Cryptography Engineering: Design Principles and Practical Applications by Niels Ferguson & Bruce Schneier & Tadayoshi Kohno
Author:Niels Ferguson & Bruce Schneier & Tadayoshi Kohno
Language: eng
Format: epub, mobi
ISBN: 9781118080917
Published: 2014-11-03T00:00:00+00:00
The lemma is a statement of fact. The proof argues why the lemma is true. The little square box signals the end of the proof. Mathematicians love to use lots of symbols.1 This is a very simple lemma, and the proof should be easy to follow, as long as you remember what the notation a | b means.
Prime numbers have been studied by mathematicians throughout the ages. Even today, if you want to generate all primes below one million, you should use an algorithm developed just over 2000 years ago by Eratosthenes, a friend of Archimedes. (Eratosthenes was also the first person to accurately measure the diameter of the earth. A mere 1700 years later Columbus allegedly used a much smaller—and wrong—estimate for the size of the earth when he planned to sail to India by going due west.) Euclid, another great Greek mathematician, gave a beautiful proof that showed there are an infinite number of primes. This is such a beautiful proof that we'll include it here. Reading through it will help you reacquaint yourself with the math.
Before we start with the real proof we will give a simple lemma.
Lemma 2 Let n be a positive number greater than 1. Let d be the smallest divisor of n that is greater than 1. Then d is prime.
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