Mathematical Ideas that Shaped the World

Prime numbers

Plan for this class

Why are prime numbers interesting? What is the Prime Number Theorem? How could prime numbers win you a

million dollars? How does public key cryptography

work?

What food did you bring for the maths picnic?!

But first...

Some mathematical mind reading!

A prime number is a number which is only divisible by itself and 1.

Prime numbers

7 6

The building blocks of numbers Primes are often known as the building

blocks of numbers, since they generate all other numbers.

The Fundamental Theorem of Arithmetic states that every number can be written uniquely as a product of primes.

28 = 22x7, 60 = 22x3x5

Sieve of Eratosthenes

This is an ancient Greek method for finding all prime numbers.

Pattern of the primes

Prime numbers seem to occur randomly.

Sometimes they come in pairs, e.g. (11,13), (29,31), (59,61)…

…and other times there are long gaps between them, e.g. (113, 127)

There is no formula that will predict where the next prime number will be.

Sequences of primes

Fermat primes

Mersenne primes

Mersenne primes

The Ulam Spiral

In 1963 the mathematician Stanislaw Ulam was doodling during a boring meeting…

37 36 35 34 33 32 31

38 17 16 15 14 13 30

39 18 5 4 3 12 29

40 19 6 1 2 11 28

41 20 7 8 9 10 27

42 21 22 23 24 25 26

43 44 45 46 47 48 49

The Ulam Spiral

200x200 Ulam spiral Sacks spiral

Distribution of the primes

Rather than trying to find patterns in primes, mathematicians started looking at the general distribution of primes among the numbers.

For example, if you pick a random number in the range of 0 to N, what is the chance that this number is prime?

The Prime Number Theorem

Prime counting

The prime number theorem

always a bit less than (x)

Enter Riemann

In 1859 Riemann gave an explicit formula for (x).

There was another function, called the Riemann zeta function, which controlled how far away the primes were from their expected positions.

(s)Get back

over here!!

Bernhard Riemann (1826 – 1866)

Born near Hanover to a poor family.

Was shy, had frequent nervous breakdowns and a fear of public speaking.

Trained to become a pastor but kept getting distracted by maths.

Bernhard Riemann (1826 – 1866)

While studying theology in Göttingen he met Gauss, and was persuaded to switch to maths.

Founded Riemannian geometry – the cornerstone of Einstein’s theory of relativity.

Died of TB at age 40.

The zeta function

The Riemann hypothesis

The Riemann Hypothesis

The Riemann Hypothesis

The position of the zeros of the zeta function dictate the positions of the prime numbers.

The Riemann Hypothesis was Problem 8 of Hilbert’s 1900 list of unsolved problems.

It is now one of 6 remaining Clay Institute Millennium Prize Problems worth $1 million.

The Riemann Hypothesis

So far 10,000,000,000,000 zeros have been checked and they all

satisfy the conjecture.

Are prime numbers useless?

"I have never done anything 'useful'. No

discovery of mine has made, or is likely to

make, directly or indirectly, for good or ill, the least difference to the amenity of the

world.“ G.H.Hardy

Modular arithmetic

What time will it be 28 hours from now? What day of the week will it be in 30

days? What is the last digit of

4538729 x 9957397632?

To answer these questions, you have used modular arithmetic!

Modular arithmetic

In modular arithmetic we have a clock with n numbers on.

Whenever we goaround we start againat zero. We say we are workingmodulo n, or mod n.

0

1

2

34

5

6

The problem with cryptography The usual problem with sending a coded

message is that you have to agree on a key for the code beforehand. A shift on a Caesar cipher The starting positions for an Enigma machine An actual key to unlock a box

If anyone finds the key, they can decrypt all your messages.

Public key cryptography

In public key cryptography, there are two different keys: A public key which can be used to

encrypt messages. A private key which can be used to

decrypt messages.

Public key cryptography

Digitally, this system requires mathematical problems which are easy to do in one direction but very difficult to do backwards without the key.

E.g. it is very easy to multiply two primes together, but very difficult to factorise the product without knowing one of the primes.

97 x 53 = 5141

One-way problems

Another example is that it is easy to compute powers modulo a prime, but difficult to find logarithms.

56 = 8 (mod 23)

Diffie-Hellman key exchange

We can use this idea to obtain a shared secret.

Alice and Bob agree on a prime p and a number g.

Alice chooses a secret number a and publishes A = ga (mod p).

Bob chooses a secret number b and publishes B = gb (mod p).

Now both can compute gab (mod p).

RSA cryptography

RSA cryptography was originally invented by the Englishman Clifford Cocks in 1973.

However, this remained secret until 1997 as the work was done for GCHQ.

It is now named after Ron Rivest, Adi Shamir and Leonard Adleman who described the method in 1977.

The RSA algorithm

1. Choose two nice big primes, p and q.2. Compute n =pq.3. Compute (n) = (p-1)(q-1)4. Choose e so that e and (n) are

coprime.5. Publish e and n; this is your public key.6. Find d = e-1 (mod (n)); this is your

private key.

Encryption

Bob wants to send Alice a message M. Bob knows Alice’s public key: (e,n). He computes c = Me (mod n).

Alice computes cd (mod n) and recovers M.

(using the magic of modular arithmetic and some number theory!)

Problems with RSA

The two primes must be chosen randomly. If many people pick the same prime then the numbers are easy to factorise.

The same plaintext message sent to many different people becomes easy to decrypt. Therefore M is randomised prior to sending.

p and q must not be too close together, and d should be large.

Breaking RSA

Currently the largest number factorised by a general algorithm is 768 bits long.

RSA keys are typically 1024 – 2048 bits long.

Note: a proof of the Riemann Hypothesis is unlikely to break our security systems!

Lessons to take home

Prime numbers are the building blocks of arithmetic.

There is no discernible pattern to the primes, although we understand how they are distributed.

Primes underlie most of our cryptosystems.

Maths Picnic

!