notes/docs/lectures/cryptography/01_intro.md

Cryptography

Cryptology

“The science and art of writing and solving codes to hide the meaning of messages.”

Symmetric

“Encryption methods in which both the encryption and decryption algorithms use the same key.”

Asymmetric

“Methods which use separate, but related, private and public keys.”

Protocols

“The application of cryptographic algorithms in secure systems.”

Cryptanalysis

“The science and art of breaking cryptosystems.”

Modern Cryptography (1970-)

Fundamentally different - a scientific and mathematical discipline

Rigorously tested - New approaches tested, justified through mathematical proofs and theory

Extremely powerful - Ciphers usually take milliseconds to use and lifetimes of the universe to break

Wider uses - including message integrity and authenticity

Civilian use - everyone benefits from cryptography now

Ciphers

• Ciphers have been used for thousands of years
• Usually based around either transposition or substitution

Caesar Cipher

• An early substitution cipher, we replace each letter of plain text with a shifted letter n letters away from the letter
• Therefore our key is an integer -25\leq n \leq 25

Modular Arithmetic

• Modular arithmetic is a system of arithmetic for finite sets of integers
• Common sets include
  • \mathbb{N} = \{1,2,3,...\}
  • \mathbb{Z} = \{..., -3, -2, -1, 0,1,2,3,...\}
  • Also \mathbb{Q}, \mathbb{R}, \mathbb{C}
• Cryptography is almost always interested in finite sets
• This is useful as it avoids overflow errors
  • When we add or multiply two 1 byte binary digits, the result will always be 1 byte

Congruence

Let a, r, m \in \mathbb{Z} and m > 0

a \equiv r (mod\space m) if \frac{m}{a-r}

Check:

a=12, m=7

a\equiv 5 (mod\space 7)

\frac{7}{12-5} ✅

This can be rewritten as: a = q\cdot m+r

Equivalence Classes

• The sets of all integers mod 5 form a series of equivalence classes
• All these numbers act the same in any modulo sum

For example

74\cdot 62 - 47 (mod \space 5) \equiv 74\%5 \cdot 62\%5 - 47\%5

Also works with exponentiation

3^8\space (mod\space 7)

3^2 = 3\cdot 3 = 9 \equiv 2\space (mod\space 7)

3^4 = 3^2\cdot 3^2 = 2\cdot 2 \equiv 4\space (mod\space 7)

3^8 = 3^4 \cdot 3^4 = 4\cdot 4 = 16 \equiv 2 \space (mod\space 7)

Integer Rings

• Modular arithmetic forms what in mathematics we would call a Ring

Ring Definition

The integer ring \mathbb{Z}_m consists of:

1. The set \mathbb{Z}_m = \{0, 1,\ldots m-1\}
2. Two operations + and \cdot for all a, b \in \mathbb{Z}_m such that:
   1. a+b \equiv c \space (mod\space m), (c\in \mathbb{Z})
   2. a\cdot b \equiv d \space (mod\space m), (d\in \mathbb{Z})

Any time you add or multiply any two numbers in the set, the result is always in the set. We use \equiv instead of = as it could be an intermediate number, e.g. 12 instead of 2.

Properties of Rings

• We can add or multiply any two numbers in the ring, and the result is in the ring
  • It is closed
• Addition and multiplication are associative
  • (a+b)+c\space =\space a+(b+c)
• There is a neutral element 0 for addition
  • a + 0 \equiv a\space mod \space m
• The additive inverse always exists
  • a + (-a) = 0\space mod \space m
• There is a neutral element for multiplication
  • a\cdot 1 \equiv a\space mod\space m
• The multiplicative inverse exists for some but not all elements
  • a\cdot a^{-1} \equiv 1 \space mod \space m

Modular Inversion

In rings, the multiplicative inverse exists for some but not all elements

• Multiplicative inverses allow us to divide by a number

\frac{b}{a} \equiv b \cdot a^{-1} \space (mod \space m)

• Not all numbers in a ring have an inverse, you can determine whether one exists quite simply:

gcd(a,m)=1

Example

3\cdot 9 \equiv 1 \space (mod\space 26)

5\cdot 9 \equiv 19 \space (mod\space 26)

19\cdot 3 \equiv 57 \equiv 5\space (mod\space 26)

Here a=3 and b=5, we can divide by 19 to get back to 5.

Shift Cipher

We can formalise the shift cipher using modular arithmetic

Let x, y, k \in \mathbb{Z}_{26}

e_k(x) = y \equiv x+k \space (mod \space 26) \\
d_k(y) = x \equiv y-k \space (mod \space 26)

Frequency Analysis

• The frequency of occurrences of each character is very consistent
• The longer a cipher text is, the easier this becomes

Affine Cipher

We can extend the shift cipher into an affine cipher

Let x,y,a,b \in \mathbb{Z}_{26}

e_k(x) = y \equiv a\cdot x+b\space (mod \space 26)\\
d_k(y) = x \equiv a^{-1}\cdot(y-b)\space (mod \space m)

where k=(a,b) and gcd(a,26)=1

This is a multiplication and an addition analogous to y=mx+c

In an Affine cipher, letters can be themselves

• The keyspace of an affine cipher
  • a can be 0-25
  • b can be 0-12
  • 25*12=300
• More secure than a Caesar cipher

Frequency analysis can still be used; in this case the columns will not only be shifted, but jumbled as well.

• This is not hard to crack

The Vigenere Cipher

• An early stream cipher, the Vigenere cipher is a shift cipher with a running key
• Unlike the Caesar cipher, the key is repeated for as long as required.
• It is the equivalent to multiple interleaved Caesar ciphers
• Spreads out occurrences of characters, making frequency analysis hard.