COMP-547A 2009 Homework set #1

COMP-547A 2009 Homework set #1Due Thursday September, 24 2009 in class at 13:00Send to cs547(at)cs(dot)mcgill(dot)ca all your Maple code before the deadline.A. THEORY: Consider an expression of the form0 ∫ a x 2 + b x + c (mod n)Show that the solutions x to this system are of the form. .x ∫ ( -b + √b 2 – 4ac )(2a) -1 (mod n). .where √b 2 – 4ac is a square root of a quadratic residue b 2 – 4ac and 2a is invertiblemodulo n.B. MAPLE: Solve the following systems of congruences.Give all the solutions. If there are no solutions, explain why.( Do not take advantage of the fact that the modulus is small and tryall possibilities; a proper reasoning is necessary in each case.However this may be useful to validate your answers… )320 ∫ 335 x + 3340 (mod 1305)320 ∫ 335 x + 3340 (mod 1307)0 ∫ 335 x 2 + 3340 x – 320 (mod 1307)0 ∫ 335 x 2 + 3340 x – 320 (mod 1305)C. THEORY: Probability ProblemCalculate a good upper bound on the probability that we mistakenly output a compositenumber instead of a prime after the following events have occurred:• pick a random m-bit integer n such that gcd(n,210)=1• the procedure Rabin-Miller prime(n,k) returns ‘prime’1) Express your bound as a function of m and k.(Assume that the prime number theorem is exactly correct for all powers of two.)2) If I want a random 2048 bits prime p, what value of k should be used in Rabin-Millerprime(p,k) to guarantee probability at most 1/2 50 of outputting a composite number?…more on back…

D. MAPLE: AUTHENTICATION CODES over finite fieldsMOVED TO HW 2.E. THEORY: from Brassard-Bratley's book8.5.13 Let p ∫1 (mod 4) be a prime, and let x be in QR p .An integer a, 0