eigenvalues for infinite matrices this paper is concerned with the problem of determining the location of eigenvalues for diagonally dominant infinite matrices upper and lower bounds for eigenvalues are established for tridiagonal matrices a numerical procedure for improving the bounds is given and the approximation of the eigenvectors is also discussed the techniques are illustrated for the solution of the well known mathieu s equation

toward a theory of pollard s rho method abstract pollard s rho method for integer factorization iterates a simple polynomial map and produces a nontrivial divisor of n when two such iterates agree modulo this divisor experience and heuristic arguments suggest that a prime divisor p should be detected in o p steps but this has never been proved indeed nothing seems to be have been rigorously proved about the probability of success that would improve the obvious lower bound of 1 p this paper shows that for fixed k this probability is at least 2 k p o p 3 2 as p this leads to an log 2 p p estimate of the success probability

three new factors of fermat numbers we report the discovery of a new factor for each of the fermat numbers f 13 f 15 f 16 these new factors have 27 33 and 27 decimal digits respectively each factor was found by the elliptic curve method after division by the new factors and previously known factors the remaining cofactors are seen to be composite numbers with 2391 9808 and 19694 decimal digits respectively

improved pollard rho method for computing discrete logarithms over finite extension fields it is clear that the distinctive feature of the normal basis representations namely the p th power of an element is just the cyclic shift of its normal basis representation where p is the characteristic of the underlying field can be used to speed up the computation of discrete logarithms over finite extension fields f p m we propose a variant of the pollard rho method to take advantage of this feature and achieve the speedup by a factor of m rather than 3p 34p 3m the previous result reported in the literature besides the theoretical analysis we also compare the performances of the new method with the previous algorithm in experiments and the result confirms our analysis due to the mov reduction our method can be applied to paring based cryptosystems over binary or ternary fields

the primality of r1031 on decrit une technique permettant de demontrer que le nombre r1031 10 1031 1 9 compose uniquement de 1 est premier

ecm using edwards curves this paper introduces eecm mpfq a fast implementation of the elliptic curve method of factoring integers eecm mpfq uses fewer modular multiplications than the well known gmp ecm software takes less time than gmp ecm and finds more primes than gmp ecm the main improvements above the modular arithmetic level are as follows 1 use edwards curves instead of montgomery curves 2 use extended edwards coordinates 3 use signed sliding window addition subtraction chains 4 batch primes to increase the window size 5 choose curves with small parameters and base points 6 choose curves with large torsion

solving structured linear systems with large displacement rank linear systems with structures such as toeplitz vandermonde or cauchy likeness can be solved in email protected a 2n operations where n is the matrix size a is its displacement rank and email protected denotes the omission of logarithmic factors we show that for such matrices this cost can be reduced to email protected a w 1n where w is a feasible exponent for matrix multiplication over the base field the best known estimate for w is w 2 38 resulting in costs of order email protected a 1 3 8n we present consequences for hermite pade approximation and bivariate interpolation

computing in the jacobian of a hyperelliptic curve in this paper we present algorithms suitable for computer use for computation in the jacobian of a hyperelliptic curve we present a reduction algorithm which is asymptotically faster than that of gauss when the genus g is very large

the multiple polynomial quadratic sieve a modification due to peter montgomery of pomerance s quadratic sieve for factoring large integers is discussed along with its implementation using it allows factorization with over an order of magnitude less sieving than the basic algorithm it enables one to factor numbers in the 60 digit range in about a day using a large minicomputer the algorithm has features which make it well adapted to parallel implementation

the desirability of adjusting for residual effects in a crossover design summary crossover experiments in which treatments are applied in sequence to the same experimental unit may give misleading results if there are carryover effects from previous treatments in each sequence and these effects are not allowed for in the analysis this paper is concerned with how to decide whether or not to allow for carryover effects the general conclusion is that despite the increased variance resulting from adjustment the safest policy is always to allow for these effects in a crossover design different treatments are applied in turn to the same experimental unit kershner and federer 1981 patterson 1951 the commonest applications are in nutritional trials on animals and in clinical trials of drugs however one difficulty associated with this design is that from the second period onwards a residual or carryover effect may be present due to the treatment applied in the previous period this problem can be mitigated by ensuring that each treatment is preceded equally often by every other treatment williams 1949 has shown how latin squares can be chosen to achieve this balance even with this balance the straightforward comparison of any two treatment means will be biased if their residual effects differ because no treatment is preceded by itself the repetition of one of the periods in order to achieve a complete balance of the residual effects and hence remove the bias is often impossible because of the time that would be required for the extended experiment when crossover designs are used for treatment comparisons the usual procedure is to test for residual effects and only if they are significantly different from zero adjust for them in the estimation of treatment effects the decision whether to adjust or not is therefore based on a significance test applied at an arbitrarily specified significance level if however the precise estimation of treatment effects is of prime interest procedures which lead to estimators that have smallest mean square errors mses are more appropriate the purpose of this paper is to investigate the effectiveness of such a procedure attention is restricted to the case in which it is reasonable to suppose that the duration of residual effects is limited to the period immediately following the direct application of the treatment we shall assume that the major interest is in the effect of the treatments during the period of their application some combination of direct and residual effects would merit consideration if in practice the treatment is to be applied for a longer period

note on irreducible diagonally dominant matrices and the convergence of the aor iterative method considering the linear systems ax b where the matrix a is irreducible and diagonally dominant we obtain bounds for the spectral radius of the 4 matrix of the aor method and we achieve the convergence conditions given in 2 by a different method if a is strictly diagonally dominant we get larger intervals for the parameter x of the sor method and we improve the results of theorems 5 6 of 3 for the aor method

a new double point multiplication algorithm and its application to binary elliptic curves with endomorphisms we present a new double point multiplication algorithm based on differential addition chains our proposed scheme has a uniform structure and has some degree of built in resistance against side channel analysis attacks we discuss deploying our scheme in a hardware implementation of single point multiplication on binary elliptic curves with efficiently computable endomorphisms based on operation counts we expect to gain accelerations of 30 and 18 for computing single point multiplication with and without availability of parallel multipliers respectively and these results are verified in our implementations

factorization of the eighth fermat number we describe a monte carlo factorization algorithm which was used to factorize the fermat number f8 2256 1 previously f8 was known to be composite but its factors were unknown comments only the abstract is given here the full paper appeared as 2 for a succinct proof of primality of the larger factor q8 of f8 see 3 at the time of this paper lenstra s elliptic curve method ecm had not been invented thus a modification of pollard s rho method 1 was used to factor f8 and several mersenne numbers 4 for the factorization of a larger fermat number by ecm see 5 the smaller factor of f8 is 1238926361552897 the epigram i am now entirely persuaded to employ the method a handy trick on gigantic composite numbers may appeal to readers who wish to memorize this factor

an iterative method with truncation for infinite linear systems abstract for the infinite linear system tx v where t is an infinite uniformly strictly diagonally dominant matrix and x and v are infinite vectors sufficient conditions are imposed on t and v for the existence and uniqueness of a bounded solution the system is rewritten as x ax b which is first iterated and then truncated to a finite system convergence of the solution of the resulting system to that of the original system is established and explicit error bounds are determined examples are given to illustrate the computation

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

cryptanalysis of short rsa secret exponents a cryptanalytic attack on the use of short rsa secret exponents is described the attack makes use of an algorithm based on continued fractions that finds the numerator and denominator of a fraction in polynomial time when a close enough estimate of the fraction is known the public exponent e and the modulus pq can be used to create an estimate of a fraction that involves the secret exponent d the algorithm based on continued fractions uses this estimate to discover sufficiently short secret exponents for a typical case where e pq gcd p 1 q 1 is small and p and q have approximately the same number of bits this attack will discover secret exponents with up to approximately one quarter as may bits as the modulus ways to combat this attack ways to improve it and two open problems are described this attack poses no threat to the normal case of rsa where the secret exponent is approximately the same size as the modulus this is because the attack uses information provided by the public exponent and in the normal case the public exponent can be chosen almost independently of the modulus

factors of fermat numbers and large primes of the form 2ⁿ 1 a new factor is given for each of the fermat numbers f52 f931 f6835 and f9448 in addition a factor of f75 discovered by gary gostin is presented the current status for all f is shown in a table primes of the form k 2 1 k odd are listed for 31 k 149 1500 ii 4000 and for 151 k 199 1000 ii 4000 some primes for even larger values of ii are included the largest one being 5 213165 1 also a survey of several related questions is given in particular values of k such that k 2 i is composite for every ii are considered as well as odd values of h such that 3h 2 1 never yields a twin prime pair

compression in finite fields and torus based cryptography we present efficient compression algorithms for subgroups of multiplicative groups of finite fields we use our compression algorithms to construct efficient public key cryptosystems called t 2 and ceilidh we disprove some conjectures and we use the theory of algebraic tori to give a better understanding of our cryptosystems the lucas based xtr and gong harn cryptosystems and conjectured generalizations

factoring large numbers with a quadratic sieve the quadratic sieve algorithm was used to factor a 47 digit number into primes a comparison with wagstaff s results using the continued fraction early abort algorithm suggests that qs should be faster than cfea when the number being factored exceeds 60 digits plus or minus ten or more digits depending on details of the hardware and software

new cullen primes numbers of the forms c n n 2 n 1 and w n n 2 n 1 are both called cullen numbers new primes c n are presented for n 4713 5795 6611 18496 for w n several new primes are listed the largest one having n 18885 furthermore all efforts made to factorize numbers c n and w n are described and the result the complete factorization for all n 300 is given in a supplement

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

a review of infinite matrices and their applications abstract infinite matrices the forerunner and a main constituent of many branches of classical mathematics infinite quadratic forms integral equations differential equations etc and of the modern operator theory is revisited to demonstrate its deep influence on the development of many branches of mathematics classical and modern replete with applications this review does not claim to be exhaustive but attempts to present research by the authors in a variety of applications these include the theory of infinite and related finite matrices such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces matrices considered here have special structures like diagonal dominance tridiagonal sign distributions etc and are frequently nonsingular moreover diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations the main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems using techniques like conformal mapping iterations truncations etc to derive estimates based solutions particular attention is paid to computable precise error estimates and explicit lower and upper bounds topics include bessel s mathieu equations viscous fluid flow simply and doubly connected regions digital dynamics eigenvalues of the laplacian etc also presented are results in generalized inverses and semi infinite linear programming

on random walks for pollard s rho method we consider pollard s rho method for discrete logarithm computation usually in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated we show that this assumption does not hold for the walk originally suggested by pollard its performance is worse than in the random case we study alternative walks that can be efficiently applied to compute discrete logarithms we introduce a class of walks that lead to the same performance as expected in the random case we show that this holds for arbitrarily large prime group orders thus making pollard s rho method for prime group orders about 20 faster than before

finding suitable curves for the elliptic curve method of factorization using the parametrizations of kubert we show how to produce in nite families of elliptic curves which have prescribed nontrivial torsion over q and rank at least one these curves can be used to speed up the ecm factorization algorithm of lenstra we also brie y discuss curves with complex multiplication in this context

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

linear recurrences with polynomial coefficients and application to integer factorization and cartier manin operator we study the complexity of computing one or several terms not necessarily consecutive in a recurrence with polynomial coefficients as applications we improve the best currently known upper bounds for factoring integers deterministically and for computing the cartier manin operator of hyperelliptic curves

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

new frameworks for montgomery s modular multiplication method we present frameworks for fast modular multiplication based on a modification of montgomery s original method for fixed large integers our algorithms may be significantly faster than conventional methods our techniques may also be extended to modular polynomial arithmetic

prime factorization using square root approximation many cryptosystems are based on the factorization of large integers the complexity of this type of factorization is still an advantage for data security developers and a challenge for both mathematicians and cryptanalysts the security of rsa relies on the difficulty of factoring large integers the factorization was studied earlier by old civilizations like the greek but their methods were extended after the emergence of computers the paradox of rsa is that in order to make rsa more efficient we use a modulus n pq which is as small as possible on the other hand it is sufficient to factor n in order to decrypt the encrypted messages in this paper we propose a new factorization method based on the square root approximation this method allows in reducing the search for candidate prime factors of a given integer by approximating each prime factor before considering it as a candidate

is the simple carry over model useful it is argued that the models usually employed for carry over effects in cross over trials are not reasonable the example of a non linear dose response for a dose finding trial arranged in a williams square is developed to show that if carry over is present to any appreciable degree the usual statistical models provide no guaranteed protection against its effects it is concluded that the most reasonably defended assumption about carry over effects is that no important carry over has taken place and that where this assumption cannot be defended statistical models provide no satisfactory substitute for it

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

high performance architecture of elliptic curve scalar multiplication a high performance architecture of elliptic curve scalar multiplication based on the montgomery ladder method over finite field gf 2m is proposed a pseudo pipelined word serial finite field multiplier with word size w suitable for the scalar multiplication is also developed implemented in hardware this system performs a scalar multiplication in approximately 6lceilm wrceil m 1 clock cycles and the gate delay in the critical path is equal to tand lceillog2 w k rceiltxor where tand and txor are delays due to two input and and xor gates respectively and 1 les k lt w is used to shorten the critical path

a practical analysis of the elliptic curve factoring algorithm much asymptotic analysis has been devoted to factoring algorithms we present a practical analysis of the complexity of the elliptic curve algorithm suggesting optimal parameter selection and run time guidelines the parameter selection is aided by a novel use of bayesian statistical decision techniques as applied to random algorithms we discuss how frequently the elliptic curve algorithm succeeds in practice and compare it with the quadratic sieve algorithm

a comparative study of several antibiotic formulations using a design based on a combination of balanced incomplete blocks and latin squares a practical application of an experimental design suitable for the comparison of several treatments and based on combining balanced incomplete blocks and latin squares balanced for carryover effects is presented in the context of comparing a number of paediatric antibiotic formulations for taste smell and colour the recommended designs originally suggested by patterson have the advantage of balanced incomplete blocks in that a single trial may be used to compare a larger number of treatments than may reasonably be given to any individual subject in addition the incorporation of suitably chosen latin squares allows for assessment of any effect of order of presentation of the treatments and for any simple first order carryover effect of one treatment into the following treatment period inclusion of such effects in the overall analysis could result in the reduction of bias in the comparisons of the treatments

on the number of elliptic pseudoprimes for an elliptic curve e with complex multiplication by an order in k q v d a point p of infinite order on e and any prime p with d i p 1 we have that p 1 p o mod p where 0 is the point at infinity and calculations are done using the addition law for e any composite number which satisfies these conditions is called an elliptic pseudoprime in this paper it is shown that assuming the generalized riemann hypothesis elliptic pseudoprimes are less numerous than primes in particular on the grh the number of elliptic pseudoprimes less than x is o x log log x log2 x for certain curves it is shown that infinitely many elliptic pseudoprimes exist

tables of fibonacci and lucas factorizations we list the known prime factors of the fibonacci numbers fn for n 999 and lucas numbers ln for n 500 we discuss the various methods used to obtain these factorizations and primality tests and give some history of the subject

succinct proofs of primality for the factors of some fermat numbers we give short and easily verified proofs of primality for the factors of the fermat numbers f5 f6 f7 and f8

two methods of directly constructing probabilistic public key encryption primitives based on third order lfsr sequences linear feedback shift registers are a commonly used method of producing pseudo random sequences with large period this paper investigated another applications of the third order linear feedback shift register sequence 3 lfsr it proposed two methods for directly constructing probabilistic public key encryption primitives the proposed probabilistic encryption schemes have properties of one wayness and semantic security

six new factors of fermat numbers in the past few years several investigators 1 11 have found many new prime factors of the fermat numbers fm 22 1 the increased availability and speed of computing equipment have allowed the search limits for factors of these numbers to be extended significantly in tables below we include the known search limits and all of the results that have been obtained since h c williams 1978 paper 12 was published using the method outlined by hallyburton and brillhart 8 we tested integers of the form k 2r 1 k odd as factors of the fermat numbers fm 9 m n 2 for the ranges shown in table 1 during our search the following new prime factors were found 1120049 23 1jf29

new regular radix 8 scheme for elliptic curve scalar multiplication without pre computation the recent advances in mobile technologies have increased the demand for high performance parallel computing schemes in this paper we present a new algorithm for evaluating elliptic curve scalar multiplication that can be used on any abelian group we show that the properties of the proposed algorithm enhance parallelism at both the point arithmetic and the field arithmetic levels then we employ this algorithm in proposing a new hardware design for the implementation of an elliptic curve scalar multiplication on a prime extended twisted edwards curve incorporating eight parallel operations we further show that in comparison to the other simple side channel attack protected schemes over prime fields the proposed design of the extended twisted edwards curve is the fastest scalar multiplication scheme reported in the literature

upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices abstract we give easily computable upper and lower bounds for the inverse elements of finite tridiagonal diagonally dominant matrices and we improve the well known upper bounds due to ostrowski the results are extended to infinite systems the theory is also applied to some earlier results and to the evaluation of bessel functions and mathieu functions by using their recurrence relations

a deterministic version of pollard s p 1 algorithm in this article we present applications of smooth numbers to the unconditional derandomization of some well known integer factoring algorithms we begin with pollard s p 1 algorithm which finds in random polynomial time the prime divisors p of an integer n such that p 1 is smooth we show that these prime factors can be recovered in deterministic polynomial time we further generalize this result to give a partial derandomization of the k th cyclotomic method of factoring k 2 devised by bach and shallit we also investigate reductions of factoring to computing euler s totient function we point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given n these sets consist roughly speaking of products of primes p satisfying with the exception of at most two certain conditions somewhat weaker than the smoothness of p 1 finally we prove that o ln n oracle queries for values of are sufficient to completely factor any integer n in less than exp 1 o 1 ln n 1 3 ln ln n 2 3 deterministic time

torus based compression by factor 4 and 6 we extend the torus based compression technique for cyclotomic subgroups and show how the elements of certain subgroups in characteristic two and three fields can be compressed by a factor of 4 and 6 respectively our compression and decompression functions can be computed at a negligible cost in particular our techniques lead to very efficient exponentiation algorithms that work with the compressed representations of elements and can be easily incorporated into pairing based protocols that require exponentiations or products of pairings

new integer factorizations new factorizations of fibonacci numbers lucas numbers and numbers of the form 2 i are presented together with the strategy a combination of known factorization methods used to obtain them

an fft extension to the p 1 factoring algorithm j m pollard in 1974 presented the p integer factoring al gorithm his paper couched the algorithm in theoretical terms based upon use of fast fourier transform techniques but he was unable to say whether the method could be made practical we discuss the mathematical basis of the al gorithm and show how it can work in practice the practical implementation depends for its success upon the use of residue number systems we also present an open problem as to how the method could be made to work for the elliptic curve factoring algorithm

factorization with genus 2 curves the elliptic curve method ecm is one of the best factorization methods available it is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower we use special hyperelliptic curves and kummer surfaces to reduce the complexity of the algorithm our implementation gmp hecm is faster than gmp ecm for factoring large numbers

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