We introduce new modulus scaling techniques for transforming a class of primes into special forms which enables eﬃcient arithmetic. The scaling technique may be used to improve multiplication and inversion in ﬁnite ﬁelds. We present an eﬃcient inversion algorithm that utilizes the structure of scaled modulus.
Let us first discuss some issues related, directly ,indirectly, to error detection and correction.
Types of ErrorsRedundancyDetection Versus CorrectionForward Error Correction Versus RetransmissionCoding
This book contains 104 of the best problems used in the training and testing of
the U.S. International Mathematical Olympiad (IMO) team. It is not a collection
of very difficult, and impenetrable questions. Rather, the book gradually builds
students’ number-theoretic skills and techniques. The first chapter provides a
comprehensive introduction to number theory and its mathematical structures.
This chapter can serve as a textbook for a short course in number theory.
Articles in this volume are based on talks given at the Gauss-Dirichlet Conference held in GÃ¶ttingen on June 20-24, 2005. The conference commemorated the 150th anniversary of the death of C.-F. Gauss and the 200th anniversary of the birth of J.-L. Dirichlet. The volume begins with a definitive summary of the life and work of Dirichlet and continues with thirteen papers by leading experts on research topics of current interest in number theory that were directly influenced by Gauss and Dirichlet.
The paper discusses some public key algorithms such as DH, RSA, DSA, ECDH and ECDSA and also gives mathematical explanations on the working of these algorithms. The paper also gives a brief introduction to modular arithmetic, which is the core arithmetic of almost all public key algorithms.
In the underlying ﬁnite ﬁeld arithmetic of an elliptic curve cryptosystem, ﬁeld multiplication is the next computational costly operation other than ﬁeld inversion. We present two novel algorithms for eﬃcient implementation of ﬁeld multiplication and modular reduction used frequently in an elliptic curve cryptosystem deﬁned over GF (2n ).
Bài giảng "Nhập môn an toàn thông tin - Chương 2c: Toán học dùng cho mật mã" có cấu trúc gồm 2 phần cung cấp cho người học các kiến thức: Số học số nguyên (Integer Arithmetic), số học đồng dư (Modular Arithmetic). Mời các bạn cùng tham khảo.
Over half of the students who enrol on economics degree courses have not studied mathematics
beyond GCSE or an equivalent level. These include many mature students whose last
encounter with algebra, or any other mathematics beyond basic arithmetic, is now a dim and
distant memory. It is mainly for these students that this book is intended. It aims to develop
their mathematical ability up to the level required for a general economics degree course (i.e.
one not specializing in mathematical economics) or for a modular degree course in economics
and related subjects, such as business studies.
Elliptic units, which are obtained by evaluating modular units at quadratic
imaginary arguments of the Poincar´e upper half-plane, provide us with a rich
source of arithmetic questions and insights. They allow the analytic construction
of abelian extensions of imaginary quadratic fields, encode special values of zeta functions through the Kronecker limit formula, and are a prototype for
Stark’s conjectural construction of units in abelian extensions of number fields.