UNIVERSITY OF MINNESOTA 
SCHOOL OF MATHEMATICS

Math 5248: Cryptology and number theory Homework

Homework Due date Problems
Homework 1 Wed Sept. 20 1.1 Shift cipher: 04, 12, 15
1.2 Reduction/division algorithm: 04, 13, 15
1.3 One-time pad: 04
1.5 Multiplicative inverses: 06, 09
1.6 The integers mod m: 10, 18, 20, 22
6.2 Euclidean algorithm: 03, 04, 09
6.3 Computing inverses: 04
Homework 2 Wed Oct. 11 1.7 The affine cipher: 11, 18, 21
3.1 Cryptograms- substitutions: 04
3.2 Anagrams- transposition: 09
4.2 LCM's and GCD's: 03, 07
4.1 The Vigenere cipher: 03, 05, 07, 09
4.4 Expected values: 01
4.5 Friedman attack: 02
5.3 Advanced encryption standard: 01 (not to be graded, but to think about)
Homework 3 Mon. Oct. 30 7.2 RSA cipher: 01, 04
7.3 Primitive roots, discrete logs: 04, 09, 13
7.5 ElGamal cipher: 01, 02, 03
9.1 Fermat's little theorem: 03, 05, 07
9.5 Exponentiation algorithm: 03, 05
23.1 Groups: 04, 07
23.2 Subgroups: 02, 05, 07
23.3 Lagrange's Theorem: 02
23.6 Finite cyclic groups: 04
Homework 4 Wed Nov.8 15.5 Primitive roots mod p: 04
10.1 Sun Ze's theorem: 01, 02
10.2 Special systems: 03
10.3 Composite moduli: 02, 03
9.6 Square roots mod p: 08
10.8 Euler's criterion: 02, 03, 04
Homework 5 Wed Nov. 29 13.1 Fermat pseudoprimes: 01, 08
13.6 Miller-Rabin test: 01, 08
18.1 Pollard's rho method: 01
18.2 Pollard's p-1 method: 02
Not to be handed in:
p.145 #10.4.01
p.212 # 16.6.01
p.186 zero knowledge proof. Let p=31 and q=11. If Vera chooses x=15, what does Vera send to Peter? Show Peter's calculation of x squared mod pq.