An Introduction to Cryptology and Coding Theory

Sarah Spence Adams

Olin College

sarah.adams@olin.edu

Gordon Prichett

Babson College

prichett@babson.edu

Communication System

Digital Source Digital Sink

Source Encoding

Source Decoding

Encryption Decryption

Error Control Encoding

Error Control 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

One-Time Pads

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

One-Time Pads

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…