Cryptography Made Easy, Summaries of Cryptography and System Security

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Cryptography Made Easy

Stuart Reges

Principal Lecturer

University of Washington

Why Study Cryptography?

 Secrets are intrinsically interesting

 So much real-life drama:

 Mary Queen of Scots executed for treason  primary evidence was an encoded letter  they tricked the conspirators with a forgery

 Students enjoy puzzles

 Real world application of mathematics

Start with an Algorithm

 The Spartans used a scytale in the fifth

century BC (transposition cipher)

 Card trick

 Caesar cipher (substitution cipher):

ABCDEFGHIJKLMNOPQRSTUVWXYZ

GHIJKLMNOPQRSTUVWXYZABCDEF

Then add a secret key

 Both parties know that the secret word

is "victory":

ABCDEFGHIJKLMNOPQRSTUVWXYZ

VICTORYABDEFGHJKLMNPQSUWXZ

 "state of the art" for hundreds of years

 Gave birth to cryptanalysis first in the

Muslim world, later in Europe

Vigenère Square (polyalphabetic)

Vigenère Cipher

 More secure than simple substitution  Confederate cipher disk shown (replica)  Based on a secret keyword or phrase  Broken by Charles Babbage

Public Key Encryption

 Proposed by Diffie, Hellman, Merkle  First big idea: use a function that cannot be reversed (a humpty dumpty function): Alice tells Bob a function to apply using a public key, and Eve can’t compute the inverse  Second big idea: use asymmetric keys (sender and receiver use different keys): Alice has a private key to compute the inverse  Key benefit: doesn't require the sharing of a secret key

RSA Encryption

 Named for Ron Rivest, Adi Shamir, and Leonard Adleman  Invented in 1977, still the premier approach  Based on Fermat's Little Theorem: ap-1^ 1 (mod p) for prime p, gcd(a, p) = 1  Slight variation: a(p-1)(q-1)^ 1 (mod pq) for distinct primes p and q, gcd(a,pq) = 1  Requires large primes (100+ digit primes)

Why does it work?

 Original message is carried to the e power, then to the d power: (msge)d^ = msged  Remember how we picked e and d: msged^ = msgk(p-1)(q-1) + 1  Apply some simple algebra: msged^ = (msg(p-1)(q-1))k^ msg^1  Applying Fermat's Little Theorem: msged^ = (1)k^ msg^1 = msg

Politics of Cryptography

 British actually discovered RSA first but

kept it secret

 Phil Zimmerman tried to bring

cryptography to the masses with PGP

and ended up being investigated as an

arms dealer by the FBI and a grand jury

 The NSA hires more mathematicians

than any other organization

Card Trick Solution

 Given 5 cards, at least 2 will be of the same suit (pigeon hole principle)  Pick 2 such cards: one will be hidden, the other will be the first card  First card tells you the suit  Hide the card that has a rank that is no more than 6 higher than the other (using modular wrap-around of king to ace)  Arrange other cards to encode 1 through 6

Encoding 1 through 6

 Figure out the low, middle, and high cards  rank (ace < 2 < 3 ... < 10 < jack < queen < king)  if ranks are the same, use the name of the suit (clubs < diamonds < hearts < spades)  Some rule for the 6 arrangements, as in: 1: low/mid/hi 3: mid/low/hi 5: hi/low/mid 2: low/hi/mid 4: mid/hi/low 6: hi/mid/low