refined analysis to the extended tower number field sieve abstract the hardness of discrete logarithm problem over finite fields is the security foundation of many cryptographic protocols when the characteristic of the finite field is medium or large the state of art algorithms for solving the corresponding problem are the number field sieve and its variants in 2016 kim and barbulescu presented the extended tower number field sieve which achieves a new complexity in the medium prime case and imposes a new estimation of the security of concrete parameters in certain cryptosystems such as pairing based cryptosystems in this paper a refined analysis to this algorithm is given as follows firstly a uniform formula is given for the total complexity of the extended tower number field sieve for a given polynomial selection method this formula can directly give the complexity in this case then a method is proposed to improve the computation in the smoothing phase by exploring subfield structures when the extension degree is composite at last the complexity of the descent phase is analyzed when sieving over degree one polynomials and high degree polynomials respectively and it is shown still negligible compared to the improved smoothing phase

strictly convex norms and topology we introduce a new topological property called and the corresponding class of topological spaces which includes spaces with g diagonals and gruenhage spaces using we characterize those banach spaces which admit equivalent strictly convex norms and give an internal topological characterization of those scattered compact spaces k for which the dual banach space c k admits an equivalent strictly convex dual norm we establish some relationships between and other topological concepts and the position of several well known examples in this context for instance we show that c k admits an equivalent strictly convex dual norm where k is kunen s compact s space also under additional axioms we provide examples of compact scattered non gruenhage spaces of cardinality 1 having

continuous functions on totally ordered spaces that are compact in their order topologies abstract the totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes the banach space of continuous functions on the lexicographic product of compact totally ordered spaces whose spaces of continuous functions have locally uniformly convex norms has an equivalent locally uniformly convex norm if and only if the product is countable the banach space of continuous functions on a compact totally ordered space always has an equivalent kadec norm those compact totally ordered spaces for which the banach space of continuous functions has an equivalent locally uniformly convex norm are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces some examples are discussed in detail

constructing nonresidues in finite fields and the extended riemann hypothesis we present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields f p n where p is prime and k is a prime divisor of p n 1 we prove under the assumption of the extended riemann hypothesis erh that for fixed n and p our algorithm runs in polynomial time unlike other deterministic algorithms for this problem this polynomial time bound holds even if k is exponentially large more generally assuming the erh in time n log p o n we can construct a set of elements that generates the multiplicative group f p n an extended abstract of this paper appeared in proc 23rd ann acm symp on theory of computing 1991

a subexponential time algorithm for computing discrete logarithms over gf p 2 an algorithm for computing discrete logarithms over gf p 2 where p is a prime in subexponential time is described the algorithm is similar to the merkle adleman algorithm for computing logarithms over gf p but it uses quadratic fields as the appropriate algebraic structure it also makes use of the idea of a virtual spanning set due to hellman and reyneri for computing discrete logarithms over gf p m for m growing and p fixed

least and greatest fixed points in intuitionistic natural deduction this paper is a comparative study of a number of intensional semantically distinct least and greatest fixed point operators that natural deduction proof systems for intuitionistic logics can be extended with in a proof theoretically defendable way eight pairs of such operators are analysed the exposition is centred around a cube shaped classification where each node stands for an axiomatization of one pair of operators as logical constants by intended proof and reduction rules and each arc for a proof and reduction preserving encoding of one pair in terms of another the three dimensions of the cube reflect three orthogonal binary options conventional style vs mendler style basic co iterative vs enhanced primitive co recursive simple vs course of value co induction some of the axiomatizations and encodings are well known others however are novel the classification into a cube is also new the differences between the least fixed point operators considered are illustrated on the example of the corresponding natural number types

the extended calculus of constructions ecc with inductive types abstract luo s extended calculus of constructions ecc is a higher order functional calculus based on coquand s and huet s calculus of constructions but has in addition strong sums and a predicative cumulative type hierarchy in this paper i introduce inductive types on the predicative type levels of ecc i also show how the set model for ecc can be extended to a model for this augmented calculus

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

semantics for classical automath and related systems l article essaie de combler la lacune concernant la semantique formelle des langages aut en fournissant une theorie modele mathematique pour automath classique la semantique proposee couvre le lambda calcul typee de l ordre de church et la theorie analogue de fonctionnalite de curry

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

using number fields to compute logarithms in finite fields we describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field we conjecture that the running time of the algorithm when restricted to finite fields of an arbitrary but fixed degree is l q 1 3 64 9 1 3 o 1 where q is the cardinality of the field l q s c exp c log q s log log q 1 s and the o 1 is for q the number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time

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

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

reducing elliptic curve logarithms to logarithms in a finite field elliptic curve cryptosystems have the potential to provide relatively small block size high security public key schemes that can be efficiently implemented as with other known public key schemes such as rsa and discrete exponentiation in a finite field some care must be exercised when selecting the parameters involved in this case the elliptic curve and the underlying field specific classes of curves that give little or no advantage over previously known schemes are discussed the main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field for the class of supersingular elliptic curves the reduction takes probabilistic polynomial time thus providing a probabilistic subexponential time algorithm for the former problem

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

on banach spaces whose norm open sets are fσ sets in the weak topology we investigate non separable banach spaces whose norm open sets are countable unions of sets closed in the weak topology and a narrower class of banach spaces with a network for the norm topology which is discrete in the weak topology in particular we answer a question of arhangel skii exhibiting various examples of non separable function spaces c k with a discrete network for the pointwise topology and consistently we answer some questions of edgar and oncina concerning borel structures and kadec renormings in banach spaces

the semantics of second order lambda calculus abstract in the second order polymorphic typed lambda calculus lambda abstraction over type variables leads to terms denoting polymorphic functions straightforward cardinality considerations show that a naive set theoretic interpretation of the calculus is impossible we give two definitions of semantic models for this language and prove them equivalent our syntactical environment model definition and a more algebraic combinatory model definition for the polymorphic calculus correspond to analogous model definitions for untyped lambda calculus soundness and completeness theorems are proved using the environment model definition we verify that some specific interpretations of the calculus proposed in the literature indeed yield models in our sense

continuous functions on compact totally ordered spaces abstract the banach space c k of continuous functions on k with its pointwise topology is shown to be fragmented by pointwise closed sets when k is a compact totally ordered set when k is a dendron and when k is a compact hausdorff space with a closed crossfree subbase for its topology in some special instances c k will have an equivalent locally uniformly convex norm

a higher order calculus and theory abstraction abstract we present a higher order calculus ecc which naturally combines coquand huet s calculus of constructions and martin lof s type theory with universes ecc is very expressive both for structured abstract reasoning and for program specification and construction in particular the strong sum types together with the type universes provide a useful module mechanism for abstract description of mathematical theories and adequate formalization of abstract mathematics this allows comprehensive structuring of interactive development of specifications programs and proofs after a summary of the meta theoretic properties of the calculus an set realizability model of ecc is described to show how its essential properties can be captured set theoretically the model construction entails the logical consistency of the calculus and gives some hints on how to adequately formalize abstract mathematics theory abstraction in ecc is discussed as a pragmatic application