Number Theory for Computing
Song Y. Yan
Language: English
Pages: 435
ISBN: 3540430725
Format: PDF / Kindle (mobi) / ePub
This book provides a good introduction to the classical elementary number theory and the modern algorithmic number theory, and their applications in computing and information technology, including computer systems design, cryptography and network security. In this second edition proofs of many theorems have been provided, further additions and corrections were made.
Kiselev's Geometry, Book 1: Planimetry
Introduction to Real Analysis (3rd Edition)
domain is a commutative ring with identity 1 -::/:- 0 that satisfies: a, b E R & ab = 0 ===? a = 0 or b = 0. (1.28) Definition 1.1.14. A division ring is a ring R with identity 1 -::/:- 0 that satisfies: for each a -::/:- 0 E R, the equation ax in R. = 1 and xa = 1 have solutions Definition 1.1.15. A field, denoted by /(, is a division ring with commutative multiplication. Example 1.1.8. The integer set Z, with the usual addition and multiplication, forms a commutative ring with identity,
Further, it can be shown that any rational number can be expressed as a finite simple continued fraction in exactly two ways, one with an odd number of terms and one with an even number of terms; we leave this as an exercise. D In what follows, we shall show that any irrational number can be expressed as an infinite simple continued fraction. Definition 1.2 .11. Let q0 , q1 , q2 , · · · be a sequence of integers, all positive except possibly q0 . Then the expression [q0 , q1 , q2 , · · ·] is
Power Residues ........................ 1. 7 Arithmetic of Elliptic Curves .......................... ....... 1. 7.1 Basic Concepts of Elliptic Curves ....................... 1.7.2 Geometric Composition Laws of Elliptic Curves .......... 1. 7.3 Algebraic Computation Laws for Elliptic Curves .......... 1.7.4 Group Laws on Elliptic Curves ......................... 1. 7.5 Number of Points on Elliptic Curves .................... 1.8 Bibliographic Notes and Further Reading ...................... 139
system of residues, then the two conditions are satisfied. To prove the converse, we note that if no two elements of S are congruent, the elements of S are in different residue classes modulo n. Since S has n elements, all the residue classes must be represented among the elements of S. Thus, Sis a complete system of residues modulo n D We now introduce one more type of systems of residues, the reduced systems of residues modulo n. Definition 1.6.8. Let [a]n be a residue class modulo ri. We say
exercise. (2) Similarly, ac = bd + bln + knd + kln 2 = bd + n(bl + k(d + ln)) = bd + n(bl + kc) = bd+ sn where s = bl + kc E Z. Thus, a· b c · d (mod n). (3) We prove Part (3) by induction. We have a b (mod n) (base step) and am bm (mod n) (inductive hypothesis). Then by Part (2) we have am+l aam bbm bm+l (mod n). D = = = = = = = Theorem 1.6.5 is equivalent to the following theorem, since a= b (mod n) a mod n b mod n {::::::::} {::::::::} {::::::::} a mod n = b mod n, [a]n, [b]n·
