Arithmetic Statistics for Algebraic Objects

Elliptic curves have been intensively studied in algebraic geometry and number theory. Recently, they have been used in devising efficient algorithms for factoring integers [8, 9], for primality proving [2, 11, 12] and for the construction of pseudorAndom bit generators [4]. In Chapter 8 we study how elliptic curves can be used in constructing efficient and secure cryptosystems, while in Chapter 10 we will discuss how good error-correcting codes may be obtained by using elliptic curves.

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2 = x3 + ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2 = x3 − 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves.In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2 = x3 − 27x + 62 by using algebraic number theory and P-adic analysis method.In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2 = x3 − 27x − 62.In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2 = x3 − 21x − 90 By using the elementary method.In 2016, Guo J solved the problem of the integer points on y2 = x3 + 27x + 62 by using the elementary method.In 2017, Guo J proved that y2 = x3 − 21x + 90 has no integer points by using the elementary method.Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 3x − 14, which is the subject of this paper.By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 3x − 14 has only one integer point: (x, y) = (2, 0).

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term “elliptic curve” in the literature seems to have been by James Thomson in 1727 in “A Poem sacred to the Memory of Sir Isaac Newton”: “He, first of Men, with awful Wing pursu’d the Comet tro’ the long Elliptic Curve.” In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

Some well-known classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z and p-adic integers Zp and maximal orders of local fields and fields of algebraic numbers go back to classical papers by S. S. Ryshkov, P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the representations, and one of the possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n. The basic idea underlying a geometric approach was given in Ryshkov’s papers on the computation of the finite subgroups of GLn(Z) and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached. In the present paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of the group G. A combinatorial construction of the series of these representations is given explicitly. In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions is constructed. We consider certain related questions including the embedding problem in Galois theory for local faithful primitive representations of supersolvable groups and integral representations arising from elliptic curves.

In this paper we describe LiDIA, a new library for computational number theory. Why do we work on a new library for computational number theory when such powerful tools as Pari [1], Kant [11], Simath [10] already exist? In fact, those systems are very useful for solving problems for which there exist efficient system routines. For example, using Pari or Kant it is possible to compute invariants of algebraic number fields and Simath can be used to find the rank of an elliptic curve over Q. However, building complicated and efficient software on top of existing systems has in our experience turned out to be very difficult. Therefore, the software of our research group is developed independently of other computer algebra systems.

In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context.

Following Lang and Trotter, we describe a probabilistic model that predicts the distribution of primes p with given Frobenius traces at p for two fixed elliptic curves over $$\mathbb {Q}$$ . In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the $$\ell $$ -adic limits determining the $$\ell $$ -factors of the universal constant, unlike the Lang–Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves, which extends the main results of Fouvry and Murty (Supersingular primes common to two elliptic curves, number theory (Paris, 1992), London Mathematical Society Lecture Note Series, vol 215, Cambridge University Press, Cambridge, 1995) and Akbary et al. (Acta Arith 111(3):239–268, 2004), following the work of David et al. (Math Ann 368(1–2):685–752, 2017).

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

In the current paper, consisting from two parts, are presented both results already published before (but hard for access) and new once. Actuality of this work is firstly in a fact that recently has been obtained a number of new results in area of applied cryptography that are needed both in a clarification and be put into practice. This is namely the main goal of the current paper. The setting problem in the first paper part concerns to a complexity of symmetric cipher breaking while in the second part of the paper is discussed, so called, keyless cryptography, namely: wiretap channel concept, execution of communication channels which allow to provide information security without of key exchange procedure between legal correspondences. In the part widely used methods of applied mathematics, namely: algebra, number, probability and information theories. Computer simulation also used there. A novelty of the first part of work consists in the following: first of all it is clarified the sense of a key lifetime limitation for different symmetric cipher modes, secondly, it is explained an approach of cipher breaking by the use of quantum computers, finely, the key authentication for the Diffie ‒ Hellman protocol based on the mobile device pairing technology is investigated in detail. In the second part of the current paper has been presented a vulnerability of Dean ‒ Goldsmith cryptosystem under some extension of attacks. The main results of this paper are: estimation of the key lifetime of single key for symmetric cipher in CBC mode, clarifying of Grover’s algorithm breaking of symmetric ciphers by brute force attack, development of a method for authentication of Diffie ‒ Hellman values based on pre-distributed sequences, selection of ciphers which allow to execute with Shamir’s protocol without any key sharing in advance, breaking of Dean ‒ Goldsmith protocol under some conclusions, proof the fact regarding of a possible breakability of the key sharing protocol over noiseless communication channels. Practical application of paper results consists in the fact of stimulation the correct choice of ciphers and their parameters in order to provide their resistance to different attacks and more attention to algorithms of keyless cryptography.

This textbook covers with the main topics in number theory as taught in universities throughout the world. Number theory deals mainly with properties of integers and rational numbers; it is not an organized theory in the usual sense, but a vast collection of individual topics and results, with some coherent sub-theories and a long list of unsolved problems. Nine main areas of the subject are treated: (i) divisibility and multiplicative functions; (ii) congruences and quadratic residues; (iii) the basics of algebraic numbers and sums of squares; (iv) continued fractions, diophantine approximations and transcendence; (v) quadratic forms; (vi) partitions; (vii) the prime numbers; (viii) diophantine equations; and (ix) elliptic curves (this part has been considerably extended for this edition). The increased use of computers in number theory is reflected in many sections (with much greater emphasis in this edition). Some results of a more advanced nature are also given, including the Gelfond-Schneider theorem, the prime number theorem, and the Mordell-Weil theorem. The latest work on fermat's last theorem is also breifly discussed. Each chapter ends with a collection of problems; hints or sketch solutions are given at the end of the book, together with various useful tables.

The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a non-constructive way, the basic building blocks of the respective algebras (the radical and the simple components of the factor by the radical). In view of the extensive computations done in such algebras, it seems important to design efficient algorithms to find these building blocks.We find polynomial time solutions to a substantial part of these problems. We restrict our attention to algebras over finite fields and over algebraic number fields. We succeed in determining the radical (the “bad part” of the algebra) in polynomial time, using (in the case of prime characteristic) some new algebraic results developed in this paper. For associative algebras we are able to determine the simple components as well. This latter result generalizes factorization of polynomials over the given field. Correspondingly, our algorithm over finite fields is Las Vegas.Some of the results generalize to fields given by oracles.Some fundamental problems remain open. An example: decide whether or not a given rational algebra is a noncommutative field.

I present this paper for publication to an American journal and in English for the following reason :The theory of linear algebras has been greatly extended through the work of American mathematicians.Of late, German mathematicians have become active in this theory.In particular, they have succeeded in obtaining some apparently remarkable results by using the theory of algebraic numbers, ideals, and abstract algebra, highly developed in Germany in recent decades.These results do not seem to be as well known in America as they should be on account of their importance.This fact is due, perhaps, to the language difference or to the unavailability of the widely scattered sources.This paper develops a new application of the above mentioned theories to the theory of linear algebras.Of particular importance is the fact that purely algebraic results are obtained from deep-lying arithmetical theorems.In the middle part, an account is given of the fundamental algebraic basis for these arithmetical methods.This account is more extended than is necessary for this paper, and should obviate an extended study of several German papers.I am very grateful to Professor F. T. Engstrom (New Haven) for going through the manuscript and proof with me and anglicising my many literal translations from the German.I. Statement of theorems to be proved!1. Cyclic algebras.In I, the reference field is assumed to be an algebraic number field of finite degree.}A cyclic algebra of degree over Q, is defined as an algebra A of the following type: * Presented to the Society, September 9, 1931; received by the editors May 29, 1931.f This section has also appeared recently in German in the Nachrichten von der Gesellschaft der Wissenschaften zu Gttingen, 1931. In the following, such notations as are only made complete by naming a definite reference field, may be implicitly understood to be with respect to unless another reference field is explicitly named. || See also Dickson (1 App. II).* It is unnecessary to distinguish between isomorphism (equivalence) and identity unless subalgebras of the same algebra are considered.t In the following, the general norm residue theory, recently developed by the author, is of the greatest importance.See therefore the papers Hasse (11,12, 13 II). This alternative form has been introduced by the editors to simplify typography. Hasse (1-4).I' a,Z\ Ia,Z\ I ,Z\(3-2) hr)-(-r)(ir> By (3.1) and (3.2), the symbol ((a, Z)/p) is indeed fixed except an arbitrary exponent prime to n and independent of a which may be attached.fIt is a pity that one is not able to-day to fix that exponent in a quite natural manner, i.e., without having to consider also the congruence behavior of a for prime spots different from p and with it, in principle, the law of reciprocity.Therefore, in order to obtain the complete definition of the symbol, one has to take the following round-about way : Let f be the conductor (Fhrer) of Z,% fp the exact power of p contained in f, and a0 a number in Q with the following properties : * Hasse (12 p. 150).t This follows immediately from the fact that Z is cyclic.X I.e., the conductor of the ideal class group to which Z belongs as class corps (Klassenkrper) See Hasse (7, 9). Hasse (7 Satz 13).Incidentally (3.5) may also be omitted; see Hasse (11, 13 6).I adjoined it here only in order to get a formal simplification in the later proof of Theorem 1, (i).* This alternative form has been introduced by the editors to simplify typography.t See Hasse (7.Erl. 30, 9 8).J See also Dickson (1 51,3 78), Artin (2). See Wedderburn (1), Artin (2).

Para>In 1882, Lindemann wrote a paper in which he sketched a general result, special cases of which imply the transcendence of e and π. This general result was later proved rigorously by K. Weierstrass in 1885. Before we begin, we make some remarks pertaining to algebraic number theory. Let K be an algebraic number field.

This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as well as a fairly complete discussion of major algebraic systems (such as rings, fields, and groups) with a view of their use in bar coding, public key cryptosystems, error-correcting codes, counting techniques, and elliptic key cryptography. This is the only entry level text for algebraic systems that includes an extensive introduction to elliptic curves, a topic that has leaped to prominence due to its importance in the solution of Fermat’s Last Theorem and its incorporation into the rapidly expanding applications of elliptic curve cryptography in smart cards. Computer science students will appreciate the strong emphasis on the theory of polynomials, algebraic geometry and Groebner bases. The combination of a rigorous introduction to abstract algebra with a thorough coverage of its applications makes this book truly unique.

The general subject of the conference was geometric group theory, that is the study of groups via suitable actions on spaces endowed with additional structures, for instance a distance, a measure (class), or both of them with natural compatibilities. The interactions with other mathematical fields such as algebraic topology (e.g. via homological methods), functional analysis (e.g. via the theory of operator algebras), probability theory (e.g. via random walks) or even representation theory and number theory, play an increasing role in the development of this part of group theory.