which utilize higher radix to reduce iterations needed without increasing complexity much thereby accelerating the process the radix 4 algorithm can reduce the average number of iterations from 1 4 n to 0 82 n and a software experiment shows the speedup is about 11 percent and iterations are 41 5 percent less on average the radix 8 algorithm can reduce the average number of iterations to 0 73 n but it is more complicated which makes it suitable only for very large numbers 2 048 bits in the experiment where the speedup can be 13 18 percent the proposed algorithms are suitable for software implementations on general purpose microprocessors

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

fast multicomputation with asynchronous strategy we propose a new strategy to improve the performance of multicomputations such as computing xa yb in an additive group or a xby in a multiplicative group this new strategy is called asynchronous strategy in each iteration it tries to compute partial results by pairing nearby zero and nonzero bits the new strategy can be applied to any binary code of x and y to further reduce the computational cost however our experiments show that sparse signed digit code can achieve minimum computational cost the algorithms using the asynchronous strategy need only o h extra registers when they try to do computation with bits which are h positions apart

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

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

existence of discrete time lqg controllers existence results for the lqg controller are investigated an infimal riccati equation based controller may potentially give closed loop eigenvalues on the unit circle assuming left and right invertibility it is shown that there exists an optimal controller if and only if the riccati equation based controller stabilizes the closed loop system after removal of all its unobservable and uncontrollable modes furthermore this reduced controller is the optimal controller and its transfer function is unique this existence condition is a considerable simplification of the more general geometric condition recently derived by trentelman and stoorvogel

algorithms for inversion mod p k p k this article describes and analyzes all existing algorithms for computing x a 1 pmod p k x a 1 mod p k for a prime p p and also introduces a new algorithm based on the exact solution of linear equations using p p adic expansions the algorithm starts with the initial value c a 1 pmod p c a 1 mod p and iteratively computes the digits of the inverse x a 1 pmod p k x a 1 mod p k in base p p the mod 2 version of the algorithm is more efficient than all existing algorithms for small values of k k moreover it stands out as being the only one that works for any p p any k k and digit by digit while the new algorithm is asymptotically worse off it requires the minimal number of arithmetic operations just a single addition per step as compared to all existing algorithms

exponentiation using division chains exponentiation may be performed faster than the traditional square and multiply method by iteratively reducing the exponent module numbers which as exponents themselves require few multiplications this mainly includes those with few nonzero bits for a suitable choice of such divisors the resulting mixed basis representation of the exponent reduces the expected number of nonsquaring multiplications by over half at the cost of a single extra register preprocessing effort depends entirely on the exponent and can be kept down to the work saved in a single exponentiation moreover no precomputed look up tables are required so the method is especially applicable where space is at a premium in particular it outperforms the instance of the m ary method which uses the same space however for 512 bit exponents it beats every instance of the m ary method achieving well under 635 multiplications on average both hardware and software implementations of the rsa crypto system can benefit from this algorithm