on the descending central sequence of absolute galois groups let p be an odd prime number and f a field containing a primitive p th root of unity we prove a new restriction on the group theoretic structure of the absolute galois group g f of f namely the third subgroup g f 3 in the descending p central sequence of g f is the intersection of all open normal subgroups n such that g f n is 1 bbb z p 2 or the extra special group m p 3 of order p 3 and exponent p 2

invertibility and inversion of linear periodic systems abstract in this paper the invertibility properties of periodic discrete time systems are studied necessary and sufficient conditions for the existence and a synthesis procedure for constructing left and right inverses are given

disturbance localization with dead beat control for linear periodic discrete time systems in this paper the classical notion of a controlled invariant subspace together with its main properties is extended to the case of linear periodic discrete time systems and used for deriving necessary and sufficient solvability conditions for the disturbance localization problem and a synthesis procedure for the solution moreover the notions of outer and inner controllable subspaces are introduced and studied for the same class of system thus allowing the derivation of necessary and sufficient solvability conditions for the disturbance localization problem with output or state dead beat control and to give synthesis procedures for the solutions

the rabin monier theorem for lucas pseudoprimes we give bounds on the number of pairs p q with 0 p q n such that a composite number n is a strong lucas pseudoprime with respect to the parameters p q

notes on some new kinds of pseudoprimes j browkin defined in his recent paper math comp 73 2004 pp 1031 1037 some new kinds of pseudoprimes called sylow p pseudoprimes and elementary abelian p pseudoprimes he gave examples of strong pseudoprimes to many bases which are not sylow p pseudoprime to two bases only where p 2 or 3 in this paper in contrast to browkin s examples we give facts and examples which are unfavorable for browkin s observation to detect compositeness of odd composite numbers in section 2 we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also sylow 2 pseudoprimes to the same bases in section 3 we give examples of sylow p pseudoprimes to the first several prime bases for the first several primes p we especially give an example of a strong pseudoprime to the first six prime bases which is a sylow p pseudoprime to the same bases for all p 2 3 5 7 11 13 in section 4 we define n to be a k fold carmichael sylow pseudoprime if it is a sylow p pseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n 1 we find and tabulate all three 3 fold carmichael sylow pseudoprimes 10 16 in section 5 we define a positive odd composite n to be a sylow uniform pseudoprime to bases b 1 b k or a syl upsp b 1 b k for short if it is a syl p psp b 1 b k for all the first w n 1 1 small prime factors p of n 1 where w n 1 is the number of distinct prime factors of n 1 we find and tabulate all the 17 syl upsp 2 3 5 s 10 16 and some syl upsp 2 3 5 7 11 s 10 24 comparisons of effectiveness of browkin s observation with miller tests to detect compositeness of odd composite numbers are given in section 6

higher order carmichael numbers we define a carmichael number of order m to be a composite integer n such that nth power raising defines an endomorphism of every z nz algebra that can be generated as a z nz module by m elements we give a simple criterion to determine whether a number is a carmichael number of order m and we give a heuristic argument based on an argument of erdos for the usual carmichael numbers that indicates that for every m there should be infinitely many carmichael numbers of order m the argument suggests a method for finding examples of higher order carmichael numbers we use the method to provide examples of carmichael numbers of order 2

on the assignment of invariant factors by time varying feedback strategies abstract in this paper the realization of arbitrary invariant factors or cyclic components of the closed loop system matrix is discussed it is shown that the use of time varying feedback strategies enables one to realize arbitrary invariant factors in contrast with time invariant feedback control

rabin miller primality test composite numbers which pass it the rabin miller primality test is a probabilistic test which can be found in several algebraic computing systems such as pari maple scratch pad because it is very easy to implement and with a reasonable amount of computing indicates whether a number is composite or probably prime with a very low probability of error in this paper we compute composite numbers which are strong pseudoprimes to several chosen bases because these bases are those used by the scratchpad implementation of the test we obtain by a method which differs from a recent one by jaeschke composite numbers which are found to be probably prime by this test 1 preliminaries

on strong pseudoprimes to several bases with y k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5 q6 q7 q8 and give upper bounds for v 9 w t 1 we discuss the methods and underlying facts for obtaining these results 1 primality tests by means of strong pseudoprimes computer algebra systems as for instance axiom 2 use strong pseudoprimes for testing primality of integers the advantage of such tests is that they are very efficient the disadvantage is that they are only probabilistic tests when the integers are not restricted to certain intervals to make such tests deterministic for integers in prescribed intervals one has to know the exact number of necessary so called strong pseudoprimality tests for this purpose we introduce the numbers v1i v 2 for which we compute lower and upper bounds these numbers are defined and discussed in this section in 2 we derive some facts which are the basis for finding bounds for the numbers v k in 3 we discuss the methods which led to our results in view of fermat s little theorem we know that n is certainly not a prime when we have bn 1 i 1 mod n for an integer b with 1 0 and when n is a composite number then n is called a strong pseudoprime to base b if either

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

on fermat s quotient base two this paper extends the search for solutions of the congruence 2p1 0 modp2 to the limit p 6 109 no solution except the well known p 1093 and p 3511 was found in 1969 brillhart tonascia and weinberger 1 reported on the search for solutions of the congruence a 1 modp2 a 2 1 199 for a 2 a special effort was made to consider all p 3 109 only the two known solutions p 1093 and p 3511 were found having the occasional opportunity to use the illiac iv one of the projects decided upon was the recalculation and extension of the above result for a 2 by a somewhat different method the calculation was pushed to twice the above limit that is top 6 109 without finding any further solutions the parallel construction of the illiac iv makes it possible to look for 64 different values of p at the same time the speed of illiac iv is such that a range of 100000 numbers can be searched in one second thus the range for p 6 109 was broken up into 60 runs of 1000 seconds each the program was run as one of a few backlog problems over the past two years let n belong to one of the 64 residue classes that are prime to 240 and let 2m am nbm mod n2 where 0 am n 0 bm n for numbers n as large as 109 the number n2 is a doubly precise integer nevertheless the calculation of am and bm can be accomplished by single precision arithmetic in only o log m operations in fact one uses one or the other of the following two recurrences if m 2h 1 then am 2a2h and bm 2b2h mod n if m 2h and if a 2 rm nqm 0 rm n and if 2ahbh dm mod n then am rm and bm qm dm received march 30 1980 1980 mathematics subject classification primary 10 04 1981 american mathematical society 0025 5718 81 0000 0027 0 1 50 289 this content downloaded from 157 55 39 58 on tue 11 oct 2016 04 26 58 utc all use subject to http about jstor org terms

disturbance localization by measurement feedback for linear periodic discrete time systems abstract the recent notions of inner controllable outer controllable and outer reconstructible subspaces together with the extension of the notion of c a b pair of subspaces to the class of linear periodic discrete time systems are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems by means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases i e with and without an additional requirement of output or state dead beat control

dead beat control of linear periodic discrete time systems in this paper the dead beat control of linear periodic discrete time systems subject to periodic disturbances is studied necessary and sufficient conditions are given for the existence of a linear periodic controller such that both output dead beat control under periodic disturbances and state dead beat control under no disturbance are performed moreover synthesis procedures of the controller as well as of state dead beat observers are given which can be applied also to a class of bilinear systems

on the distribution of pseudoprimes let v x denote the pseudoprime counting function with l x exp log x log log log x log log x we prove vp x x l x 2 for large x an improvement on the 1956 work of erdos we conjecture that vp x x l x i l

finding strong pseudoprimes to several bases ii define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic efficient primality testing algorithm which is easy to implement thanks to pomerance et al and jaeschke the m are known for 1 m 8 upper bounds for g 10 and 11 were first given by jaeschke and those for 10 and 11 were then sharpened by the first author in his previous paper math comp 70 2001 863 872 in this paper we first follow the first author s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes spsp s n 1024 to the first five or six prime bases which have the form n pq with p q odd primes and q 1 k p 1 k 4 3 5 2 3 2 6 then we tabulate all carmichael numbers 1020 to the first six prime bases up to 13 which have the form n q1q2q3 with each prime factor qi 3 mod 4 there are in total 36 such carmichael numbers 12 numbers of which are also spsp s to base 17 5 numbers are spsp s to bases 17 and 19 one number is an spsp to the first 11 prime bases up to 31 as a result the upper bounds for 9 10 and 11 are lowered from 20 and 22 decimal digit numbers to a 19 decimal digit number 9 10 11 q11 3825 12305 65464 13051 19 digits 149491 747451 34233211 we conjecture that 9 10 11 3825 12305 65464 13051 and give reasons to support this conjecture the main idea for finding these carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases comparisons of effectiveness with arnault s bleichenbacher s jaeschke s and pinch s methods for finding carmichael numbers with three prime factors which are strong pseudoprimes to the first several prime bases are given

pole placement for linear discrete time systems by periodic output feedbacks abstract necessary and sufficient conditions for the existence of a solution to the pole placement problem by periodic output feedbacks for single input single output linear discrete time systems are established an algorithm for finding a sequence of output feedback gains is presented and illustrated by a simple numerical example

multivariable multirate sampled data systems state space description transfer characteristics and nyquist criterion first a discrete time state space description is derived for multivariable multirate sampled data systems then the multirate impulse modulation mim and the symmetric coordinate multirate impulse modulation smim of a transfer matrix are defined and their relation to the discrete time state space description is clarified based on the above results four stability criteria are derived the contents of the paper will be useful to extend the modern results about continuous time and single rate sampled data control systems to the multirate sampled data case especially extensions of the ina method and of the robust stability conditions will be straightforward

two kinds of strong pseudoprimes up to 10 36 let n 1 be an odd composite integer write n 1 2 s d with d odd if either b d 1 mod n or b 2r d 1 mod n for some r 0 1 s 1 then we say that n is a strong pseudoprime to base b or spsp b for short define t to be the smallest strong pseudoprime to all the first t prime bases if we know the exact value of t we will have for integers n 10 36 and to give reasons and numerical evidence of k2 and c 3 spsp s 12 where t resp t is the smallest k2 resp c 3 strong pseudoprime to all the first t prime bases for this purpose we describe procedures for computing and enumerating the two kinds of spsp s 10 36 to the first 9 prime bases the entire calculation took about 4000 hours on a pc pentium iv 1 8ghz recall that a k2 spsp is an spsp of the form n pq with p q primes and q 1 2 p 1 and that a c 3 spsp is an spsp and a carmichael number of the form n q 1 q 2 q 3 with each prime factor q i 3 mod 4

robust stabilization of uncertain systems by periodic feedback an uncertain control system described by a family parametrized by an unknown parameter that takes values in a known constraint set is considered the robust stabilization problem is defined as finding a dynamic output controller that globally stabilizes the uncertain system that is no matter what the parameter value chosen by nature the closed loop system is globally asymptotically stable it is shown that in the class of dynamic periodic controllers the solution to this problem exists under reasonable assumptions given for a general family of abstract non linear models those assumptions in the case of a family of linear finite dimensional models are equivalent to suitable stabilizability and detectability assumptions for this case insensitivity of the design with respect to small errors in the data is proved

finding c 3 strong pseudoprimes let q 1 10 9 and so that 12 3186 65857 83403 11511 67461 24 digits 399165290221 798330580441 which was found by the author in an earlier paper we give reasons to support the conjecture the main idea of our method for finding those 21978 c 3 spsp 2 s is that we loop on candidates of signatures and kernels with heights bounded subject those candidates n q 1 q 2 q 3 of c 3 spsp 2 s and their prime factors q 1 q 2 q 3 to miller s tests and obtain the desired numbers at last we speed our algorithm for finding larger c 3 spsp s say up to 10 50 with a given signature to more prime bases comparisons of effectiveness with arnault s and our previous methods for finding c 3 strong pseudoprimes to the first several prime bases are given

primality testing and jacobi sums we present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by adleman and rumely the new algorithm performs well in practice it is the first pnmality test in existence that can routinely handle numbers of hundreds of decimal digits

disturbance localization for left invertible linear periodic discrete time systems the disturbance localization problem for left invertible linear periodic discrete time systems is solved using periodic state feedback controllers the proposed technique is of algebraic nature and has the following two main characteristics i it yields simple algebraic criteria for testing the solvability of the problem as compared to known geometric criteria which may not be so easy to check ii it derives analytically the general expressions of all periodic controllers admissible for disturbance localization as compared to known techniques which lead to nonanalytic parametrizations of the admissible controllers via constructive procedures moreover for the aforementioned class of periodic systems the state feedback simultaneous disturbance localization and stabilization or pole placement problem is treated and conditions for its solvability are established on the basis of a decentralized control approach that makes use of the equivalence between the above problem and the stabilization or pole placement problem of a general proper multichannel system by decentralized static output feedback

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

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

minimal weight digit set conversions we consider the problem of recoding a number to minimize the number of nonzero digits in its representation that is to minimize the weight of the representation a general sliding window scheme is described that extends minimal binary sliding window conversion to arbitrary radix and to encompass signed digit sets this new conversion expresses a number of known recoding techniques as special cases proof that this scheme achieves minimal weight for a given digit set is provided and results concerning the theoretical average and worst case weight are derived

strong pseudoprimes to the first eight prime bases define m to be the smallest strong pseudoprime to the firstm prime bases the exact value of m is known for 1 m 8 z zhang have found a 19decimal digit number q11 3825 12305 65464 13051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that 9 10 11 q11 we prove the conjecture by algorithms

constructing carmichael numbers through improved subset product algorithms we have constructed a carmichael number with 10 333 229 505 prime factors and have also constructed carmichael numbers with k prime factors for every k between 3 and 19 565 220 these computations are the product of implementations of two new algorithms for the subset product prob lem that exploit the non uniform distribution of primes p with the property that p 1 divides a highly composite

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

galois module structure of pth power classes of cyclic extensions of degree pn in the mid 1960s borevi setminus v c and faddeev initiated the study of the galois module structure of groups of p th power classes of cyclic extensions k f of p th power degree they determined the structure of these modules in the case when f is a local field in this paper we determine these galois modules for all base fields f

average case error estimates for the strong probable prime test consider a procedure that chooses k bit odd numbers independently and from the uniform distribution subjects each number to t independent iterations of the strong probable prime test miller rabin test with randomly chosen bases and outputs the first number found that passes all t tests let pk t denote the probability that this procedure returns a composite number we obtain numerical upper bounds for pk t for various choices of k t and obtain clean explicit functions that bound pk t for certain infinite classes of k t for example we show pioo 10 2 in addition we characterize the worst case numbers with unusually many false witnesses and give an upper bound on their distribution that is probably close to best possible

geometric and algebraic structure at infinity of discrete time linear periodic systems a notion of structure at infinity for linear discrete time periodic system is proposed and investigated in particular two equivalent characterizations of such notion are given a geometric one and an algebraic one the first is based on an extension to the periodic case of the controlled invariant subspace algorithm and the second is based on a newly developed periodic version of the structure algorithm an application to a model matching problem is described

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

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

primality testing using elliptic curves we present a primality proving algorithm a probablistic primality test that produces short certificates of primality on prime inputs we prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes as a corollary we obtain an algorithm for generating large certified primes with distribution statistically close to uniform under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size the test is shown to run in expected polynomial time for all primes yielding a las vegas primality test our test is based on a new methodology for applying group theory to the problem of prime certification and the application of this methodology using groups generated by elliptic curves over finite fields we note that our methodology and methods have been subsequently used and improved upon most notably in the primality proving algorithm of adleman and huang using hyperelliptic curves and in practical primality provers using elliptic curves

a new algorithm for constructing large carmichael numbers we describe an algorithm for constructing carmichael numbers n with a large number of prime factors p 1 p 2 p k this algorithm starts with a given number lcm p 1 1 p 2 1 p k 1 representing the value of the carmichael function n we found carmichael numbers with up to 1101518 factors

some new kinds of pseudoprimes we define some new kinds of pseudoprimes to several bases which generalize strong pseudoprimes we call them sylow p pseudoprimes and elementary abelian p pseudoprimes it turns out that every n 10 12 which is a strong pseudoprime to bases 2 3 and 5 is not a sylow p pseudoprime to two of these bases for an appropriate prime p n 1 we also give examples of strong pseudoprimes to many bases which are not sylow p pseudoprimes to two bases only where p 2 or 3

linear function dead beat observers with disturbance localization for linear periodic discrete time systems abstract in this paper the classical notion of a conditioned invariant subspace is extended to the case of linear periodic discrete time systems properly characterized and suitably specialized to the notion of an outer reconstructible subspace these ideas are used in order to find necessary and sufficient conditions for the existence of a linear function dead beat observer for such systems under unknown and unmeasurable disturbances and a synthesis procedure for such an observer

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

the kummerian property and maximal pro p galois groups abstract for a prime number p we give a new restriction on pro p groups g which are realizable as the maximal pro p galois group g f p for a field f containing a root of unity of order p this restriction arises from kummer theory and the structure of the maximal p radical extension of f we study it in the abstract context of pro p groups g with a continuous homomorphism g 1 p z p and characterize it cohomologically and in terms of 1 cocycles on g this is used to produce new examples of pro p groups which do not occur as maximal pro p galois groups of fields as above

sharpening primes is in p for a large family of numbers we present algorithms that are deterministic primality tests for a large family of integers namely integers n 1 mod 4 for which an integer a is given such that the jacobi symbol a n 1 and integers n 1 mod 4 for which an integer a is given such that a n 1 a n 1 the algorithms we present run in 2 min k 2 log log n o log n 6 time where k v 2 n 1 is the exact power of 2 dividing n 1 when n 1 mod 4 and k v 2 n 1 if n 1 mod 4 the complexity of our algorithms improves up to o log n 4 when k 2 log log n we also give tests for a more general family of numbers and study their complexity

infinitely generated pseudocompact modules for finite groups and weiss theorem abstract one of the most beautiful results in the integral representation theory of finite groups is a theorem of a weiss that detects a permutation r lattice for the finite p group g in terms of the restriction to a normal subgroup n and the n fixed points of the lattice where r is a finite extension of the p adic integers using techniques from relative homological algebra we generalize weiss theorem to the class of infinitely generated pseudocompact lattices for a finite p group allowing r to be any complete discrete valuation ring in mixed characteristic a related theorem of cliff and weiss is also generalized to this class of modules the existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result the permutation cover is explicitly described

geometric tools for solving the fdi problem for linear periodic discrete time systems this paper studies the problem of detecting and isolating faults in linear periodic discrete time systems the aim is to design an observer based residual generator where each residual is sensitive to one fault whilst remaining insensitive to the other faults that can affect the system making use of the geometric tools and in particular of the outer observable subspace notion the fault detection and isolation fdi problem is formulated and necessary and solvability conditions are given an algorithmic procedure is described to determine the solution of the fdi problem

divisors in residue classes in this paper the following result is proved let r s and n be integers satisfying t i l 3 gcd r s 1 then there exist at most 11 positive divisors of n that arc congruent to r modulo s moreover there exists an efficient algorithm for determining all these divisors the bound 11 is obtained by means of a combinatorial model related to coding theory it is not known whether 11 is best possible in any case it cannot be replaced by 5 nor is it known whether similar results are true for significantly smaller values of log s log n the algorithm trcated in the paper has applications in computational number theory in this paper we prove the following theorem theorem let r s and n be integers satisfying o n n3 gcd r s 1 then there exist at most 11 positive divisors of n that are congruent to r niodulo s and there is a polynomial algorithm for determining all these divisors the algorithm referred to in the theorem is described in section 1 it is polynomial in the sense that the number of bit operations required by the algorithm is bounded by a polynomial function of the binary length of n more precisely we shall see that this number of bit operations is o log n 3 employing fast multiplication techniques we can improve this bound to o log n 2 e for every e 0 we mention two applications of the algorithm in several primality testing algorithms see 3 7 the number n to be tested is subjected to a collection of pseudo prime tests if n does not pass all these tests it is composite if n does pass all these tests one knows that each divisor of n lies in one of a small and explicitly known set of residue classes modulo an auxiliary number s in the latter case all divisors of n can easily be found if s satisfies the condition s n 2 our algorithm shows that the same can be done if s satisfies the weaker condition s nl 3 in special cases this observation was already made in 2 theorems 5 and 17 the second application is to the related problem of factoring n choosing s to be a suitable integer exceeding n 3 and applying our algorithm to all residue classes rmod s we obtain an algorithm that factors n in time o n l 3 e for every e 0 the same bound was achieved by lehman 6 and conjecturally by pinter 9 by methods that are similar in spirit there exist better factoring methods both in theory and in practice see 7 but this application indicates at least that it may be difficult to extend the algorithm to significantly smaller values of s received april 27 1983 1980 mathematics subject classificcation primary loh20 ioa25 94b25

a one parameter quadratic base version of the baillie psw probable prime test the well known baillie psw probable prime test is a combination of a rabin miller test and a true i e with d n 1 lucas test arnault mentioned in a recent paper that no precise result is known about its probability of error grantham recently provided a probable prime test rqft with probability of error less than 1 7710 and pointed out that the lack of counter examples to the baillie psw test indicates that the true probability of error may be much lower in this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter tu t mod t2 ut 1 and define the base counting functions b n u 0 u n n is a psp tu and sb n u 0 u n n is an spsp tu then we give explicit formulas to compute b n and sb n and prove that for odd composites n b n n 2 and sb n n 8 and point out that these are best possible finally based on one parameter quadratic base pseudoprimes we provide a probable prime test called the one parameter quadratic base test opqbt which passed by all primes 5 and passed by an odd composite n p1r1 p2r2 psrs p1 p2 ps odd primes with probability of error n we give explicit formulas to compute n and prove that n 1 n4 3 for n nonsquare free with s 1 1 n2 3 for n square free with s 2 1 n2 7 for n square free with s 3 1 8s 4 166 p1 1 for n square free with s even 4 1 16s 5 119726 for n square free with s odd 5 1 4s i 1s 1 pi2 ri 1 otherwise i e for n nonsquare free with s 2 the running time of the opqbt is asymptotically 4 times that of a rabin miller test for worst cases but twice that of a rabin miller test for most composites we point out that the opqbt has clear finite group field structure and nice symmetry and is indeed a more general and strict version of the baillie psw test comparisons with gantham s rqft are given

primality test for numbers m with a large power of 5 dividing m 4 1 the quintic reciprocity law is used to produce an algorithm that runs in polynomial time and that determines the primality of numbers m such that m4 1 is divisible by a power of 5 which is larger that m provided that a small prime p p 1 mod 5 is given such that m is not a fifth power modulo p the same test equations are used for all such m a sufficiency test together with its probability of succeeding in determining primality is given when the condition on m modulo p is omitted

implementing the asymptotically fast version of the elliptic curve primality proving algorithm the elliptic curve primality proving ecpp algorithm is one of the current fastest practical algorithms for proving the primality of large numbers its running time cannot be proven rigorously but heuristic arguments show that it should run in time o log n 5 to prove the primality of n an asymptotically fast version of it attributed to j o shallit runs in time o log n 4 the aim of this article is to describe this version in more details leading to actual implementations able to handle numbers with several thousands of decimal digits

the carmichael numbers up to 10 15 we extend our previous computations to show that there are 1401644 carmichael numbers up to 10 18 as before the numbers were generated by a back tracking search for possible prime factorisations together with a large prime variation we present further statistics on the distribution of carmichael numbers

implementation of a new primality test an implementation of the cohen lenstra version see ibid vol 42 p 297 330 1984 of the adleman pomerance rumely primality test see l m adleman c pomerance and r s rumely ann of math vol 117 p 173 206 1983 is presented primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes

factoring with two large primes the study of integer factoring algorithms and the design of faster factoring algorithms is a subject of great importance in cryptology cf 1 and a constant concern for cryptographers in this paper we present a new technique that proved to be extremely useful not only to achieve a considerable speed up of an older and widely studied factoring algorithm but also and more importantly to make practical application of a new factoring algorithm feasible while this first application does not pose serious threats to factorization based cryptosystems the consequences of the second application could be very encouraging from the cryptanalysts point of view

fast exponentiation using data compression we present the first exponentiation algorithm that uses the entropy of the source of the exponent to improve on existing exponentiation algorithms when the entropy is smaller than 1 w s l s 1 where w s is the hamming weight of the exponent and l s is its length for entropy 1 it is comparable to the best known general purpose exponentiation algorithms

finding isomorphisms between finite fields we show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time every finite field has cardinality p for some prime number p and some positive integer conversely if p is a prime number and a positive integer then there exists a field of cardinality p and any two fields of cardinality p are isomorphic these results are due to e h moore 1893 10 in the present paper we are interested in an algorithmic version of his theorem in particular of the uniqueness part we say that a finite field is explicitly given if for some basis of the field over its prime field we know the product of any two basis elements expressed in the same basis let more precisely p be a prime number and a positive integer then by explicit data for a finite field of cardinality p we mean a system of elements ajjk k x of the prime field fp z pz suchthat f becomes a field with the ordinary addition and multiplication by elements of fp and the multiplication determined by n

the sylow subgroups of the absolute galois group gal q abstract we describe the l sylow subgroups of gal q for an odd prime l by observing and studying their decomposition as f z l where f is a free pro l group and z l are the l adic integers we determine the finite z l quotients of f and more generally show that every split embedding problem of z l groups for f is solvable moreover we analyze the z l action on generators of f

finding strong pseudoprimes to several bases define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic primality testing algorithm which is not only easier to implement but also faster than either the jacobi sum test or the elliptic curve test thanks to pornerance et al and jaeschke m are known for 1 m 8 upper bounds for 9 10 and 1 were given by jaeschke in this paper we tabulate all strong pseudoprimes spsp s n 10 24 to the first ten prime bases 2 3 29 which have the form n pq with p q odd primes and q 1 k p 1 k 2 3 4 there are in total 44 such numbers six of which are also spsp 31 and three numbers are spsp s to both bases 31 and 37 as a result the upper bounds for 10 and 11 are lowered from 28 and 29 decimal digit numbers to 22 decimal digit numbers and a 24 decimal digit upper bound for 12 is obtained the main tools used in our methods are the biquadratic residue characters and cubic residue characters we propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases comparisons of effectiveness with both jaeschke s and arnault s methods are given