Homework23 - (p-1). 6) A Carmichael number n is a composite...

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Introduction to Cryptography Summer ’04 Homework #2, 3 Assigned: Tuesday, 6/29/04 Due: Monday, 7/12/04 1) Write a Java program (using the BigInteger class) to find the first five prime numbers greater than 10 30 . Use the Miller-Rabin primality test instead of trial division or the BigInteger method isProbablePrime(). Include your code with the assignment you turn in. 2) Compute φ (187072301). (Hint: All prime factors of this value are less than 200.) 3) Find all integer solutions x for the following system of equations: 12x 37 mod 91 5x 62 mod 72 11x 3 mod 25 4) Find all the primitive roots of 211. For each of these primitive roots, g, solve the equation 203 = g x mod 211 for x. Attach the code you use to do this. 5) Let p be a prime with a primitive root g. Then any element y Z p * can be written as y=g i , where 0 i < p-1, in an unique way. Prove that a) The order of y=g i , is ) , 1 gcd( 1 i p p - - . b) The number of primitive elements modulo p is
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Unformatted text preview: (p-1). 6) A Carmichael number n is a composite number such that for all values of b with gcd(b,n) = 1, b n-1 1 (mod n). Prove: If n=p 1 p 2 ...p k where each p i for 1 i k is a distinct prime that satisfies (p i- 1) | (n -1) for all i, then n is a Carmichael number. 7) We proved that for RSA encryption, where e(x) = x e mod n and d(y) = y d mod n, that d(e(x)) = x if gcd(n,x)=1. Now prove that this statement is true for all x, even if gcd(n,x)&gt;1. 8) Supposed three users in a network, say Angie, Bob and Carmen, all have public encryption exponents e = 3. Let their moduli be denoted by n A , n B , and n C . Now suppose David encrypts the same plaintext x to send to Angie, Bob and Carmen. That is, Alice computes y A = x 3 mod n A , y B = x 3 mod n B , and y C = x 3 mod n C . Describe how Oscar can compute x, given y A , y B and y C . 9) Do 9.10 in the textbook (page 281)....
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