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Unformatted text preview: variste Galois was a French mathematician who lived in the early nineteenth century and did a lot of work in number theory before he was killed at age 20 in a duel.) In a Galois field, addition, subtraction, multiplication, and division by nonzero elements are all welldefined. There is an additive identity, 0, and a multiplicative identity, 1. Every nonzero number has a unique inverse (this would not be true if p were not prime). The commutative, associative, and distributive laws are true. Arithmetic in a Galois field is used a great deal in cryptography. All of the number theory works; it keeps numbers a finite size, and division doesn’t have any rounding errors. Many cryptosystems are based on GF(p), where p is a large prime. To make matters even more complicated, cryptographers also use arithmetic modulo irreducible polynomials of degree n whose coefficients are integers modulo q, where q is prime. These fields are called GF(qn). All arithmetic is done modulo p (x), where p (x) is an irreducible polynomial of degree n. The mathematical theory behind this is far beyond the scope of the book, although I will describe some cryptosystems that use it. If you want to try to work more with this, GF(23) has the following elements: 0, 1, x, x + 1, x2, x2 + 1, x2 + x, x2 + x + 1. There is an algorithm for computing inverses in GF(2n) that is suitable for parallel implementation [421]. When talking about polynomials, the term “prime” is replaced by “irreducible.” A polynomial is irreducible if it cannot be expressed as the product of two other polynomials (except for 1 and itself, of course). The polynomial x2 + 1 is irreducible over the integers. The polynomial x3 + 2x2 + x is not; it can be expressed as x (x + 1)(x + 1). A polynomial that is a generator in a given field is called primitive; all its coefficients are relatively prime. We’ll see primitive polynomials again when we talk about linearfeedback shift registers (see Section 16.2). Computation in GF(2n) can be quickly implemented in hardware with linearfeedback shift registers. For that reason, c...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
 Fall '10
 ALIULGER
 Cryptography

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