Due Tuesday October, 3 2000 in class at 14:30 SHARP
Exercises (from Stinsonís book)
4.3 Solve the following system of congruences:
13x = 4 (mod 99)
15x = 56 (mod 101).
HINT:
First use the Extended Euclidean algorithm, and then apply the Chinese remainder theorem.
Exercises (from Brassard-Bratley's book)
8.5.13 Let p = 1 (mod 4) be a prime, and let x be in QR_{p}.
An integer a, 0<a<p, gives the key to x^{1/2} if (a^{2}-x) mod p is in QNR_{p}.
Prove that
Exercises (from Crépeau's brain)
1. Assume that the prime number theorem is exactly correct for all powers of two. Calculate a good upper bound on the probability that we mistakenly output a composite number instead of a prime after the following events have occurred:
ï pick a random m-bit odd integer n
ï the procedure Rabin-Miller prime(n,k) returns ëprimeí
Express your bound as a function of m and k.
If I want a random 512 bits prime p, what value of k should be used in Rabin-Miller prime(p,k) to guarantee probability at most 1/2^{20} of outputting a composite number?
2. Let B=2^{k} be an upper limit on a set of candidate primes. Show that their exists a set of integers A={a_{i}}, such that #A is in O(k) such that any n, 1<n<B, is prime if and only if pseudo(a_{i},n)="pseudo" for all a_{i} in A.
WARNING: the following two problems are surprisingly different !!!
3. Let p be a odd prime number. We can define a notion of fourth residues mod p similar to quadratic residues mod p, by replacing square roots by fourth roots.
ï What are the fourth residues mod 7?
ï What are the fourth residues mod 17?
ï Find an efficient algorithm to decide whether a value a is a fourth residue mod p or not.
ï Find an efficient algorithm to extract the fourth root mod p of a fourth residue a.
WARNING: the following problem contains some fairly difficult cases!!!
4. Let p be a prime number greater than 3. We can define a notion of cubic residues mod p similar to quadratic residues mod p, by replacing square roots by cube roots.
ï What are the cubic residues mod 7?
ï What are the cubic residues mod 17?
ï Find an efficient algorithm to decide whether a value a is a cubic residue mod p or not.
ï Find an efficient algorithm to extract the cube root mod p of a cubic residue a.
HINT: use Fermatís little theorem.
5. Consider the number n=10403.
Show all your calculations.