A Concrete Introduction to Higher Algebra, 2nd Edition

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Unformatted text preview: Columbia University Handout 2 E6291: Topics in Cryptography January 29, 1999 Professor Luca Trevisan Notes on Algebra This section contains as little theory as possible, and most results are stated without proof. Any introductory book on algebra will contain proofs and put the results in a more general, and more beautiful framework. For example, a book by Childs [C95] covers all the required material without getting too abstract. It also points out the cryptographic applications. 1 Prime Numbers By integer , we mean a positive or negative integer. We denote by Z the set Z = { . . . , 3 , 2 , 1 , , 1 , 2 , 3 , . . . } . A natural number is a non-negative integer. We denote by N the set N = { , 1 , 2 , 3 . . . } . We also denote by Z + the set Z + = { 1 , 2 , 3 , . . . } of positive integers. For an integer n , we denote by || n || the length of n , i.e. the number of bits needed to represent it, i.e. || n || = d log 2 n e . Logarithms will always be to the base 2, so we will omit the base hereafter. We will denote by ln n the natural logarithm of n , i.e. the logarithm taken to the base e = 2 . 71828 . . . For integers k, n , we say that k divides n (or that k is divisor of n ) if n is a multiple of k . For example 5 divides 35. We write k | n when k divides n . A prime number is a positive integer p 2 whose only divisors are 1 and p . Notice that 2 is the only even prime number. The first few prime numbers are 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , . . . . When a number is not prime, it is called composite . A composite can always be written (in a unique way) as a product of primes, possibly with repetitions. E.g. 300 = 2 2 5 5. There are infinitely many prime numbers (which is very easy to prove), and in fact there are quite a lot of them (which is harder to prove). Specifically, if we define ( n ) to be the number of prime numbers p such that 2 p n , then ( n ) is about n/ ln n . Formally Theorem 1.1 (Prime Numbers Theorem) lim n ( n ) n/ ln n = 1 The following bounds are also known ( n ) n ln n and, for n 17, 2 Handout 2: Notes on Algebra ( n ) 1 . 10555 n ln n There is an ecient randomized algorithm that on input an integer tests whether it is prime or not. Therefore if we want to generate a large prime (in the interval from 1 to n , where n can be thought of as a number around 10 200 ) we can just pick a random number in the set { 1 , . . . , n = 10 200 } and then test whether it is prime. If it is not, we try again. Each time we have a probability 1 / ln n 1 / 460 of succeeding, so we expect to succeed after less than 500 attempts. Big prime numbers are very important in applied cryptography, and the Prime Number Theorem is a very useful tools to analyze certain cryptographic protocols....
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This note was uploaded on 02/04/2008 for the course CS 276 taught by Professor Trevisan during the Spring '02 term at University of California, Berkeley.

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notes algebra - Columbia University Handout 2 E6291: Topics...

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