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(BQ) Part 2 book "Cryptography engineering" has contents: Primes, Diffie-Hellman, introduction to cryptographic protocols, key negotiation, implementation issue, the clock, key servers, the dream of PKi, storing secrets, storing secrets,. and other contents. | HAPTER 10 Primes The following two chapters explain public-key cryptographic systems. This requires some mathematics to get started. It is always tempting to dispense with the understanding and only present the formulas and equations, but we feel very strongly that this is a dangerous thing to do. To use a tool, you should understand the properties of that tool. This is easy with something like a hash function. We have an "ideal" model of a hash function, and we desire the actual hash function to behave like the ideal model. This is not so easy to do with public-key systems because there are no "ideal" models to work with. In practice, you have to deal with the mathematical properties of the public-key systems, and to do that safely you must understand these properties. There is no shortcut here; you must understand the mathematics. Fortunately, the only background knowledge required is high school math. This chapter is about prime numbers. Prime numbers play an important role in mathematics, but we are interested in them because some of the most important public-key crypto systems are based on prime numbers. 1 0.1 Divisibility and Primes A number a is a divisor of b (notation a I b, pronounced "a divides b") if you can divide b by a without leaving a remainder. For example, 7 is a divisor of 35 so we write 7 I 35. We call a number a prime number if it has exactly two positive divisors, namely 1 and itself. For example, 13 is a prime; the two 1 63 1 64 Part III • Key Negotiation 1 and 13. The first few primes are easy to find: 2, 3, 5, 7, 1 1, 13, . . Any integer greater than 1 that is not prime is called a composite. The number divisors are . . 1 is neither prime nor composite. We will use the proper mathematical notation and terminology in the chapters ahead. This will make it much easier to read other texts on this subject. The notation might look difficult and complicated at first, but this part of mathematics is really easy. Here is a simple .
