The unit

• Number Theory (Dr Killian O'Brien)
• Cryptography (Dr Jon Borresen)
• 3 hours lecture + 1 hour tutorial per week
• Coursework problems (30%), Examination (70%)
• Popular unit (49 students in 14/15) with good ISS results

In a nutshell

Number Theory

• The study of the integers
  ℤ = {…−3, −2, −1, 0, 1, 2, 3, …}
  
• Of fundamental importance are the primes
  2, 3, 5, 7, …
  
• Nice mixture of proof oriented theoretical work and algorithmic methods

Cryptography

• The science/art of transforming text so that it can only be read by selected recipients.
• Often in connection to military/industrial/political/... secrets.
• Universally and intensively used in modern computer network communications.

Software

• Interesting use of mathematical software for various aspects of the unit
  • Matlab
  • SageMath, an open source software system aimed at pure mathematics
  • CoCalc, SageMath, and other maths software, in the cloud
  • Python, a general purpose programming language with many advanced math libraries

A quick tour of some highlights from the unit

Integers in the news

What's so special about ... ?

• 77, 232, 917
• 2^{77, 232, 917} − 1

Open conjecture or tutorial problem?

• There are infinitely many prime numbers.
• 2ⁿ − 1 can only be prime when n is prime.
• There are infinitely many 2ⁿ − 1 which are prime.

Some Python SageMath code

The following SageMath code finds the first few Mersenne primes. See the results

for p in prime_range(1,10^2):
    if is_prime(2^p - 1):
        print(p)
        print(2^p -1)

Open conjecture or tutorial problem?

• There are infinitely many primes p for which p + 2 is also prime, i.e. narrow steps on the π staircase.
• For any integer n ≥ 1 there are infinitely many gaps of at least length n between consecutive primes, i.e. wide steps on the π staircase.

Classical vs. modern cryptography

Classical

• Topics include mono- and poly-alphabetic substitution ciphers.
• Classical cryptography required prearranged secrets between sender and recipient.
• Can be broken with the aid of frequency analysis and these vulnerabilities.

Modern

• Crypto systems based on number theoretic concepts.
• Prearranged secrets no longer required, so called Public Key Cryptography.
• Enables secure mass communication between anyone across public non-secure networks.
• Increasingly important application in crypto-currency such as Bitcoin

Who is this?

Who was that?

• Edward Snowden, former worker for the CIA and NSA.
• In 2013, he fled the USA, briefly staying in Hong Kong before securing temporary asylum in Russia.
• Has passed many thousands of classified files from the NSA, GCHQ and other intelligence agencies to journalists.
• These revelations concern global surveillance programs carried out by these agencies on the public.

CitizenFour movie

How did Snowden make secure contact with journalists he had never met?