Cryptography and eCommerce

1
Cryptography and e-Commerce

• To carry out e-commerce requires secure
  communications between parties who do not have a
  secure line to agree prior keys
• Secure communication required secure prior
  agreement of keys for all encryption systems used
  before 1970s
• This weakness helped the Bletchley Park
  codebreakers

2
Trapdoor Functions

• Trapdoor functions are functions that are easy to
  compute, but hard to undo, except if you know a
  special piece of information
• It is not known whether trapdoor functions
  actually exist, but there are several functions
  that are thought to be trapdoors

3
A Physical Trapdoor

• Imagine you want to receive a message securely
  using a physical system

Send box with open padlock but no key
Insert message and lock box with padlock then
return box
Open lock with key
4
Electronic Analog

• Send an electronic padlock
• Receiver locks message using padlock and
  returns it to sender
• Sender unlocks the padlock to read the message
• Electronic padlocks can be copied easily, so
  everyone can use them
• Only the owner of the key can unlock the padlocks

5
A Possible Trapdoor Function

• It is very easy to multiply numbers even large
  numbers using a computer
• It turns out that it is very hard to factorise
  numbers in general
• For example
• 61158437 is the product of just two primes

• but it is much harder to find them than to
  multiply 7723
  and 7919

6
Prime Factorisation

• In general, with large primes (eg 150 decimal
  digits or more), it is very hard to factorise a
  product of pairs of primes
• It is surprisingly easy to find large primes and
  to multiply them together

7
Computation Modulo n

• Recall that arithmetic modulo an integer n means
  that all results are reduced to their remainder
  on division by n
• For example
• 2014 modulo 29 280 modulo 29 19

8
Public Key Cryptography

• Find two large primes, p and q
• Find x (p-1)(q-1) and n pq
• Choose E that shares no divisors with x (usually
  a good sized prime is used)
• Find D such that DE 1 modulo x
• Make public the padlock, n and E

9
Encoding and Decoding

• To encode a message, m, (which must be smaller
  than n) calculate
• c mE modulo n
• Send c
• To decode calculate
• d cD modulo n

10
Simple Example

• Let p 7, q 11
• Then x (7-1)(11-1) 60 and n 711 77
• Choose E 13
• Find D such that D13 1 modulo 60
• D 37
• Key E 13, n 77
• To send message m 15, say, calculate
• c 1513 modulo 77 64
• To decode encrypted message (64) calculate
• d 6437 modulo 77 15

11
RSA

• Rivest, Shamir and Adelman published this
  approach in 1977
• Now the most common way to send keys securely for
  a simpler encryption scheme
• Currently 1024 bit numbers used for p and q, and
  larger values will be required soon
• If we ever discover how to factorise numbers
  quickly then this technique will be broken

12
Remember!

• Assessment today
• Information theory and data compression
• Open book
• Key thing to remember Shannons definition for
  information content
• Message m has content -log(p(m)) where p(m) is
  probability of message m and log is taken base 2
  (use log(p(m))/log(2) to find this)
• Huffman codes