Cryptography: Theory and Practice

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ICS 268: Cryptography and Communication Security 10/7/2004 Homework 2 Due Thursday , 10/14/2004 1 Prime Modular Arithmetics Let p be a prime. Recall groups Z p and Z * p from lecture and the handout on modular arithmetics. Recall the fact that Z * p is cyclic and has a generator (in fact it has many of them). Recall set QR p of squares modulo p . 1.1 Write out the elements of group Z 11 . How many elements are there? Pick an element g which is a generator of this group. For every element a Z 11 , write down DL ( g, 11) ( a ). 1.2 Let g be a generator of Z * p . Prove that if g x = g y mod p then x = y mod p - 1. 1.3 Let g be a generator of Z * p . Prove that y QR p if and only if x = DL ( g,p ) ( y ) is even. (Note that since g is a generator then for every y Z * p there is x = DL ( g,p ) ( y ).) Which of the elements of Z 11 are squares? 1.4 Prove that y QR p if and only if y ( p - 1) / 2 = 1 mod p . 1.5 Prove that if y, z are quadratic residues modulo p then so is yz mod p and y - 1 mod p . (In fact this holds for any p , not necessarily prime one.) 1.6 Recall that an order ord p ( a
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