R. A. Mollin

An Introduction to Cryptography, Second Edition

As the title states the book by Johannes Buchmann provides an introduction to cryptography. Buchmann’s text in only 324 pages (excluding the appendices) presents a stand alone introduction to some modern cryptographic methods. The book begins with the mathematical background that will be used as foundation for the cryptographic methods discussed in this book. Chapter one explains the important properties of integers and the extended Euclidean algorithm. Chapter two includes some important algebraic definitions (groups, residue class, ring, fields,...). Also, it contains algorithms for fast evaluation of power products and the Chinese remainder theorem. Probability theory and Shannon’s view of perfect secrecy are presented in chapter 4. There are three chapters discussing symmetric cryptography and to be more specific they are devoted to block ciphers. Chapter 3 explains the meaning of cryptosystems and gives some different encryption schemes. For the symmetric cryptography the author just defines the structure of stream ciphers and provides two examples, whereas he explains the block cipher in more details. Moreover chapter 5 and chapter 6 represent a complete study to the most famous block ciphers DES and AES, respectively. In the next four chapters, the author discusses asymmetric cryptography (public key cryptography). Since many public key cryptosystems use large prime numbers, the author gives more additional mathematical preliminaries for the prime number generation and some algorithms used for testing the primality of large numbers in chapter 7. The idea of public key cryptosystems and a description to the most important schemes are given in chapter 8, for examples RSA and ElGamal. The security of RSA is based on the difficulty of the factorization problem and is studied in chapter 9. It is focusing on the quadratic sieve algorithm and providing an estimation to the efficiency of it and some other factoring algorithms. Similarly, in chapter 10 the security analysis of ElGamal cryptosystem is discussed by analyzing the algorithms that solve the discrete logarithm problem.

Diophantine equations and class numbers

Abstract The goals of this paper are to provide: (1) sufficient conditions, based on the solvability of certain diophantine equations, for the non-triviality of the class numbers of certain real quadratic fields; (2) sufficient conditions for the divisibility of the class numbers of certain imaginary quadratic fields by a given integer; and (3) necessary and sufficient conditions for an algebraic integer (which is not a unit) to be the norm of an algebraic integer in a given extension of number fields.

A conjecture of S. Chowla via the generalized Riemann hypothesis

S. Chowla conjectured that if p = m2 + 1 is prime and m > 26, then hK, the class number of K = Q(,JFp), is greater than 1. We prove this conjecture under the assumption of the Riemann hypothesis for fK, the zeta function of K, i.e. the generalized Riemann hypothesis (GRH). It is the purpose of this note to prove the following result. THEOREM. Let K = Q(Q/;f), where p = m2 + 1 is prime and m > 26. If the Riemann hypothesis holds for fK then hK > 1. Without the GRH hypothesis, this is known as the Chowla conjecture given in [1]. We note that it is an easy consequence of the celebrated Brauer-Siegel theorem. that there are only finitely many such p for which hK = 1. In [3] Mollin reduced the problem to the case where m = 2r and r > 13 is prime. In what follows we make use of an idea of Cornell and Washington [2] to show how to use the GRH to get an effective bound (p > 1023) for which the Chowla conjecture holds. The remaining finite cases are then handled by a simple sieve process. Throughout the remainder of the paper p will denote a prime of the form 4r2 + 1 where r is an odd prime, and K will denote Q( /Fp). The following result contains facts which are either well known or trivial. Therefore we state it without proof. LEMMA. Let y be a real number. (1) If y > 1, then > l/q 0, then E(1/q2) y, (3) If jyj y Received by the editors August 13, 1986 and, in revised form, December 8, 1986. 1980 Mathematics Subject Classtfication (1985 Revision). Primary 12A50, 12A25; Secondary 10B05.

On the Insolubility of a Class of Diophantine Equations and the Nontriviality of the Class Numbers of Related Real Quadratic Fields of Richaud-Degert Type

Many authors have studied the relationship between nontrivial class numbers h(n ) of real quadratic fields and the lack of integer solutions for certain diophantine equations. Most such results have pertained to positive square-free integers of the form n = l 2 + r with integer >0 , integer r dividing 4l and — l For n of this form, is said to be of Richaud-Degert (R-D) type (see [3] and [8]; as well as [2], [6], [7], [12] and [13] for extensions and generalizations of R-D types.)

Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla

Based on the fundamental unit of Q(\?), an arbitrary real quadratic field, we provide a necessary condition for the class number h(n) to be 1. For n = 4m2 +1 we prove the equivalence of three necessary and sufficient conditions for h(n) to be 1. One of these conditions is that -x2 + x + m2 is prime for all integers x such that 1 < x < m. This is the exact analogue of the complex quadratic field case. We discuss the connection with a conjecture of S. Chowla as well as with other related topics.

On prime valued polynomials and class numbers of real quadratic fields

Gauss conjectured that there are infinitely many real quadratic fields with class number one. Today this is still an open problem. Moreover, as Dorian Goldfeld, one of the recipients of the 1987 Cole prize in number theory (for his work on another problem going back to Gauss) recently stated in his acceptance of the award: “This problem appears quite intractible at the moment.” However there has recently been a search for conditions which are tantamount to class number one for real quadratic fields. This may be viewed as an effort to shift the focus of the problem in order to understand more clearly the inherent difficulties, and to reveal some other beautiful interrelationships.

Algebras with uniformly distributed invariants

Let K be a finite abelian extension of the rational field Q. If A is a central simple algebra over K then we let [A] denote the class of A in the Brauer group B(K) of K. The Schur subgrozcp S(K) of B{K) consists of those algebra classes which contain a simple component of the group algebra K[GJ for some finite group G. M. Benard and M. Schacher [2, Theorem 1, p. 3801 have shown that if [A] is in S(K) then:

Uniform distribution and the Schur subgroup

In this paper we continue the investigation into the group of algebras with uniformly distributed invariants U(K), and its relation to the Schur subgroup, undertaken in [9]. The notation is the same as in [9]. In the first section we investigate the index 1 U(K), : S(K), 1 where q is an odd prime. We obtain [9, Theorem 2.71 as a special case of Theorem 1.2, wherein we obtain that the above index is infinite when q 1 / K : Q(+)], where +, is the highest q-power root of unity in K, a > 0 provided qa+h +’ /L : K 1, where L is the smallest root of unity field containing K, and ++b is the highest q-power root of unity in L. In the case where aatb 1 / L : K /, the author’s conjecture made in [9] is validated. In Section 2, we show that 1 U(K), : S(K), / is infinite where K/Q is finite, imaginary, and abelian. As an illustration of this result we calculate generators of U(Q(E~,,))~ for primes p + 1 (mod 4) explicitly.

CLASS NUMBERS OF REAL QUADRATIC FIELDS, CONTINUED FRACTIONS, REDUCED IDEALS, PRIME-PRODUCING QUADRATIC POLYNOMIALS AND QUADRATIC RESIDUE COVERS

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in (10)—(31) and is related to work in (3)-(4).

On the Cyclotomic Polynomial

For a given positive integer m and an algebraic number field K necessary and sufficient conditions for the mth cyclotomic polynomial to have K-integral solutions modulo a given integer of K are given. Among applications thereof are: that the solvability of the cyclotomic polynomial mod an integer yields information about the class number of related number fields; and about representation of integers by binary quadratic forms. The latter extends previous work of the author. Moreover some information is obtained pertaining to when an integer of K is the norm of an integer in a given quadratic extension of K. Finally an explicit determination of the pqth cyclotomic polynomial for distinct primes p and q is provided, and known results in the literature as well as generalizations thereof are obtained.