# An Introduction to Cryptology and Coding Theory

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```					An Introduction to Cryptology
and Coding Theory

Olin College

Gordon Prichett
Babson College
prichett@babson.edu
Communication System

Digital Source             Digital Sink
Source                     Source
Encoding                   Decoding
Encryption                 Decryption
Error Control              Error Control
Encoding                   Decoding
Modulation       Channel   Demodulation
Cryptology

   Cryptography
   Inventing cipher systems; protecting
communications and storage

   Cryptanalysis
   Breaking cipher systems
Cryptography
Cryptanalysis
What is used in Cryptology?

   Cryptography:
   Linear algebra, abstract algebra, number
theory
   Cryptanalysis:
   Probability, statistics, combinatorics,
computing
Caesar Cipher
   ABCDEFGHIJKLMNOPQRSTUVWXYZ
   Key = 3
   DEFGHIJKLMNOPQRSTUVWXYZABC

   Example
   Plaintext:    OLINCOLLEGE
   Encryption:   Shift by KEY = 3
   Ciphertext:   ROLQFROOHJH
   Decryption:   Shift backwards by KEY = 3
Cryptanalysis of Caesar

   Try all 26 possible shifts

   Frequency analysis
Substitution Cipher

 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.

 Example
   Plaintext: OLINCOLLEGE
   Key:       PSEOAPSSIFI
Cryptanalysis of Substitution Ciphers

   Try all 26! permutations – TOO MANY!
Bigger than Avogadro's Number!
   Frequency analysis
   Map A, B, C, … Z to 0, 1, 2, …25
   A B… M N … T U
   0 1 … 13 14 … 20 21
   Plaintext:   MATHISUSEFULANDFUN
   Key:         NGUJKAMOCTLNYBCIAZ
   Encryption: “Add” key to message mod 26
   Ciphertext: BGO…..
   Decryption: “Subtract” key from ciphertext
mod 26
Modular Arithmetic

   Unconditionally secure

   Problem: Exchanging the key

   There are some clever ways to
exchange the key – we will study some
of them!
Public-Key Cryptography

   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
WWII Folly:
The Weather-
Beaten Enigma
Need more than secrecy….

   Need reliability!

   Enter coding theory…..
What is Coding Theory?

   Coding theory is the study of error-
control codes
   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
   Linear algebra
   Abstract algebra (groups, rings, fields)
   Probability&Statistics
   Signals&Systems
   Implementation issues
   Optimization issues
   Performance issues
General Problem
   We want to send data from one place to
another…
   channels: telephone lines, internet cables, fiber-optic
lines, microwave radio channels, cell phone
channels, etc.

   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.
General Solution

   Add controlled redundancy to the
message to improve the chances of
being able to recover the original
message

   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
errors
ISBN Example

   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
is 8!!
UPC (Universal Product Code)

   x1 x2… x12
 x12 is a check digit chosen so that
S = 3x1 + 1x2 + … + 3x11 + 1x12 = 0 mod
10
 Can detect all single and most
transposition errors
 What transposition errors go
undetected?
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
as 00000
The Repetition Code

   The distance between the two
codewords is 5, because they differ in
5 spots
   Large distance between codewords is
good!

   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)
   Length (n)
   Number of codewords (M)
   Minimum Hamming distance (d): Directly
related to probability of decoding correctly
   Code rate: Ratio of information bits to
codeword bits
How Good Does It Get?

   What are the ideal trade-offs between rate,
error-correcting capability, and number of
codewords?

   What is the biggest distance you can get
given a fixed rate or fixed number of
codewords?

   What is the best rate you can get given a
fixed distance or fixed number of
codewords?
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
sophisticated mathematics…

   Picture transmitted via Reed-Solomon codes
Summary
   From Caesar to Public-Key…. from Repetition
Codes to Reed-Solomon Codes….
   More sophisticated mathematics  better
ciphers/codes

   Cryptology and coding theory involve abstract
algebra, finite fields, rings, groups, probability,
linear algebra, number theory, and additional
exciting mathematics!
Who Cares?
   You and me!
   Shopping and e-commerce
   ATMs and online banking
   Satellite TV & Radio, Cable TV, CD
players
   Corporate/government espionage
   Who else?
   NSA, IDA, RSA, Aerospace, Bell Labs,
AT&T, NASA, Lucent, Amazon, iTunes…

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