Islander Math Circles
Saturday, April 27, 2013
Cryptography
1. The Die Hard Jug Problem: Fill a jug with exactly 4 gallons of water using only a 3
gallon and a 5 gallon jug. Why? Solving the problem will defuse a bomb!
Discussion of the mathematics leading to general question:
Given positive integers a and b, for which integers c can you find integers x and y
with
c = ax + by
2. Modular Arithmetic:
 With a fixed number n, take the set {0, 1, 2, … , n-1}, the remainders (mod n)
 Given any integer, we can subtract a multiple of n to get one of the numbers
in the set.
 Stated another way, given any integer, we can subtract one of those to get a
multiple of n.
 We can do addition and multiplication in meaningful ways with the those
remainders.
A couple of problems:
 Are there any perfect squares in the sequence 3, 8, 13, 18, …?
 Are there any integers x, y satisfying x2 – 7y2 – 24 = 0?
3. Cyptography: The shift-cypher
The letters A to Z are labled 0 to 25, the remainders (mod 26). We could have more
characters, including spaces. Note that simply writing letters as numbers is
encoding but not encryption.
We will do encryption using a shift-cypher. Calculations done on the encoded
remainders, c, are of the form E(c) = a*c + b (mod 26). E(c) is called the encryption
of c. Can E(c) be unencrypted? Can you find d, e with U(c) = E(c) *d + e with
U(E(c))) = c?
See the worksheet on the next page.
 Write a statement of at least 10 characters
 Encode it using the table at the top of the page
 Encrypt each encoding using the given formulas based on a, b.
 Create encrypted text using the coding to go back from numbers to letters.
 Can you unencrypt your numbers? Can you find d, e as above?
For discussion, what are the essential criteria for a good encryption?
A
0
B
1
C
2
D
3
E
4
F
5
G
6
H
7
I
8
J
9
K
10
L
11
Parameters for encryption: a = _______ and b = _________
Clear text
Coded
Numbers
Encrypted
numbers
Encrypted
Text
Decrypted
Numbers
M
12
N
13
O
14
P
15
Q
16
R
17
S
18
T
19
U
20
V
21
W
22
X
23
Y
24
Z
25
Parameters for decryption: d= __________, e = ____________
4. Finding multiplicative inverses (mod n).
 The remainder b is the multiplicative inverse of a means a*b = 1 (mod n)
 As an equation of integers, this means a*b = 1 + a multiple of n, or
 Given a, n, can you solve the equation a*b + n*y = 1 for b, y
 Possible problems that can be solved:
o The jug problem, but which ones?
o Can you use a 4 gallon and a 6 gallon jug to get exactly 5 gallons?
o I have nickels and quarters adding to 65 cents. How many of each can
I have?
o If you have one solution, can you find others?
5. Cancellation of remainders (mod n)
 For which numbers n (and a,b,c) can you guarantee that ac = bc (mod n)
implies a = b (mod n)
6. Fermat’s Little Theorem
 What is it?
 Prove that 22225555+55552222 is a multiple of 7
7. RSA Coding
 Suppose p and q are prime numbers and e is relatively prime to (p-1)(q-1).
 The numbers pq (the product, not factors) and e are publicly known.
 Then E(x) = xe (mod pq) is an encryption of the integers relatively prime to
pq.