Transcript Mathematical Ideas that Shaped the World

Mathematical Ideas that
Shaped the World
Prime numbers
Plan for this class
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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!
Prime numbers
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A prime number is a number which is only
divisible by itself and 1.
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The building blocks of numbers
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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
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This is an ancient Greek method for finding
all prime numbers.
Pattern of the primes
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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
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Fermat primes
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Mersenne primes
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Mersenne primes
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The Ulam Spiral
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In 1963 the mathematician Stanislaw Ulam
was doodling during a boring meeting…
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The Ulam Spiral
200x200 Ulam spiral
Sacks spiral
Distribution of the primes
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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
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Prime counting
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The prime number theorem
always a bit less
than (x)
Enter Riemann
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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)
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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)
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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
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The Riemann hypothesis
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The Riemann Hypothesis
The Riemann Hypothesis
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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
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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.
0
 Whenever we go
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1
around we start again
at zero.
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modulo n, or mod n.
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The problem with cryptography
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The usual problem with sending a coded
message is that you have to agree on a key
for the code beforehand.
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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
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In public key cryptography, there are two
different keys:
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A public key which can be used to encrypt
messages.
A private key which can be used to decrypt
messages.
Public key cryptography
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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
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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
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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
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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
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Choose two nice big primes, p and q.
Compute n =pq.
Compute (n) = (p-1)(q-1)
Choose e so that e and (n) are coprime.
Publish e and n; this is your public key.
Find d = e-1 (mod (n)); this is your private
key.
Encryption
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Bob wants to send Alice a message M.
Bob knows Alice’s public key: (e,n).
He computes c = Me (mod n).
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Alice computes cd (mod n) and recovers M.
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(using the magic of modular arithmetic and
some number theory!)
Problems with RSA
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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
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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
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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!