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Unformatted text preview: Exam 2 Review Sheet Solutions Math 4023 The second exam will be on Friday, October 28, 2011. It will cover Sections 0.7, 0.8, 3.1, 3.2, 3.4 (except 3.4.1), 4.1 and 4.2 plus the handout on calculation of high powers of an integer modulo n via successive squaring. Of course, the material is cumulative, and the listed sections depend on earlier sections, which it is assumed that you still know. Following are some of the concepts and results you should know: ² Know the meaning of the basic concepts: ring, fleld, characteristic of a ring, the ring of polynomials R [ x ]. ² The characteristic of a fleld is either 0 or a prime. ² The number of elements in a flnite fleld is p r , where p is a prime (equal to the characteristic of F ) and r is a positive integer. ² Know the division algorithm for polynomials (Theorem 0.8.2). ² Know how to use successive division (the Euclidean algorithm) to flnd the greatest common divisor of two polynomials with coe–cients in a fleld. ² Know the meaning of reducible and irreducible for polynomials. ² If F is a fleld, and p ( x ) is a polynomial of degree 2 or 3, then p ( x ) is irreducible if and only if p ( x ) has no roots in F . This is not true if deg( p ( x )) ‚ 4. ² Know how to use congruence arithmetic to make the set of congruence classes K = F [ x ] = ( p ( x )) into a fleld when the polynomial p ( x ) is irreducible. (Theorem 0.8.4) ² Know how to use an irreducible polynomial of degree r over the fleld Z p for a prime p to construct a fleld with p r elements. (Theorem 0.8.6) ² If F is a flnite fleld, the multiplicative group F / of nonzero elements of F is a cyclic group. (Theorem 0.8.8) ² An element a of a flnite fleld F is a primitive element of F is a is a generator. An irreducible polynomial p ( x ) 2 Z p [ x ] is primitive if x = x +( p ( x )) = the congruence class of x in the fleld Z p [ x ] = ( p ( x )), is a primitive element of Z p [ x ] = ( p ( x )). ² Know the Caesar cipher, a–ne cipher, and the Hill cipher. Know how to encipher and decipher with each. ² Know how the RSA Cryptosystem is deflned. Know the relationship between the public and private keys. ² Know how to compute powers of integers modulo n using Euler’s theorem and successive squaring....
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- Fall '09