Cryptography

What is cryptography?

kryptos – “hidden” grafo – “write”

Keeping messages secretUsually by making the message unintelligible

to anyone that intercepts it

The Problem

Bob Alice

Eve

Private Message

Eavesdropping

The Solution

Bob Alice

Eve

Scrambled Message

Eavesdropping

Encryption Decryption

Private Message Private Message

What do we need?

Bob and Alice want to be able to encrypt/decrypt easily

But no one else should be able to decrypt How do we do this?

Keys!

Using Keys

Plaintext

Ciphertext DecryptionEncryption

Plaintext

Nonsense

The Shift Cipher

We “shift” each letter over by a certain amount

ILYH UHG EDOORRQV

five red balloons

f + 3 = Ii + 3 = Lv + 3 = Y

…

Plaintext

Ciphertext

EncryptionKey = 3

The Shift Cipher cont.

To decrypt, we just subtract the key

five red balloons

I - 3 = fL - 3 = iY - 3 = v

…

Plaintext

DecryptionKey = 3

ILYH UHG EDOORRQV Ciphertext

What’s wrong with the shift cipher?

Not enough keys! If we shift a letter 26 times, we get the

same letter backA shift of 27 is the same as a shift of 1, etc.So we only have 25 keys (1 to 25)

Eve just tries every key until she finds the right one

The Substitution Cipher

Rather than having a fixed shift, change every plaintext letter to an arbitrary ciphertext letter

a G

b X

c N

d S

e D

… …

z Q

Plaintext Ciphertext

The Substitution Cipher cont.

a G

b X

c N

d S

e D

f A

g F

h V

i L

j M

k C

l O

m EALRD HDS XGOOYYBW

five red balloons

f = Ai = Lv = R

…

Plaintext

Ciphertext

Encryption

Key =

n B

o Y

p Z

q P

r H

s W

t I

u J

v R

w U

x K

y T

z Q

The Substitution Cipher cont.

To decrypt we just look up the ciphertext letter in the table and then write down the matching plaintext letter

How many keys do we have now? A key is just a permutation of the letters of the

alphabet There are 26! permutations

403291461126605635584000000

Frequency Analysis

In English (or any language) certain letters are used more often than others

If we look at a ciphertext, certain ciphertext letters are going to appear more often than others

It would be a good guess that the letters that occur most often in the ciphertext are actually the most common English letters

Letter Frequency

This is the letter frequency for English

The most common letter is ‘e’ by a large margin, followed by ‘t’, ‘a’, and ‘o’

‘J’, ‘q’, ‘x’, and ‘z’ hardly occur at all

Frequency Analysis in Practice

Suppose this is our ciphertext dq lqwurgxfwlrq wr frpsxwlqj surylglqj d eurdg vxuyhb

ri wkh glvflsolqh dqg dq lqwurgxfwlrq wr surjudpplqj. vxuyhb wrslfv zloo eh fkrvhq iurp: ruljlqv ri frpsxwhuv, gdwd uhsuhvhqwdwlrq dqg vwrudjh, errohdq dojheud, gljlwdo orjlf jdwhv, frpsxwhu dufklwhfwxuh, dvvhpeohuv dqg frpslohuv, rshudwlqj vbvwhpv, qhwzrunv dqg wkh lqwhuqhw, wkhrulhv ri frpsxwdwlrq, dqg duwlilfldo lqwhooljhqfh.

0

0.02

0.04

0.06

0.08

0.1

0.12

a b c d e f g h i j k l m n o p q r s t u v w x y z

Letter

Rel

ativ

e F

req

uen

cy

Ciphertext distribution English distribution

In our ciphertext we have one letter that occurs more often than any other (h), and 6 that occur a good deal more than any others (d, l, q, r, u, and w)

There is a good chance that h corresponds to e, and d, l, q, r, u, and w correspond to the 6 next most common English letters

Frequency Analysis cont.

If we replace ‘e’ with ‘h’ and the 6 next most common letters with their matches, the ciphertext becomes an intro???tion to ?o?p?tin? pro?i?in? a ?roa? ??r?e? o?

t?e ?i??ip?ine an? an intro???tion to pro?ra??in?. ??r?e? topi?? ?i?? ?e ??o?en ?ro?: ori?in? o? ?o?p?ter?, ?ata repre?entation an? ?tora?e, ?oo?ean a??e?ra, ?i?ita? ?o?i? ?ate?, ?o?p?ter ar??ite?t?re, a??e???er? an? ?o?pi?er?, operatin? ???te??, net?or?? an? t?e internet, t?eorie? o? ?o?p?tation, an? arti?i?ia? inte??i?en?e.

Classical to Modern Cryptography

Classical cryptographyEverything up until around WWIIEncryption/decryption done by hand

Modern cryptographyComputers to encrypt and decryptSame principles, but automation allows

ciphers to become much more complex

The Enigma Machine

German encryption and decryption machine used in WWII

Essentially a complex, automated substitution cipher

How did Enigma work?

Rotors have different wiring connecting input to output

Rotors move after each keypress

The key is the initial position of the three rotors

Breaking the Enigma

Britain set up its cryptanalysis team in Bletchley Park

They consistently broke German codes throughout the war

Provided the intelligence codenamed ULTRA

Important location in the history of computing Alan Turing COLOSSUS

Cryptography in the Computer Age

Working with binary instead of letters

We can do things many, many times Think of an Enigma machine that has 2128 pairs of

symbols on each rotor, and 20 rotors

Other than that, the basic principles are the same as classical cryptography

Modern Ciphers

We design one relatively simple scrambling method (called a round) and repeat it many times Think of each round as a rotor on the Enigma One round may be easy to break, but when you put them all

together it becomes very hard

Almost all ciphers follow one of two structures SPN (Substitution Permutation Network) Feistel Network These describe the basic structure of a round

Modern Ciphers in Practice

Follow SPN/Feistel structure in general, but with added twists for security

There are two important ciphers in the history of modern cryptographyDES (Data Encryption Standard)AES (Advanced Encryption Standard)

DES

U.S. Government recognized the need to have a standardized cipher for secret documents

DES was developed by IBM in 1976

Analysis of DES was the beginning of modern cryptographic research

Controversy Surrounding DES

Development process was hidden from publicSuspicions that the government had put in a

“backdoor”

Government attempted to shut down research in cryptography

Breaking DES

The key length of DES was too short If a key is 56 bits long, that means there are

256 possible keys“DES Cracker” machines were designed to

simply try all possible keys

Breaking DES cont.

DES was further weakened by the discovery of differential cryptanalysis Biham and Shamir in 1990 The most significant advance in cryptanalysis since frequency analysis

Ideally a ciphertext should be completely random, there should be no connection to its matching plaintext Differential analysis exploits the fact that this is never actually the case Uses patterns between plaintext and ciphertext to discover the key

There is evidence that IBM knew about differential cryptanalysis back when they were designing DES in 1976

Developing the AES

With DES effectively broken, a new standard was needed

U.S. Government made it an open application/review process this time, and received many submissions

In 2001, after five years, the Rijndael cipher was selected to become the Advanced Encryption Standard

The Problem of Symmetric Key Cryptography Up until now we’ve been talking about symmetric

key cryptography Alice and Bob are using the same key to

encrypt/decrypt

Problem: How does Bob get the key to Alice when Eve is eavesdropping?

Up until 1976 the only solution was to physically give Alice the key in a secure environment

Public Key Cryptography

Diffie and Hellman published a paper in 1976 providing a solution

We use one key for encryption (the public key), and a different key for decryption (the private key)

Everyone knows Alice’s public key, so they can encrypt messages and send them to her But only Alice has the key to decrypt those messages

No one can figure out Alice’s private key even if they know her public key

Using Public Keys

Plaintext

Ciphertext DecryptionEncryption

Plaintext

Nonsense

Public Key Cryptography in Practice The problem is that public key algorithms are too

slow to encrypt large messages Instead Bob uses a public key algorithm to send Alice

the symmetric key, and then uses a symmetric key algorithm to send the message

The best of both worlds! Security of public key cryptography Speed of symmetric key cryptography

Sending a Message What’s your public key?

Bob picks a symmetric key and encrypts it using Alice’s public key

Alice decrypts the symmetric key using her private key

Bob encrypts his message using the symmetric key

Then sends the key to Alice

Then sends the message to Alice

Alice decrypts the message using the symmetric key

hi

The RSA Public Key Cipher

The most popular public key cipher is RSA, developed in 1977 Named after its creators: Rivest, Shamir, and Adleman

Uses the idea that it is really hard to factor large numbers Create public and private keys using two large prime numbers Then forget about the prime numbers and just tell people their

product Anyone can encrypt using the product, but they can’t decrypt

unless they know the factors If Eve could factor the large number efficiently she could get the

private key, but there is no known way to do this

Are we all secure now?

Unfortunately not, there are still many problems that need to be dealt with How does Bob know that he’s really talking to Alice? How does Alice know that the message she receives

hasn’t been tampered with? How does Alice know the message was sent by Bob?

These are questions addressed by other areas of cryptography

The End