Number Theory in Cryptography and its Applications
CIMPA-UNESCO-MICINN, NEPAL
Dhulikhel, Nepal, July 19-31.
Assignments.
The following Exercises should be done by hand
1. Use the fact that X = 19 is a solution of X 2 ≡ 4 mod 119 to factor 119.
2. Show that n = 33 is not prime without attempting to factor it nor mentioning any of
its factors.
3. Legendre Symbols.
a) Compute all quadratic residues in Z/31Z
37
b) Compute 101
.
4. GCD. Compute the GCD of (2345, 1085) first with the binary algorithm and then with
the Extended Euclidean Algorithm. In the latter case also write the Bezout identity.
5. Roots of Polynomials. Find all solutions of X 3 + X + 2 ≡ 0(mod20) by computing first all solutions modulo 4 and 5, then using the Chinese Remainder Theorem
to compute the solutions modulo 20.
The following Exercises should be done with Mathematica
1. RSA. Produce two (random) primes with 15 decimal digits that are suitable to produce
an RSA module with encryption exponent e = 3
ˆ Compute the decription exponent
ˆ Encode 2067
ˆ Decode 5425254974713583142451954123;
2. List all primitive roots of (Z/2069Z)∗ and those of (Z/202Z)∗ ;
3. DL. Choose a random prime p with 30 decimal digits, compute a primitive roots
(Z/pZ)∗ and
a) Generate a key using the Diffie–Hellmann method
b) Use the Massey Omura Cryptosystem to encode and decode 2067
c) Use the ElGamal system to encode and decode 2067
√
4. Compute the roots k 2067 by Newton’s method (with 10 iterations), for k = 2, . . . , 10
5. Find all solutions of X 2 ≡ 6(mod22067 + 2949) and X 2 ≡ 10(mod22067 + 2949).