a geometric approach to root finding in gt q sup m the problem of finding roots in f of polynomials in f x for f gf q sup m where q is a prime or prime power and m is a positive integer greater than 1 is considered the problem is analyzed by making use of the finite affine geometry ag m q a new method is proposed for finding roots of polynomials over finite extension fields it is more efficient than previous algorithms when the degree of the polynomial whose roots are to be found is less than dimension m of the extension field implementation of the algorithm can be enhanced in cases in which optimal normal bases for the coefficient field are available

on binary cyclic codes of odd lengths from 101 to 127 all binary cyclic codes of odd lengths are checked from 101 to 127 to find codes which are better than those in a table by t verhoeff 1989 there are five such cases namely 117 36 32 117 37 29 117 42 26 117 49 24 and 127 36 35 cyclic codes according to verhoeff s table the previously known ranges of the highest minimum distance were 28 40 28 40 25 37 22 32 and 32 46 respectively applying constructions x and y1 120 37 32 and 108 28 32 codes were found moreover the highest minimum distances that cyclic codes of length 127 can attain are determined

polynomial factorization over f 2 we describe algorithms for polynomial factorization over the binary field f2 and their implementation they allow polynomials of degree up to 250 000 to be factored in about one day of cpu time distributing the work on two processors

factoring multivariate polynomials over large finite fields a simple probabilistic algorithm is presented to find the irreducible factors of a bivariate polynomial over a large finite field for a polynomial f x y over fq of total degree n our algorithm takes at most 4 89 2 n log n log q operations in fq to factor f x y completely this improves a probabilistic factorization algorithm of von zur gathen and kaltofen which takes o n 1 l log n log q operations to factor f x y completely over fq the algorithm can be easily generalized to factor multivariate polynomials over finite fields we shall give two further applications of the idea involved in the algorithm one is concerned with exponential sums the other is related to permutational polynomials over finite fields a conjecture of chowla and zassenhaus

computing in groups of lie type we describe two methods for computing with the elements of untwisted groups of lie type using the steinberg presentation and using highest weight representations we give algorithms for element arithmetic within the steinberg presentation conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction

nonbinary quantum codes we present several results on quantum codes over general alphabets that is in which the fundamental units may have more than two states in particular we consider codes derived from finite symplectic geometry assumed to have additional global symmetries from this standpoint the analogs of calderbank shor steane codes and of gf 4 linear codes turn out to be special cases of the same construction this allows us to construct families of quantum codes from certain codes over number fields in particular we get analogs of quadratic residue codes including a single error correcting code encoding one letter in five for any alphabet size we also consider the problem of fault tolerant computation through such codes generalizing ideas of gottesman see phys rev a vol 57 no 1 p127 37 1998

the asymptotic probability distribution of the relative distance of additive quantum codes abstract for given information rate r it is proved as n tends to infinite that almost all additive n n r quantum codes pure and impure are the codes with their relative distance tending to h 1 1 r where h x x log 2 x 3 1 x log 2 1 x is an entropy function

using the theory of cyclotomy to factor cyclotomic polynomials over finite fields we examine the problem of factoring the rth cyclotomic polynomial r x over fp r and p distinct primes given the traces of the roots of r x we construct the coefficients of r x in time o r 4 we demonstrate a deterministic algorithm for factoring r x in time o r 1 2 log p 9 when r x has precisely two irreducible factors finally we present a deterministic algorithm for computing the sum of the irreducible factors of r x in time o r 6

a note on grobner bases and berlekamp s algorithm it will be shown that it is possible to factorize square free polynomials over finite fields using grobner bases and a maple implementation of this method will be described

zero aliasing for modeled faults when using built in self test bist for testing vlsi circuits the circuit response to an input test sequence which may consist of thousands to millions of bits is compacted into a signature which consists of only tens of bits usually a linear feedback shift register lfsr is used for response compaction via polynomial division the compacting function is a many to one function and as a result some erroneous responses may be mapped to the same signature as the good response this is known as aliasing in this paper we deal with the selection of a feedback polynomial for the compacting lfsr such that an erroneous response resulting from any modeled fault is mapped to a signature that is different from that for the good response such lfsrs are called zero aliasing lfsrs only zero aliasing lfsrs with primitive or irreducible feedback polynomials are considered due to their suitability for bist test pattern generation

on bivariate polynomial factorization over finite fields this paper shows that a recently proposed approach of d q wan to bivariate factorization over finite fields the univariate factoring algorithm of v shoup and the new bound of this paper for the average number of irreducible divisors of polynomials of a given degree over a finite field can be used to design a bivariate factoring algorithm that is polynomial for almost all bivariate polynomials

approximate solution methods for linear stochastic difference equations i moments the cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations we consider a vector difference equation which contains a deterministic matrix a0 and a random perturbation matrix a1 t the expansion proceeds in powers of c where c is the correlation time of the fluctuations in a1 t and a measure for their strength compared to the differential case additional cumulants occur in the expansion moreover one has to distinguish between a nonsingular and a singular a0 we also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation the derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

an efficient basis conversion algorithm for composite fields with given representations we describe an efficient method for constructing the basis conversion matrix between two given finite field representations where one is composite we are motivated by the fact that using certain representations e g low hamming weight polynomial or composite field representations permits arithmetic operations such as multiplication and inversion to be computed more efficiently an earlier work by paar defines the conversion problem and outlines an exponential time algorithm that requires an exhaustive search in the field another algorithm by sunar et al provides a polynomial time algorithm for the limited case where the second representation is constructed rather than initially given the algorithm we present facilitates existing factorization algorithms and provides a randomized polynomial time algorithm to solve the basis conversion problem where the two representations are initially given we also adapt a fast trace based factorization algorithm to work in the composite field setting which yields a subcubic complexity algorithm for the construction of the basis conversion matrix

factoring polynomials over finite fields with drinfeld modules in the following we describe a way of factoring polynomials in fq x with drinfeld modules we furthermore analyse the complexity of the algorithm and compare it to the well known cantor zassenhaus algorithm

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

recognition of finite exceptional groups of lie type let q be a prime power and let g be an absolutely irreducible subgroup of gld f where f is a finite field of the same characteristic as fq the field of q elements assume that g g q a quasisimple group of exceptional lie type over fq which is neither a suzuki nor a ree group we present a las vegas algorithm that constructs an isomorphism from g to the standard copy of g q if g 6 3d4 q with q even then the algorithm runs in polynomial time subject to the existence of a discrete log oracle

some new results on the minimum length of binary linear codes of dimension nine let n k d be the smallest integer n for which a binary linear code of length n dimension k and minimum distance d exists using the residual code technique the macwilliams identities and the weight distribution of appropriate reed muller codes we prove that n 9 64 133 n 9 120 spl ges 244 n 9 124 252 and n 9 184 371 we also show that puncturing a known 322 9 160 code yields length optimal codes with parameters 319 9 158 315 9 156 and 312 9 154

nonbinary quantum stabilizer codes we define and show how to construct nonbinary quantum stabilizer codes our approach is based on nonbinary error bases it generalizes the relationship between self orthogonal codes over f sub 4 and binary quantum codes to one between self orthogonal codes over f q sup 2 and q ary quantum codes for any prime power q

identification of matrix generators of a chevalley group abstract we give a polynomial time algorithm that replaces an arbitrary module of a finite chevalley group by the adjoint module written with respect to a chevalley basis

the black box niederreiter algorithm and its implementation over the binary field the most time consuming part of the niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix this paper describes the so called black box niederreiter algorithm in which these elements are found by using a method developed by wiedemann the main advantages over an approach based on gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower the black box niederreiter algorithm for factoring polynomials over the binary field was implemented in the c programming language and benchmarks for factoring high degree polynomials over this field are presented these benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations in addition the wan algorithm which was recently introduced is described and connections between implementation aspects of wan s and niederreiter s algorithm are given

factorization of multivariate polynomials over finite fields we present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size i e in the degree of the polynomial and log cardinality of field the algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs a deterministic version of the algorithm is also discussed whose running time is polynomial in the degree of the input polynomial and the size of the field

an algorithm for lang s theorem we give an efficient las vegas type algorithm for lang s theorem in split connected reductive groups defined over finite fields of characteristic greater than 3 this algorithm can be used to construct many important structures in finite groups of lie type we use an algorithm for computing a chevalley basis for a split reductive lie algebra which is of independent interest for our time analysis we derive that the proportion of reflection derangements in a weyl group is less than 2 3

inhomogeneous linear random differential equations with mutual correlations between multiplicative additive and initial value terms abstract the cumulant expansion for linear stochastic differential equations is extended to the general case in which the coefficient matrix the inhomogeneous part and the initial condition are all random and moreover statistically interdependent the expansion now involves not only the autocorrelation functions of the coefficient matrix as in the homogeneous case but also crosscorrelation functions of the coefficient matrix with the inhomogeneous part and with the initial value term as a first illustration we consider an exactly solvable stochastic differential equation with initial correlations and compare the exact solution with that of the cumulant expansion secondly we show in general how the method can be used for the calculation of second moments and treat the harmonic oscillator with random frequency and driving term as an example

computing chevalley bases in small characteristics abstract let l be the lie algebra of a simple algebraic group defined over a field f and let h be a split maximal toral subalgebra of l then l has a chevalley basis with respect to h if char f 2 3 it is known how to find it in this paper we treat the remaining two characteristics to this end we present a few new methods implemented in magma which vary from the computation of centralizers of one root space in another to the computation of a specific part of the lie algebra of derivations of l

multipartite entangled states symmetric matrices and error correcting codes a pure quantum state is called k uniform if all its reductions to k qudit are maximally mixed we investigate the general constructions of k uniform pure quantum states of n subsystems with d levels we provide one construction via symmetric matrices and the second one through the classical error correcting codes there are three main results arising from our constructions first we show that for any given even n ge 2 there always exists an n 2 uniform n qudit quantum state of level p for sufficiently large prime p second both constructions show that there exist k uniform n qudit pure quantum states such that k is proportional to n i e k omega n although the construction from symmetric matrices in general outperforms the one by error correcting codes third our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3 uniform n qudit pure quantum state for all n ge 8 in fact we can further prove that for every k there exists a constant m k such that there exists a k uniform n qudit quantum state for all n ge m k in addition by using the concatenation of algebraic geometry codes we give an explicit construction of k uniform quantum state when k tends to infinity

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

writing projective representations over subfields let g x be an absolutely irreducible subgroup of gl d k and let f be a proper subfield of the finite field k we present a practical algorithm to decide constructively whether or not g is conjugate to a subgroup of gl d f k where k denotes the centre of gl d k if the derived group of g also acts absolutely irreducibly then the algorithm is las vegas and costs o x d3 d2log f arithmetic operations in k this work forms part of a recognition project based on aschbacher s classification of maximal subgroups of gl d k

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

distance verification for classical and quantum ldpc codes the techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes then these techniques are considered for classical and quantum stabilizer low density parity check ldpc codes new complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of ldpc codes these bounds are expressed in terms of the erasure correcting capacity of the corresponding ensemble we also present a new irreducible cluster technique that can be applied to any ldpc code and takes advantage of parity checks sparsity for both the classical and quantum ldpc codes this technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances which also include all known families of the quantum stabilizer ldpc codes

computing discrete logarithms in the jacobian of high genus hyperelliptic curves over even characteristic finite fields we describe improved versions of index calculus algorithms for solving discrete logarithm problems in jacobians of high genus hyperelliptic curves de ned over even characteristic elds our rst improvement is to incorporate several ideas for the low genus case by gaudry and theriault including the large prime variant and using a smaller factor base into the large genus algorithm of enge and gaudry we extend the analysis in 24 to our new algorithm allowing us to predict accurately the number of random walk steps required to nd all relations and to select optimal degree bounds for the factor base our second improvement is the adaptation of sieving techniques from flassenberg and paulus and jacobson to our setting the new algorithms are applied to concrete problem instances arising from the weil descent attack methodology for solving the elliptic curve discrete logarithm problem demonstrating signi cant improvements in practice

asymptotic bounds on quantum codes from algebraic geometry codes we generalize a characterization of p ary p is a prime quantum codes given by feng and xing to q ary q is a prime power quantum codes this characterization makes it possible to convert an asymptotic bound of stichtenoth and xing for nonlinear algebraic geometry codes to a quantum asymptotic bound besides we also investigate the asymptotic behavior of quantum codes

ordered cumulant technique and differential equations for probability density abstract this paper is concerned with stochastic differential equations of langevin type in which the stochastic forces are gaussian and not of white noise type the problem of finding explicitly the differential equation for the probability density of the process is solved by means of an expansion in the small correlation time of the noise for this purpose we will use the cumulant technique different situations are studied such as markovian or fokker planck equation first correction to the markovian approximation but with the fokker planck form and the corrections of non fokker planck form

some additional remarks on the cumulant expansion for linear stochastic differential equations we summarize our previous results on cumular expasions for linear stochastic differential equations with correlated multipliclative and additive noise the application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail because our earlier results for this case are only valid in special situations moreover a second partially time ordered cumulant is introduced which is appropriate for the calculation of multi time averages of the solution process and which differs from the one to be used for single time averages by the two supplements just mentioned we complete our earlier investigations on the subject

factoring polynomials and primitive elements for special primes abstract for those prime numbers p for which all prime factors of p 1 are small the two problems of finding a primitive element modulo p and of factoring univariate polynomials over finite fields of characteristic p are deterministically polynomial time equivalent assuming the extended riemann hypothesis they can be solved in polynomial time

lengthening and the gilbert varshamov bound we use lengthening and an enhanced version of the gilbert varshamov lower bound for linear codes to construct a large number of record breaking codes our main theorem may be seen as a closure operation on databases

univariate polynomial factorization over finite fields abstract this paper is a tutorial introduction to univariate polynomial factorization over finite fields we recall the classical methods that induced most factorization algorithms berlekamp s and the cantor zassenhaus ones and some refinements which can be applied to these methods explicit algorithms are presented in a form suitable for almost immediate implementation we give a detailed description of an efficient implementation of the cantor zassenhaus algorithm used in the release 2 of the axiom computer algebra system

concatenated quantum codes can attain the quantum gilbert varshamov bound a family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant the asymptotic quantum gilbert varshamov gv bound states that there exist q ary quantum codes of sufficiently long block length n having fixed rate r with distance at least n h 1 q 2 1 r 2 where h q 2 is the q 2 ary entropy function for q only random quantum codes are known to asymptotically attain the quantum gv bound however random codes have little structure in this paper we generalize the classical result of thommesen to the quantum case thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum gv bound the outer codes are quantum generalized reed solomon codes and the inner codes are random independently chosen stabilizer codes where the rates of the inner and outer codes lie in a specified feasible region

classical and quantum evaluation codes at the trace roots we introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function we give conditions for self orthogonality of these codes and their subfield subcodes with respect to the hermitian inner product they allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature for the binary case we obtain records at http codetables de moreover we obtain several classical linear codes over the field mathbb f 4 which are records at http codetables de

a refinement of h c williams th root algorithm let p and q be primes such that p 1 mod q let a be an integer such that a p l q 1 mod p in 1972 h c williams gave an algorithm which determines a solution of the congruence xq a mod p in o q3 logp steps once an integer b has been found such that bq a p1 q 0 0 1 mod p a step is an arithmetic operation mod p or an arithmetic operation on q bit integers we present a refinement of this algorithm which determines a solution in o q4 o q2 logp steps once b has been determined thus the new algorithm is better when q is small compared with p

fidelity lower bounds for stabilizer and css quantum codes in this paper we estimate the fidelity of stabilizer and css codes first we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator next we find the average quantum enumerators of the ensembles of finite length stabilizer and css codes we use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles we further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and css codes finally we derive fidelity bounds in the asymptotic regime when the code length tends to infinity these results tell us which code rate we can afford for achieving a target fidelity with codes of a given length the results also show that in symmetric depolarizing channel a typical stabilizer code has better performance in terms of fidelity and code rate compared with a typical css codes and that balanced css codes significantly outperform other css codes asymptotic results demonstrate that css codes have a fundamental performance loss compared with stabilizer codes

forward and backward equations for a nonmarkovian process a cumulant approach abstract this paper deals with nonmarkovian processes induced by colored noise the forward equations of motion for both the single and joint probability distribution functions are derived at the same time the backward equation for the probability density of a nonmarkovian process is for the first time deduced these results are obtained by means of the systematic use of the ordered cumulant technique

on the covariance completion problem under a circulant structure covariance matrices with a circulant structure arise in the context of discrete time periodic processes and their significance stems also partly from the fact that they can be diagonalized via a fourier transformation this note deals with the problem of completion of partially specified circulant covariance matrices the particular completion that has maximal determinant i e the so called maximum entropy completion was considered in carli where it was shown that if a single band is unspecified and to be completed the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data i e it has the dempster property the purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements more specifically we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant a description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables facilitated by the circulant structure is highlighted

using partial smoothness of p 1 for factoring polynomials modulo p let an arbitrarily small positive constant less than 1 and a poly nomial f with integer coefficients be fixed we prove unconditionally that j modulo p can be completely factored in deterministic polynomial time if p 1 has a lnp o 1 smooth divisor exceeding p we also address the issue of factoring f modulo p over finite extensions of the prime field f p and show that p 1 can be replaced by p k 1 k n for explicit classes of primes p

computer construction of split cartan subalgebras abstract we give a polynomial time algorithm that finds a split cartan subalgebra of a finite chevalley lie algebra

short presentations for finite groups we conjecture that every finite groupghas a short presentation in terms of generators and relations in the sense that the totallengthof the relations is log g o 1 we show that it suffices to prove this conjecture for simple groups motivated by applications in computational complexity theory we conjecture that for finite simple groups such a short presentation is computable in polynomial time from the standard name ofg assuming in the case of lie type simple groups overgf pm that an irreducible polynomialfof degreemovergf p and a primitive root ofgf pm are given we verify this stronger conjecture for all finite simple groups except for the three families of rank 1 twisted groups we do not handle the unitary groupspsu 3 q 2a2 q the suzuki groupssz q 2b2 q and the ree groupsr q 2g2 q in particular all finite groups g without composition factors of these types have presentations of length o log g 3 for groups of lie type normal or twisted of rank 2 we use a reduced version of the curtis steinberg tits presentation

factoring multivariate polynomials via partial differential equations a new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic it is based on a simple partial differential equation that gives a system of linear equations as in berlekamp s and niederreiter s algorithms for factoring univariate polynomials the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd s and by factoring univariate polynomials over the ground field the new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields local fields number fields and the complex number field the theory of the new method allows an effective hilbert irreducibility theorem thus an efficient reduction of polynomials from multivariate to bivariate

trading grh for algebra algorithms for factoring polynomials and related structures in this paper we develop techniques that eliminate the need of the generalized riemann hypothesis grh from various almost all known results about deterministic polynomial factoring over finite fields our main result shows that given a polynomial f x of degree n over a finite field k we can find in deterministic poly n log n log k time either a nontrivial factor of f x or a nontrivial automorphism of k x f x of order n this main tool leads to various new grh free results most striking of which are 1 given a noncommutative algebra a of dimension n over a finite field k there is a deterministic poly n logn log k time algorithm to find a zero divisor in a this is the best known deterministic grh free result since r 1990 first studied the problem of finding zero divisors in finite algebras and showed that this problem has the same complexity as factoring polynomials over finite fields 2 given a positive integer r 4 such that either 4 r or r has two distinct prime factors there is a deterministic polynomial time algorithm to find a nontrivial factor of ther th cyclotomic polynomial over a finite field this is the best known deterministic grh free result since evdokimov 1989 showed that cyclotomic polynomials can be factored over finite fields in deterministic polynomial time assuming grh in this paper following the seminal work of lenstra 1991 on constructing isomorphisms between finite fields we further generalize classical galois theory constructs

on the minimum distance of some quadratic residue codes corresp we find the minimum distances of the binary 113 57 and ternary 37 19 61 31 71 36 and 73 37 quadratic residue codes and the corresponding extended codes these distances are 15 10 11 17 and 17 respectively for the nonextended codes and are increased by one for the respective extended codes we also characterize the minimum weight codewords for the 113 57 binary code and its extended counterpart

computing conjugating sets and automorphism groups of rational functions abstract let and be endomorphisms of the projective line of degree at least 2 defined over a field f from a dynamical perspective a significant question is to determine whether and are conjugate or to answer the related question of whether a given rational function has a nontrivial automorphism we construct efficient algorithms for computing the set of conjugating maps resp the group of automorphisms with an emphasis on the case where f is a finite field or a number field each of our algorithms takes advantage of different dynamical structures so context e g field of definition and degree of the map determines the preferred algorithm

is every matrix similar to a polynomial in a companion matrix abstract given a field f an integer n 1 and a matrix a m n f are there polynomials f g f x with f monic of degree n such that a is similar to g c f where c f is the companion matrix of f for infinite fields the answer is easily seen to positive so we concentrate on finite fields in this case we give an affirmative answer provided f n 2 moreover for any finite field f with f m we construct a matrix a m m 3 f that is not similar to any matrix of the form g c f of use above but also of independent interest is a constructive procedure to determine the similarity type of any given matrix g c f purely in terms of f and g without resorting to polynomial roots in f or in any extension thereof this in turn yields an algorithm that given g and the invariant factors of any a returns the elementary divisors of g a it is a rational procedure as opposed to the classical method that uses the jordan decomposition of a to find that of g a finally extending prior results by the authors we show that for an integrally closed ring r with field of fractions f and companion matrices c d the subalgebra r c d of m n r is a free r module of rank n n m n 1 where m is the degree of gcd f g f x and a presentation for r c d is given in terms of c and d a counterexample is furnished to show that r c d need not be a free r module if r is not integrally closed the preceding information is used to study m n r and others as r x modules

on weak odd domination and graph based quantum secret sharing a weak odd dominated wod set in a graph is a subset b of vertices for which there exists a distinct set of vertices c such that every vertex in b has an odd number of neighbors in c we point out the connections of weak odd domination with odd domination domination and perfect codes we introduce bounds on g the maximum size of wod sets of a graph g and on g the minimum size of non wod sets of g moreover we prove that the corresponding decision problems are np complete the study of weak odd domination is mainly motivated by the design of graph based quantum secret sharing protocols a graph g of order n corresponds to a secret sharing protocol whose threshold is q g max g n g these graph based protocols are very promising in terms of physical implementation however all such graph based protocols studied in the literature have quasi unanimity thresholds i e q g n o n where n is the order of the graph g underlying the protocol in this paper we show using probabilistic methods the existence of graphs with smaller q i e q g 0 811 n where n is the order of g we also prove that deciding for a given graph g whether q g k is np complete which means that one cannot efficiently double check that a graph randomly generated has actually a q smaller than 0 811n

computing special powers in finite fields we study exponentiation in nonprime finite fields with very special exponents such as they occur for example in inversion primitivity tests and polynomial factorization our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time

fast polynomial factorization and modular composition we obtain randomized algorithms for factoring degree n univariate polynomials over mathbb f q requiring o n 1 5 o 1 rm log 1 o 1 q n 1 o 1 rm log 2 o 1 q bit operations when rm log q n this is asymptotically faster than the best previous algorithms j von zur gathen and v shoup comput complexity 2 1992 pp 187 224 e kaltofen and v shoup math comp 67 1998 pp 1179 1197 for rm log q ge n it matches the asymptotic running time of the best known algorithms the improvements come from new algorithms for modular composition of degree n univariate polynomials which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields the best previous algorithms for modular composition use o n omega 1 2 field operations where omega is the exponent of matrix multiplication r p brent and h t kung j assoc comput mach 25 1978 pp 581 595 with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication x huang and v y pan j complexity 14 1998 pp 257 299 we show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent alpha implies an algorithm for the other with exponent alpha o 1 and vice versa we then give two new algorithms that solve the problem near optimally an algebraic algorithm for fields of characteristic at most n o 1 and a nonalgebraic algorithm that works in arbitrary characteristic the latter algorithm works by lifting to characteristic 0 applying a small number of rounds of multimodular reduction and finishing with a small number of multidimensional ffts the final evaluations are reconstructed using the chinese remainder theorem as a bonus this algorithm produces a very efficient data structure supporting polynomial evaluation queries which is of independent interest our algorithms use techniques that are commonly employed in practice in contrast to all previous subquadratic algorithms for these problems which relied on fast matrix multiplication

connections between the algorithms of berlekamp and niederreiter for factoring polynomials over fq abstract in this paper we determine an explicit isomorphism between the solution spaces of berlekamp s and niederreiter s algorithm for factoring polynomials over finite fields this can be used for the implementation of a procedure combining the benefits of both methods

inverses of cartan matrices of lie algebras and lie superalgebras abstract the inverses of indecomposable cartan matrices are computed for finite dimensional lie algebras and lie superalgebras over fields of any characteristic and for hyperbolic almost affine complex lie super algebras this enables one to express the fundamental weights in terms of simple roots corresponding to the cartan matrix we discovered three yet inexplicable new phenomena of which a and b concern hyperbolic almost affine complex lie super algebras a several of the inverses of cartan matrices have all their elements negative not just non positive as they should be according to an a priori characterization due to zhang hechun b the 0s only occur on the main diagonals of the inverses c the determinants of inequivalent cartan matrices of the simple lie super algebra of any dimension may differ in any characteristic we interpret most of the results of wei and zou 2017 31 as inverses of the gram matrices of non degenerate invariant symmetric bilinear forms on the super algebras considered not of cartan matrices and give more adequate references in particular the inverses of cartan matrices of simple lie algebras were already published starting with dynkin s paper in 1952 see also table 2 in springer s book by onishchik and vinberg 1990

approximate solution methods for linear stochastic difference equations ii inhomogeneous equations multi time averages biological application abstract the cumulant expansion for linear stochastic difference equations introduced in part i is applied to the general case where the equation contains multiplicative additive and initial value terms which are all random and statistically interdependent also the two time correlation functions of the solution are discussed finally the expansion for the probability density functions is studied in the case that the coefficient matrix constitutes a markov process an exact equation for the joint probability density function of the solution of the difference equation and the random coefficient matrix is derived from this equation the moments of the solution can be obtained in a simple way as an application we consider the growth of a biological population with two age classes in a random environment which itself is modelled by a two state markov chain the exact results are compared with those of the cumulant expansion and with previous findings of tuljapurkar

a cumulant expansion for the time correlation functions of solutions to linear stochastic differential equations it is shown that the cumulant expansion for linear stochastic differential equations hitherto used to compute one time averages of the solution process is also capable of yielding the two time correlation and probability density functions the general case with a coefficient matrix an inhomogeneous part and an initial condition which are all random and mutually correlated is discussed two examples are given the latter of which treats the harmonic oscillator with stochastic frequency and driving term studied before finally we investigate the relation of our method with the so called smoothing method

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

fast evaluation of logarithms in fields of characteristic two a method for determining logarithms in gf 2 n is presented its asymptotic running time is o exp cn 1 3 log 2 3 n for a small constant c while by comparison adleman s scheme runs in time o exp c n 1 2 log 1 2 n the ideas give a dramatic improvement even for moderate sized fields such as gf 2 127 and make barely possible computations in fields of size around 2 400 the method is not applicable to gf q for a large prime q

a finite gilbert varshamov bound for pure stabilizer quantum codes a finite gilbert varshamov gv bound for pure stabilizer binary and nonbinary quantum error correcting codes is presented in analogy to the gv bound for classical codes by using several enumerative results in finite unitary geometry from this quantum gv bound we obtain several new binary quantum codes in a nonconstructive way having better parameters than the known codes