Secret Codes, CD players, and Missions to Mars: An Introduction to Cryptology and Coding Theory Sarah Spence Adams Associate Professor of Mathematics Franklin.

Secret Codes, CD players, and Missions to Mars:An Introduction to Cryptology and Coding Theory

Sarah Spence AdamsAssociate Professor of Mathematics

Franklin W. Olin College of Engineering

AKA

• Sit back, relax, and learn something new while your combinatorics knowledge marinates for the test

Cryptology Cryptography

Inventing cipher systems

CryptanalysisBreaking cipher systems

BobAlice

Eve

Hidden Messages Through the Ages

440 BC – The Histories of Herodotus messages concealed beneath wax on wooden tablets tattoos on a slave's head concealed by regrown hair

WWII – microdots

Modern day – hidden information within digitial pictures

The Scytale of Ancient Greece

Used by the Spartan military

5th Century B.C.

Caesar’s Substitution Cipher

Example Plaintext: OLINCOLLEGE Encryption: Shift forward by 3 Ciphertext: ROLQFROOHJH Decryption: Shift backwards by 3 (or forwards by 23)

= 3

1st Century B.C.

Modular Arithmetic

15 mod 12 = 3 27 mod 26 = 1 90 mod 10 = 0

a mod m is the remainder obtained upon dividing a by m

Can be obtained by subtracting off multiples of m from a until the result is between 0 and m-1

Caesar Using Modular Arithmetic Map A,B,C,…Z to 0,1,2,…,25

View cipher as function e(pi) = pi + k (mod 26)

Example with key = k = 15 Plaintext: OLINCOLLEGE=14,11,8,13,2,14,11,11,4,6,4

Encryption: e(pi) = pi + 15 (mod 26)

Ciphertext:

3,0,23,2,17,3,0,0,19,21,19 = DAXCRDAATVT

Decryption: d(ci) = ci -15 (mod 26) or

d(ci) = ci +11 (mod 26)p

Kerckhoff’s Principle

Must assume the enemy knows the system

Also known as Shannon’s Maxim

Cryptanalysis of Substitution Ciphers

Caesar’s 26 shifts to test

Generally 26! permutations to test

If parsed, guess at common words or cribs

Frequency analysis

Frequency Analysis (Al-Kindi ~850A.D.)

www.wikipedia.com

The Letter E

Vigenère Cipher (1586)

Polyalphabetic cipher

Involves multiple Caesar shifts

Example Plaintext: O L I N C O L L E G EKey: S U N S U N S U N S UEncryption: e(pi) = pi + ki (mod 26)Decryption: d(ci) = ci - ki (mod 26)

One-Time Pads: The Ultimate Substitution Cipher

Plaintext: MATHISUSEFULANDFUN

Key: NGUJKAMOCTLNYBCIAZ

Encryption: e(pi) = pi + ki (mod 26)

Ciphertext: BGO…..

Decryption: d(ci) = ci - ki (mod 26)

One-Time Pads

Unconditionally secure

Problem: Exchanging the key

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 (traditionally)Factoring large integers (RSA)The discrete logarithm problem (ElGamal)

WWII: The Weather-Beaten Enigma 3 x 10114 ciphering possibilities;

polyalphabetic

Destroyed frequency counts

Cracked thanks to traitors, captured machines, mathematicians, and human error

Not only do you want secrecy…

…but you also want reliability!

Enter coding theory…..

Communication System

Digital Source Digital Sink

Source Encoding

Source Decoding

Encryption Decryption

Error Control Encoding

Error Control Decoding

Modulation Channel Demodulation

What is Coding Theory?

Coding theory is the study of error-control codes, which 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, telecommunicationsLinear algebraAbstract algebra (groups, rings, fields)Probability&StatisticsSignals&Systems Implementation issuesOptimization issuesPerformance issues

General Problem

We want to send data from one place to another over some channel 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 may be corrupted hardware malfunction, atmospheric disturbances,

attenuation, interference, jamming, etc.

General Solution

Introduce controlled redundancy to the message to improve the chances of recovering the original message

Trivial example: The telephone game

Introductory Example

We want to communicate YES or NO

Message 1 represents YES

Message 0 represents NO

If I send my message and there is an error, then you will decode incorrectly

Repetition Code of Length 2

Encode message 1 as codeword 11 Encode message 0 as codeword 00

If one error occurs, you can detect that something went wrong

Repetition Code of Length 5

Encode message 1 as codeword 11111 Encode message 0 as codeword 00000

If up to two errors occur, you can correct them using a majority vote

Evaluating and Comparing Codes

Important Code Parameters Minimum distance

Determines error-control capabilityCode rate

Ratio of information bits to codeword bits Measure of efficiency

Length of codewordsNumber of codewords

Complicated Problem

WantLarge minimum distance for reliabilityLarge number of codewords High rate for efficiency

Conflicting goals Require trade-offs Inspire more sophisticated mathematical

solutions

Inherent Trade-offs

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?

The ISBN-10 Code

x1 x2… x10

x10 is a check digit chosen so that

S = x1 + 2x2 + … + 9x9 + 10x10 = 0 mod 11

If check digit should be 10, use X instead

Can detect all single and all transposition errors

ISBN-10 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

The check digit must be 8

New ISBN-13

x1 x2… x13

x13 = 10-((1x1 + 3x2 + … + 1x11 + 3x12) mod 10)

If check digit should be 10, use 0 instead

Convert ISBN-10 to ISBN-13 using 978 prefix and new check digit

Universal Product Code (UPC)

Universal Product Code (UPC)

First number: Type of product 0, 6, 7: Standard 2: Random-weight items (fruits, meats, etc) 4: In-store 5: Coupons

Next chunk: Manufacturer identification Next chunk: Item identification Last number: Check digit

UPC Check Digit

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

Which transposition errors go undetected?

Hadamard Code

Used in NASA Mariner Missions

Pictures divided into pixels

Pixels are assigned a level a darkness on a scale of 0 to 63

Binary Representation of Messages

Express 64 levels of darkness (our messages) using binary strings

0 0000001 0000012 0000103 000011….63 111111

Hadamard Matrices Map length 6 messages to length 32

codewords obtained from rows of Hadamard matrices

Hadamard matrices (Sylvester, 1867) have special properties that give these codewords a minimum distance of 16

HHT = n I

Compare with Length 5 Repetition Code

Send a message of length 6

Probability of bit error p= .01

Length 5 Repetition Code Sending 6 info bits requires 30 coded bits: Rate = 6/30 P(decode incorrectly) = .00006

Hadamard Code Sending 6 info bits requires 32 coded bits: Rate = 6/32 P(decode incorrectly) = .0000000008

Voyager Missions (1980’s-90’s) Reed-Solomon codes use abstract algebra to get

even better results

Ideals of polynomial rings (Dedekind,1876)

Same codes used to protect CDs from scratches!

Summary Cryptology and coding theory are all around us

From Caesar to RSA…. from Repetition to Reed-Solomon Codes…. More sophisticated mathematics better ciphers/codes

New uses for old mathematics, motivation for new mathematics

Cryptology has existed for thousands of years… what ciphers and codes will be next?

Jefferson Disk - Bazeries Cylinder

Alice rotates wheels to spell

message in one row sends any other row of text

Used by US Army from early 1920’s to early 1940’s

Alice and Bob agree on order of disks – the key

Bob spells out the ciphertext on wheel looks around the rows until he sees the (coherent)

message

The Code Talkers Refers primarily to Navajo speakers in WWII

Only unbroken wartime cipher

Fast and efficient

Interesting connections between cracking secret ciphers and cracking ancient languages

Major Questions 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?

A Parity Check Code

Suppose we want to send 4 messages 00, 01, 10, 11

Form codewords by appending a parity check bit to the end of each message 00 000 01 011 10 101 11 110

Compare with length 2 repetition code Both detect all single errors using minimum distance 2 Now rate 2/3 compared with rate 1/2 Now 4 codewords compared with 2 codewords