# notes - A Primer on Algebra and Number Theory for Computer...

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A Primer on Algebra and Number Theory for Computer Scientists (version 0.1) Victor Shoup 1 September 6, 2002 1 Courant Institute, New York University, [email protected]

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Preface This is a very preliminary version of some notes that are intended to introduce computer science students to elementary notions of algebra and number theory, especially those notions that are relevant to the study of cryptography. These notes are meant to be quite self contained — the only prerequisites are some programming experience, a little knowledge of probabilities, and, most importantly, the ability to read and write mathematical proofs (or at least the willingness and ability to learn to do so). Many proofs of theorems that are relatively straightforward applications of definitions and other theorems are left as exercises for the reader. There are also many examples that the reader can verify as well. Other than these, there are no “formal” exercises in these notes. As stated above, these notes are in a very preliminary form. They certainly need more proof reading and polishing, and I may eventually want to add some new material as well. Also, there are currently no bibliographic references, which is a serious problem that I will have to correct very soon. If you find any typos or more serious problems, please let me know, so that I can correct the problem in a future revision. i
Remarks on Notation log( x ) denotes the natural logarithm of x . For any function f from a set A into a set B , if A 0 A , then f ( A 0 ) := { f ( a ) B : a A 0 } . For b B , f - 1 ( b ) := { a A : f ( a ) = b } , and more generally, for B 0 B , f - 1 ( B 0 ) := { a A : f ( a ) B 0 } . f is called one to one or injective if f ( a ) = f ( b ) implies a = b . f is called onto or surjective if f ( A ) = B . f is called bijective if it is both injective and surjective; in this case, f is called a bijection . Suppose f and g are functions from either the non-negative integers or non-negative reals into the non-negative reals. More generally, we may allow the domain of definition of f and g to consist all off integers or reals above some fixed bound. Then we write f = O ( g ) if there exist constants c > 0 and x 0 0 such that f ( x ) cg ( x ) for all x x 0 . We write f = Ω( g ) if g = O ( f ), and we write f = Θ( g ) if f = O ( g ) and g = O ( f ). We write f g if lim x →∞ f ( x ) /g ( x ) = 1, and we write f = o ( g ) if lim x →∞ f ( x ) /g ( x ) = 0. One also may write O ( g ) in an expression to denote an implicit function f such that f = O ( g ). For example, one may write ( n + 1)( n + 2) = n 2 + O ( n ). Similarly for Ω( g ), Θ( g ), and o ( g ). Note that f g is equivalent to f = g (1 + o (1)). ii

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Contents 1 Basic Properties of the Integers 1 1.1 Divisibility and Primality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Ideals and Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Finishing the Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Further Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Congruences 5 2.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Solving Linear Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Residue Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Computing with Large Integers 10 3.1 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Basic Integer Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Computing in Z n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Abelian Groups 19 4.1 Definitions, Basic Properties, and Some Examples . . . . . . . . . . . . . . . . . . . 19 4.2
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