Public Key Cryptography

ShHHHHhh!
It's a secret!

Humans invented ways to send secret messages to one another over 4000 years ago (ca 1900BCE). Since then, we have not stopped trying to come up with better and better ways to keep our communications private and/or ways to ensure that things, once written down, can't be altered by anyone.

 

Why do you think that is?

"Slave power crushes freedom of speech and of opinion. Slave power degrades labor. Slave power is arrogant, is jealous and intrusive, is cruel, is despotic, not only over the slave but over the community, the state."
 

Elizabeth Van Lew

Spy and Anti-Slavery Activist

Code has been around... even before computers...

Code-breaking was even the inspiration and motivation behind the creation of one of the very first working computers...

TRUST

Why is it necessary?

CRYPTOGRAPHY

DANGER!!!

A flawed implementation of an Android random number generator allowed hackers access to several bitcoin wallets in early 2013!

In 2010, a group of hackers calling themselves failOverflow figured out how to get the PlayStation 3 private key based on a similar security flaw.

It's important to understand this stuff if you are implementing the tech!!!

BRIEF HIstory of Cryptography

• Writing
  (Mesopotamia, circa 3500 BCE)
• Ciphertext
  (Egypt, circa 1900 BCE)
• Cryptanalysis
  (Muslims, circa 800 CE)
• Electromechanical ciphers
  (circa 1917)
• Mathematical cryptography (Shannon, 1949~)

Early

• DES (1974)
• Diffie-Hellman key exchange
  (1977)
• RSA Algorithm
  (1977)
• Elliptic Curve Algorithms
  (1985)
• MD5 (1992), SHA-0 (1993)
• SHA-1 (1995)
• AES replaces DES (2001)
• SHA-2 (2004)

MOdern

Ciphertext

• The most ancient method of encrypting a message
• ALL cryptography up until the early 20th century was more or less some form of ciphertext
• Essentially involves some scheme where every letter of a message is replaced with some other letter or symbol; Recipient must have a "key" to decipher it

PROblem

• For ciphertext to work, both sender and recipient must have the key
• Sending the key across a public network (i.e. the internet) defeats the purpose since it could be intercepted at any point along the way
• Secure communication via the web required a new solution → asymmetric cryptography, aka public key cryptography

Trapdoor Algorithms

An IMportant Step Forward:

Trapdoor Algorithms

are like...

INteger Factorization

It turns out that while it's very easy to multiply two numbers together, it's VERY HARD to discover their prime factors.

The RSA algorithm is widely used, but some fear it might be vulnerable to quantum computing.

RSA-1024 is a 309-digit number that is the sum of two large prime numbers. Even using ALL of the computing power on earth, it could take over 1000 years to discover these factors.

Elliptic
Curve
Cryptography

Currently, one of the more popular families of trapdoor function being used in public key encryption.

Allows smaller (faster) keys to be used for the same level of security as RSA.

How RSA Works:

To generate keys:

1. Pick 2 large prime numbers p and q that differ in length by a few digits
2. Compute n = pq
3. Calculate the totient
   φ(n) = (p -1)(q - 1) 
4. Pick a number e such that
   1 < e < φ(n) and e and φ(n) are coprime
5. Use the extended Euclidean algorithm to calculate d, the modular multiplicative inverse,
   d = e-1 mod φ(n)
6. e is the public key
   d is the private key

To encrypt  a value:

1. The encrypted value c of message m is: c = me mod n

To decrypt  a value:

1. The decrypted message m given c is:
   m = cd mod n

Public Key Encryption

photo credit: US Naval Academy https://www.usna.edu/CyberDept/sy110/lec/cryptDig/img/bobaliceAsymmetric.png

How does Public Key Encryption work?

STep 1: Generate a Private Key

A private key is just a random number, but it is generated from a HUGE, COLOSSAL, GI-NORMOUS number of possibilities, so the chances two people will have the same key, or that someone might guess your key are infinitesimally small 

Example

bitcoin private key space = 2 256 ≈ 1077

number of atoms in visible universe ≈ 1080

STep 2: Generate a
PUBLIC Key

A public key is calculated from your private key using a one-way hashing function

knot metaphor

dictionary metaphor