testing by verifying walsh coefficients testing of logic networks by verifying the walsh coefricients of the outputs is explored measurement of one of these can detect arbitrarily many input leads stuck and just two measurements requiring little hardware can detect any single stuck at fault in appropriately designed networks

low complexity normal elements over finite fields of characteristic two in this paper we extend previously known results on the complexities of normal elements using algorithms that exhaustively test field elements we are able to provide the distribution of the complexity of normal elements for binary fields with degree extensions up to 39 we also provide current results on the smallest known complexity for the remaining degree extensions up to 512 by using a combination of constructive theorems and known exact values we give an algorithm to exhaustively search field elements by using gray codes which allows us to reuse previous computations we compare this with a standard method we analyze this algorithm and show both experimentally and asymptotically that the gray code optimization gives substantial savings the total computation of the distribution of the complexity of normal elements for degrees up to 39 in our experiments allows us to draw several conjectures in particular our data provides remarkable evidence for the conjecture that the complexity of normal elements follows a normal distribution finally we conjecture that there is no linear bound on the minimum complexity with respect to the degree of the extension

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

supremum weighted log rank test and sample size for comparing two stage adaptive treatment strategies summary in two stage adaptive treatment strategies patients receive an induction treatment followed by a maintenance therapy given that the patient responded to the induction treatment they received to test for a difference in the effects of different induction and maintenance treatment combinations a modified supremum weighted log rank test is proposed the test is applied to a dataset from a two stage randomized trial and the results are compared to those obtained using a standard weighted log rank test a sample size formula is proposed based on the limiting distribution of the supremum weighted log rank statistic the sample size formula reduces to eng and kosorok s sample size formula for a two sample supremum log rank test when there is no second randomization monte carlo studies show that the proposed test provides sample sizes that are close to those obtained by standard weighted log rank test under a proportional hazards alternative however the proposed test is more powerful than the standard weighted log rank test under non proportional hazards alternatives

the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

restricted mean survival time for interval censored data restricted mean survival time rmst evaluates the mean event free survival time up to a prespecified time point it has been used as an alternative measure of treatment effect owing to its model free structure and clinically meaningful interpretation of treatment benefit for right censored data in clinical trials another type of censoring called interval censoring may occur if subjects are examined at several discrete time points and the survival time falls into an interval rather than being exactly observed the missingness of exact observations under interval censored cases makes the nonparametric measure of treatment effect more challenging employing the linear smoothing technique to overcome the ambiguity we propose a new model free measure for the interval censored rmst as an alternative to the commonly used log rank test we further construct a hypothesis testing procedure to assess the survival difference between two groups simulation studies show that the bias of our proposed interval censored rmst estimator is negligible and the testing procedure delivers promising performance in detecting between group difference with regard to size and power under various configurations of survival curves the proposed method is illustrated by reanalyzing two real datasets containing interval censored observations

upper bounds for the prime divisors of wendt s determinant let c 2 be an even integer 3 c 1 the resultant wc of the polynomials tc 1 and 1 t c 1 is known as wendt s determinant of order c we prove that among the prime divisors q of wc only those which divide 2c 1 or lc 2 can be larger than c 4 where 2 2487338 and ln is the nth lucas number except when c 20 and q 61 using this estimate we derive criteria for the nonsolvability of fermat s congruence

improved techniques for lower bounds for odd perfect numbers if n is an odd perfect number and qk n q prime k even then it is almost immediate that n q2k we prove here that subject to certain conditions verifiable in polynomial time in fact n q5k 2 using this and related results we are able to extend the computations in an earlier paper 1 to show that n 10300 comments only the abstract is given here the full paper appeared as 2 the main part of the proof that there is no odd perfect number n less than 10300 is a very large tree each of whose 12655 leaves gives either a contradiction or a sufficiently large lower bound on n the proof tree is available by anonymous ftp in a separate file rpb116p dvi z the integer factorizations used in the proof are also available by anonymous ftp see 3 references 1 r p brent and g l cohen a new lower bound for odd perfect numbers mathematics of computation 53 1989 431 437 supplement ibid s7 s24 mr 89m 11008 also appeared as report tr cs 88 05 computer sciences laboratory anu february 1988 50 pp rpb100 2 r p brent g l cohen and h j j te riele improved techniques for lower bounds for odd perfect numbers mathematics of computation 57 1991 857 868 mr 92c 11004 also appeared as report nm r8921 centrum voor wiskunde en informatica amsterdam october 1989 13 pp a longer version was submitted to the mathematics of computation umt file and appeared as report cma r50 89 cma anu october 1989 198 pp rpb116 3 r p brent factor an integer factorization program for the ibm pc report tr cs 89 23 computer sciences laboratory anu october 1989 7 pp rpb117 brent computer sciences laboratory australian national university canberra e mail address rpb cslab anu edu au cohen school of mathematical sciences university of technology sydney po box 123 broadway nsw 2007 australia te riele centrum voor wiskunde en informatica kruislaan 413 1098 sj amsterdam the nether lands e mail address herman cwi nl 1991 mathematics subject classification primary 11a25 secondary 11 04 11y05 11y70

on permutation tests for comparing restricted mean survival time with small sample from randomized trials between group comparison based on the restricted mean survival time rmst is getting attention as an alternative to the conventional logrank hazard ratio approach for time to event outcomes in randomized controlled trials rcts the validity of the commonly used nonparametric inference procedure for rmst has been well supported by large sample theories however we sometimes encounter cases with a small sample size in practice where we cannot rely on the large sample properties generally the permutation approach can be useful to handle these situations in rcts however a numerical issue arises when implementing permutation tests for difference or ratio of rmst from two groups in this article we discuss the numerical issue and consider six permutation methods for comparing survival time distributions between two groups using rmst in rcts setting we conducted extensive numerical studies and assessed type i error rates of these methods our numerical studies demonstrated that the inflation of the type i error rate of the asymptotic methods is not negligible when sample size is small and that all of the six permutation methods are workable solutions although some permutation methods became a little conservative no remarkable inflation of the type i error rates were observed we recommend using permutation tests instead of the asymptotic tests especially when the sample size is less than 50 per arm

figures of merit for digital multistep pseudorandom numbers the statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p s i which depends on s and the characteristic polynomial f of the recursion used in the generation procedure we extend previous work for s 2 and describe how to obtain large figures of merit for s 2 thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties tables of figures of merit for s 3 4 5 and degrees 32 are included

fast arithmetic with general gaus periods we show how to apply fast arithmetic in conjunction with general gaus periods in finite fields this is an essential ingredient for some efficient exponentiation algorithms

factoring high degree polynomials over f 2 with niederreiter s algorithm on the ibm sp2 a c implementation of niederreiter s algorithm for factoring polynomials over f 2 is described the most time consuming part of this algorithm which consists of setting up and solving a certain system of linear equations is performed in parallel once a basis for the solution space is found all irreducible factors of the polynomial can be extracted by suitable gcd computations for this purpose asymptotically fast polynomial arithmetic algorithms are implemented these include karatsuba ofman multiplication cantor multiplication and newton inversion in addition a new efficient version of the half gcd algorithm is presented sequential run times for the polynomial arithmetic and parallel run times for the factorization are given a new world record for polynomial factorization over the binary field is set by showing that a pseudo randomly selected polynomial of degree 300000 can be factored in about 10 hours on 256 nodes of the ibm sp2 at the cornell theory center

weighted kaplan meier statistics a class of distance tests for censored survival data a class of statistics based on the integrated weighted difference in kaplan meier estimators is introduced for the two sample censored data problem with positive weight functions these statistics are intuitive for and sensitive against the alternative of stochastic ordering the standard weighted log rank statistics are not always sensitive against this alternative particularly if the hazard functions cross qualitative comparisons are made between the weighted log rank statistics and these weighted kaplan meier wkm statistics a statement of null asymptotic distribution theory is given and the choice of weight function is discussed in some detail results from small sample simulation studies indicate that these statistics compare favorably with the log rank procedure even under the proportional hazards alternative and may perform better than it under the crossing hazards alternative

gauss periods orders and cryptographical applications experimental results on the multiplicative orders of gauss periods in finite fields are presented these results indicate that gauss periods have high order and are often primitive self dual normal elements in finite fields it is shown that gauss periods can be exponentiated in quadratic time an application is an efficient pseudorandom bit generator

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

restricted mean life with adjustment for covariates abstract although the mean is the most widely used measure of location in comparing two distributions its use is essentially precluded in the presence of substantial censoring typical in survival analysis of medical therapies to accommodate censoring irwin 1949 proposed estimation of the expectation of life restricted to a time t using the actuarial estimator of the survival function in this article we examine the restricted mean as an index for comparing survival in two groups and then extend the method to incorporate covariates into the analysis for this purpose it is convenient to depart slightly from the actuarial approach and to consider piecewise exponential models asymptotic properties are explored and the assumption of piecewise exponentiality is shown to be rather mild as departures from the model have a relatively small effect on estimation of the restricted mean the methodology is illustrated using published data from a study conducted by the radiation therapy oncology group rtog

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

a table of primitive binary polynomials for those n 5000 for which the factorization of 2n 1 is known the first primitive trinomial if such exists and a randomly generated primitive 5 and 7 nomial of degree n in gf 2 are given a primitive polynomial of degree n over gf 2 is useful for generating a pseudo random sequence of n tuples of zeros and ones see 8 if the polynomial has a small number k of terms then the sequence is easily computed but for cryptological applications correlation attack see 5 it is often necessary to have the primitive polynomials with k larger than one can find in the existing tables for example zierler and brillhart 10 11 have calculated all irreducible trinomials of degree n 1000 with the period for some for which the factorization of 2n 1 is known stahnke 7 has listed one example of a trinomial or pentanomial of degree n 168 zierler 12 has listed all primitive trinomials whose degree is a mersenne exponent 11213 m23 here mj denotes the jth mersenne exponent rodemich and rumsey 6 have listed all primitive trinomials of degree mj 12 j 17 kurita and matsumoto 2 have listed all primitive trinomials of degree mj 24 j 28 and one example of primitive pentanomials of degree mj 8 j 27 1980 mathematics subject classification 1985 revision primary 11t06 11t71

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

estimation of the average hazard ratio summary simple estimators of the average hazard ratio are obtained for the two sample problem with censored failure time data these estimators are compared with estimators arising out of the partial likelihood within the proportional hazards class efficiency results are found to be generally quite favourable provided the hazard ratio is not too large some comments are made on extensions to the k sample problem

table of primitive binary polynomials ii for those n 5000 for which the factorization of 2n 1 is known the first primitive trinomial if such exists and a randomly generated primitive 5and 7 nomial of degree n in gf 2 are given if the respective entry is absent from the previously published table after our paper with a list of primitive binary polynomials 3 had been accepted for publication s s wagstaff jr made available to us reference 2 with some new factorizations of 2n 1 updates to 1 obtained up to april 5 1993 this enabled us to make an update to our list in which the smallest degree of missing primitive binary polynomial is now 51 1 here like in our previous table the degrees of nonzero coefficients except 0 and n of primitive k nomials if there are any of degree n are given for some missing 2 n 5000 table 1 primitive binary polynomials

on the empirical choice of the time window for restricted mean survival time the t year mean survival or restricted mean survival time rmst has been used as an appealing summary of the survival distribution within a time window 0 t rmst is the patient s life expectancy until time t and can be estimated nonparametrically by the area under the kaplan meier curve up to t in a comparative study the difference or ratio of two rmsts has been utilized to quantify the between group difference as a clinically interpretable alternative summary to the hazard ratio the choice of the time window 0 t may be prespecified at the design stage of the study based on clinical considerations on the other hand after the survival data have been collected the choice of time point t could be data dependent the standard inferential procedures for the corresponding rmst which is also data dependent ignore this subtle yet important issue in this paper we clarify how to make inference about a random parameter moreover we demonstrate that under a rather mild condition on the censoring distribution one can make inference about the rmst up to t where t is less than or even equal to the largest follow up time either observed or censored in the study this finding reduces the subjectivity of the choice of t empirically the proposal is illustrated with the survival data from a primary biliary cirrhosis study and its finite sample properties are investigated via an extensive simulation study

prime clusters and cunningham chains we discuss the methods and results of a search for certain types of prime clusters in particular we report specific examples of prime 16 tuplets and cunningham chains of length 14

a unified view of test compression methods a unified treatment of the various techniques to reduce the output data from a unit under test is given the characteristics of time compression schemes with respect to errors detected are developed the use of two or more of these methods together is considered methods to design efficient test compression structures for built in tests are proposed the feasibility of the proposed approach is demonstrated by simulation results

subquadratic space complexity binary field multiplier using double polynomial representation this paper deals with binary field multiplication we use the bivariate representation of binary field called double polynomial system dps presented in this concept generalizes the composite field representation to every finite field as shown in the main interest of dps representation is that it enables to use lagrange approach for multiplication and in the best case fast fourier transform approach which optimizes lagrange approach we use here a different strategy from to perform reduction and we also propose in this paper some new approaches for constructing dps we focus on dps which provides a simpler and more efficient method for coefficient reduction this enables us to avoid a multiplication required in the montgomery reduction approach of and thus to improve the complexity of the dps multiplier the resulting algorithm proposed in the present paper is subquadratic in space o n1 31 and logarithmic in time the space complexity is 33 percent better than in and 18 percent faster it is asymptotically more efficient than the best known method specifiably more efficient than when n 3 000 furthermore our proposal is available for every n and not only for n a power of two or three

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

normal bases via general gauss periods gauss periods have been used successfully as a tool for constructing normal bases in finite fields starting from a primitive r th root of unity one obtains under certain conditions a normal basis for mathbb f q n over f q where r is a prime and nk r 1 for some integer k we generalize this construction by allowing arbitrary integers r with nk varphi r and find in many cases smaller values of k than is possible with the previously known approach

the versatility of function indexed weighted log rank statistics abstract two sample weighted log rank statistics are used in the presence of right censoring to test whether failure times from two populations have different survival distributions kosorok has showed that large families of these statistics form stochastic processes indexed by weight functions and that these function indexed statistics can be used to construct versatile test procedures simultaneously sensitive to a wide array of both ordered hazards and stochastic ordering alternatives the complexity of the asymptotic distribution of these statistics precludes obtaining p values through analytical means in this article we develop a monte carlo method for accurately obtaining these p values and we evaluate the moderate sample size properties of this method and compare the power of function indexed statistics with previously developed weighted log rank tests these statistics are also examined in a data analysis of the beta blocker heart attack trial bhat the results of this article demonstrate that

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

factors of generalized fermat numbers generalized fermat numbers have the form f b m b 2m 1 their odd prime factors are of the form k 2 n 1 k odd n m it is shown that each prime is a factor of some f b m for approximately 1 k bases b independent of n divisors of generalized fermat numbers of base 6 base 10 and base 12 are tabulated three new factors of standard fermat numbers are included

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

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

a fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377 the standard algorithm for testing reducibility of a trinomial of prime degree r over gf 2 requires 2r o 1 bits of memory we describe a new algorithm which requires only 3r 2 o 1 bits of memory and significantly fewer memory references and bit operations than the standard algorithm if 2r 1 is a mersenne prime then an irreducible trinomial of degree r is necessarily primitive we give primitive trinomials for the mersenne exponents r 756839 859433 and 3021377 the results for r 859433 extend and correct some computations of kumada et al the two results for r 3021377 are primitive trinomials of the highest known degree

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

supremum versions of the log rank and generalized wilcoxon statistics abstract considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics the most commonly used generalizations are the log rank test statistic which extends the savage exponential scores statistic cox 1972 mantel 1966 and the generalized wilcoxon statistics gehan 1965 peto and peto 1972 gill 1980 showed that in the two sample problem the asymptotic efficiency properties of the savage and wilcoxon statistics are maintained in censored data by the log rank and peto and peto wilcoxon statistics respectively when censoring is the same in both samples heuristics and simulations have implied that some recently proposed supremum statistics closely related to these two popular linear rank statistics can be used to construct tests that provide good power against a much wider range of nonlocal alternatives see e g fleming and harrington 1981 fleming o brien o fallon and harrington 1980 gill 1980 schumacher 1984 in this article we consider n

aliasing probability calculations for arbitrary compaction under independently selected random test vectors this paper discusses a systematic methodology for calculating the exact aliasing probability associated with schemes that use an arbitrary finite state machine to compact the response of a combinational circuit to a sequence of independently selected random test input vectors the proposed approach identifies the strong influence of fault activation probabilities on the probability of aliasing and uses an asymmetric error model to simultaneously track the states of two fictitious compactors one driven by the response of the fault free combinational circuit and one driven by the response of the faulty combinational circuit by deriving the overall markov chain that describes the combined behavior of these two compactors we are able to calculate the exact aliasing probability for any test sequence length in particular for long enough sequences the probability of aliasing is shown to only depend on the stationary distribution of the markov chain the insights provided by our analysis are used to evaluate the testing performance of simple examples of nonlinear compactors and to demonstrate regimes where they exhibit lower aliasing probability than linear compactors with the same number of states finally by establishing connections with previous work that evaluated aliasing probability in linear compactors our analysis clarifies the role played by the entropy of the stationary distribution of the compactor states

efficiency of two sample tests via the restricted mean survival time for analyzing event time observations in comparing two treatments with the event time observations the hazard ratio hr estimate is routinely used to quantify the treatment difference however this model dependent estimate may be difficult to interpret clinically especially when the proportional hazards ph assumption is violated an alternative estimation procedure for treatment efficacy based on the restricted means survival time or t year mean survival time t mst has been discussed extensively in the statistical and clinical literature on the other hand a statistical test via the hr or its asymptotically equivalent counterpart the logrank test is asymptotically distribution free in this article we assess the relative efficiency of the hazard ratio and t mst tests with respect to the statistical power under various ph and non ph models theoretically and empirically when the ph assumption is valid the t mst test performs almost as well as the hr test for non ph models the t mst test can substantially outperform its hr counterpart on the other hand the hr test can be powerful when the true difference of two survival functions is quite large at end but not the beginning of the study unfortunately for this case the hr estimate may not have a simple clinical interpretation for the treatment effect due to the violation of the ph assumption

weighted kaplan meier statistics large sample and optimality considerations summary we propose a cumulative weighted difference in the kaplan meier estimates as a test statistic for equality of distributions in the two sample censored data survival analysis problem for stability of such a statistic the absolute value of the possibly random weight function must be bounded above by a multiple of c 1 2 6 where 1 c is the left continuous censoring distribution function and 6 0 for these weighted kaplan meier wkm statistics asymptotic distribution theory is presented along with expressions for the efficacy under a sequence of local alternatives a simple censored data generalization of the two sample difference in means test z test is a member of this class and in large samples is seen to be quite efficient relative to the popular log rank test under a range of alternatives including the proportional hazards alternative optimal weight functions are also calculated the optimal wkm statistic is as efficient as the optimal weighted log rank statistic for any particular sequence of local alternatives stratified statistics and trend statistics are also presented

efficient montgomery based semi systolic multiplier for even type gnb of gf 2 m efficient finite field multiplication is crucial for implementing public key cryptosystem to achieve this multipliers using gaussian normal basis have been widely explored in previous works in this paper based on proposed gaussian normal basis montgomery gnbm representation a semi systolic even type gnbm multiplier is developed analysis shows that the proposed architecture saves about 57 percent space complexity and 50 percent time complexity when compared with the only existing semi systolic even type gnb multiplier moreover due to properties of regularity and modularity the proposed multiplier is very suitable for vlsi implementation

the loss in efficiency from misspecifying covariates in proportional hazards regression models summary this note considers partial likelihood score tests for association between a covariate process and survival time based on a proportional hazards regression model we derive the asymptotic efficiency of the test based on a misspecified form of the covariate process relative to the test based on the correct form the results can be used to assess the consequences of various types of misspecification

spectral fault signatures for single stuck at faults in combinational networks a method is described for the derivation of fault signatures for the detection of stuck at faults in single output combinational networks these signatures consist of a set of values derived from the network any single stuck at fault causes at least one value to change the fault signatures developed are a generalization of syndrome testing the technique is developed in the rademacher walsh spectral domain but is easily implemented using conters and basic gates

a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases abstract this paper proposes a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases normal bases have the following useful property in the case that an element x in gf 2 m is represented by normal bases 2 k power operation of an element x in gf 2 m can be carried out by k times cyclic shift of its vector representation c c wang et al proposed an algorithm for computing multiplicative inverses using normal bases which requires m 2 multiplications in gf 2 m and m 1 cyclic shifts the fast algorithm proposed in this paper also uses normal bases and computes multiplicative inverses iterating multiplications in gf 2 m it requires at most 2 log 2 m 1 multiplications in gf 2 m and m 1 cyclic shifts which are much less than those required in the wang s method the same idea of the proposed fast algorithm is applicable to the general power operation in gf 2 m and the computation of multiplicative inverses in gf q m q 2 n

a new framework for designing and analyzing bist techniques and zero aliasing compression a general framework for shift register based signature analysis is presented and a mathematical model for this framework based on coding theory is developed there are two key features of this formulation first it allows for uniform treatment of lfsr misr and multiple misr based signature analyzer in addition using this formulation a new compression scheme for multiple output cut is proposed this scheme referred to as multiinput lfsr has the potential to achieve better aliasing than other schemes such as the multiple misr scheme of comparable hardware complexity several results on aliasing are presented and certain known results are shown to be direct consequences of the formulation also developed are error models that take into account the circuit topology and the effect of faults at the outputs using these models exact closed form expressions for aliasing probability are developed a closed form aliasing expression for misr under an independent error model is provided

spectral fault signatures for internally unate combinational networks a method is described for the derivation of fault signatures for certain classes or irredundant combinational networks these signatures consist of a set of values derived from the network any stuck at fault causes at least one of the values to change the signatures provide complete fault detection for all single stuck at faults they are usually short and never contain more than n 1 values for an n input network

syndrome and transition count are uncorrelated logic circuit testing in the testing of logic circuits two proposed data compression methods use the number of ones syndrome and the number of sequence changes transition count an enumeration n m k t of the number of length m binary sequences having syndrome value k and transition count t is developed examination of this result reveals that the parallel compression of these two methods has small overlap in error masking an asymptotic expression for n m k t is developed

restricted mean life with covariates modification and extension of a useful survival analysis method abstract karrison has presented a method for covariate adjusted comparison of two groups with respect to survival when the group effect is expected to not be of proportional hazards form this article describes a simplified procedure for implementing karrison s basic approach along with an extended version designed to achieve robustness against misspecification in the underlying analytical model a simulation study of these techniques is presented and the techniques are illustrated with an example taken from karrison

some observations on parallel algorithms for fast exponentiation in gf 2 n a normal basis representation of operatorname gf 2 n allows squaring to be accomplished by a cyclic shift algorithms for multiplication in operatorname gf 2 n using a normal basis have been studied by several researchers in this paper algorithms for performing exponentiation in operatorname gf 2 n using a normal basis and how they can be speeded up by using parallelization are investigated

the computational complexity of continued fractions the knuth scho nhage algorithm for expanding a quolynomial into a continued fraction is shown to be essentially optimal with respect to the number of multiplications divisions used uniformly in the inputs

a generalization of addition chains and fast inversions in binary fields in this paper we study a generalization of addition chains where k previous values are summed together on each step instead of only two values as in traditional addition chains such chains are called k chains and we show that they have applications in finding efficient parallelizations in problems that are known to be difficult to parallelize in particular 3 chains improve computations of inversions in finite fields using hybrid double multipliers recently it was shown that this operation can be efficiently computed using a ternary algorithm but we show that 3 chains provide a significantly more efficient solution

a versatile test for equality of two survival functions based on weighted differences of kaplan meier curves with censored event time observations the logrank test is the most popular tool for testing the equality of two underlying survival distributions although this test is asymptotically distribution free it may not be powerful when the proportional hazards assumption is violated various other novel testing procedures have been proposed which generally are derived by assuming a class of specific alternative hypotheses with respect to the hazard functions the test considered by pepe and fleming 1989 is based on a linear combination of weighted differences of the two kaplan meier curves over time and is a natural tool to assess the difference of two survival functions directly in this article we take a similar approach but choose weights that are proportional to the observed standardized difference of the estimated survival curves at each time point the new proposal automatically makes weighting adjustments empirically the new test statistic is aimed at a one sided general alternative hypothesis and is distributed with a short right tail under the null hypothesis but with a heavy tail under the alternative the results from extensive numerical studies demonstrate that the new procedure performs well under various general alternatives with a caution of a minor inflation of the type i error rate when the sample size is small or the number of observed events is small the survival data from a recent cancer comparative study are utilized for illustrating the implementation of the process copyright 2015 john wiley sons ltd

new architectures for digit level single hybrid double hybrid triple field multiplications and exponentiation using gaussian normal bases gaussian normal bases gnbs are special set of normal bases nbs which yield low complexity gf left 2 m right arithmetic operations in this paper we present new architectures for the digit level single hybrid double and hybrid triple multiplication of gf left 2 m right elements based on the gnb representation for odd values of m 1 the proposed fully serial in single multipliers perform multiplication of two field elements and offer high throughput when the data path capacity for entering inputs is limited the proposed hybrid double and hybrid triple digit level gnb multipliers perform respectively two and three field multiplications using the same latency required for a single digit level multiplier at the expense of increased area in addition we present a new eight ary field exponentiation architecture which does not require precomputed or stored intermediate values

algorithms and arithmetic operators for computing the ηt pairing in characteristic three since their introduction in constructive cryptographic applications pairings over hyper elliptic curves are at the heart of an ever increasing number of protocols with software implementations being rather slow the study of hardware architectures became an active research area in this paper we discuss several algorithms to compute the etat pairing in characteristic three and suggest further improvements these algorithms involve addition multiplication cubing inversion and sometimes cube root extraction over f3m we propose a hardware accelerator based on a unified arithmetic operator able to perform the operations required by a given algorithm we describe the implementation of a compact coprocessor for the field f397 given by f3 x x97 x12 2 which compares favorably with other solutions described in the open literature

finding a basis conversion matrix using a polynomial basis derived by a small multiplicative cyclic group several methods for finding a basis conversion matrix between two different bases in an extension field have been proposed among them the one based on gauss period normal basis gnb is on average the most efficient however since it needs to construct a certain tower field some inefficient cases in which the towering degree becomes large have been reported this paper first determines that such inefficient cases are caused by the gnb condition in order to overcome this inefficiency we propose a method that does not use any gnb in the target extension field but instead uses a certain polynomial basis in derived by a certain small cyclic group in this causes re laxation of the condition for the towering degree in addition our experimental results show that the proposed method substantially accelerates the computation time for finding a basis conversion matrix

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

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

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

on orders of optimal normal basis generators in this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in f 2 n over f 2 for n 1200 whenever the complete factorization of 2 n 1 is known our results show that a subclass of optimal normal basis generators always have high multiplicative orders at least o 2 n 1 n and are very often primitive for a given optimal normal basis generator a in f 2 n and an arbitrary integer e we show that e can be computed in o n v e bit operations where v e is the number of 1 s in the binary representation of e