An Introduction to Cryptology
and Coding Theory
Sarah Spence Adams
Digital Source Digital Sink
Error Control Error Control
Modulation Channel Demodulation
Inventing cipher systems; protecting
communications and storage
Breaking cipher systems
What is used in Cryptology?
Linear algebra, abstract algebra, number
Probability, statistics, combinatorics,
Key = 3
Encryption: Shift by KEY = 3
Decryption: Shift backwards by KEY = 3
Cryptanalysis of Caesar
Try all 26 possible shifts
Permute A-Z randomly:
A B C D E F G H I J K L M N O P… becomes
H Q A W I N F T E B X S F O P C…
Substitute H for A, Q for B, etc.
Cryptanalysis of Substitution Ciphers
Try all 26! permutations – TOO MANY!
Bigger than Avogadro's Number!
Map A, B, C, … Z to 0, 1, 2, …25
A B… M N … T U
0 1 … 13 14 … 20 21
Encryption: “Add” key to message mod 26
Decryption: “Subtract” key from ciphertext
Problem: Exchanging the key
There are some clever ways to
exchange the key – we will study some
Diffie & Hellman (1976)
Known at GCHQ years before
Uses one-way (asymmetric) functions,
public keys, and private keys
Public Key Algorithms
Based on two hard problems
Factoring large integers
The discrete logarithm problem
Need more than secrecy….
Enter coding theory…..
What is Coding Theory?
Coding theory is the study of error-
Error control codes are used to detect
and correct errors that occur when
data are transferred or stored
What IS Coding Theory?
A mix of mathematics, computer science,
electrical engineering, telecommunications
Abstract algebra (groups, rings, fields)
We want to send data from one place to
channels: telephone lines, internet cables, fiber-optic
lines, microwave radio channels, cell phone
or we want to write and later retrieve data…
channels: hard drives, disks, CD-ROMs, DVDs, solid
state memory, etc.
BUT! the data, or signals, may be corrupted
additive noise, attenuation, interference, jamming,
hardware malfunction, etc.
Add controlled redundancy to the
message to improve the chances of
being able to recover the original
Trivial example: The telephone game
The ISBN Code
x1 x2… x10
x10 is a check digit chosen so that
S = x1 + 2x2 + … + 9x9 + 10x10 = 0 mod 11
Can detect all single and all transposition
Cryptology by Thomas Barr: 0-13-088976-?
Want 1(0) + 2(1) + 3(3) + 4(0) + 5(8) + 6(8) +
7(9) + 8(7) + 9(6) + 10(?) = multiple of 11
Compute 1(0) + 2(1) + 3(3) + 4(0) + 5(8) + 6(8)
+ 7(9) + 8(7) + 9(6) = 272
Ponder 272 + 10(?) = multiple of 11
Modular arithmetic shows that the check digit
UPC (Universal Product Code)
x1 x2… x12
x12 is a check digit chosen so that
S = 3x1 + 1x2 + … + 3x11 + 1x12 = 0 mod
Can detect all single and most
What transposition errors go
The Repetition Code
Send 0 and 1
Noise may change 0 to 1 or change 1 to 0
Instead, send codewords 00000 and 11111
If noise corrupts up to 2 bits, decoder can
use majority vote and decode received word
The Repetition Code
The distance between the two
codewords is 5, because they differ in
Large distance between codewords is
The “rate” of the code is 1/5, since for
every bit of information, we need to
send 5 coded bits
High rate is good!
When is a Code “Good”?
Important Code Parameters (n, M, d)
Number of codewords (M)
Minimum Hamming distance (d): Directly
related to probability of decoding correctly
Code rate: Ratio of information bits to
How Good Does It Get?
What are the ideal trade-offs between rate,
error-correcting capability, and number of
What is the biggest distance you can get
given a fixed rate or fixed number of
What is the best rate you can get given a
fixed distance or fixed number of
1969 Mariner Mission
We’ll learn how Hadamard matrices
were used on the 1969 Mariner Mission
to build a rate 6/32 code that is
approximately 100,000x better at
correcting errors than the binary
repetition code of length 5
1980-90’s Voyager Missions
Better pictures need better codes need more
Picture transmitted via Reed-Solomon codes
From Caesar to Public-Key…. from Repetition
Codes to Reed-Solomon Codes….
More sophisticated mathematics better
Cryptology and coding theory involve abstract
algebra, finite fields, rings, groups, probability,
linear algebra, number theory, and additional
You and me!
Shopping and e-commerce
ATMs and online banking
Satellite TV & Radio, Cable TV, CD
NSA, IDA, RSA, Aerospace, Bell Labs,
AT&T, NASA, Lucent, Amazon, iTunes…