At our last meeting on Wednesday 22nd September 1999, Sean treated us to a feast of prime number `hokey-poke' with a tour through the world of `The Planck Spectrum, Riemann Zeta Functions, and Modern Cryptography'.
After showing how prime numbers can be identified as the holes in the `Sieve of Eratosthenes' Sean reminded us that the expression for the energy of a Harmonic Oscillator,
E = (1/2m)p*p + (k/2)x*x,
can be written as an equation describing an ellipse:
1 = (1/2mE)p*p + (k/2E)x*x,
In determining the area enclosed by this ellipse it was shown that a quantum state occupies a `volume' in phase space of h (Planck's Constant) per degree of freedom. In the early 1920's Bose used this fact to calculate the total energy of a gas of photons by summing the probability density weighted energy over the states of the system. Performing the integration it is found that the total energy is proportional both to T^4 (temperature to the fourth power) and the Riemann Zeta Function for s=4 where the Riemann Zeta function with argument s is
1 + 1/2^s + 1/3^s + 1/4^s....
Sean then showed via a Taylor expansion that the product of 1/(1-1/p^s) over the prime numbers p is exactly equal to the Riemann Zeta Function, thus demonstrating the curious connection between prime numbers and the Planck Spectrum !
Sean then showed how modern cryptography incorporates prime numbers with an exposition of the RSA algorithm (where RSA acronymously encodes its inventors). Beginning with two prime numbers p and q, further integers are defined such that
n = pq,
f = (p-1)(q-1),
e is relatively prime with f but otherwise randomly selected and
d = (fk+1)/e
(for a generally unique pair d, k). The message text, represented by the integer t, is encoded into the cyphertext c with the prescription
c = (t^e)mod(n)
and can be decoded with
t = (c^d)mod(n).
We tried an example with p=2 and q=5 to check that the maths worked out, and Sean also showed us a more `gnarly' case with p=47, q=71, to indicate the complications involved. Once Bob has selected his set of integers he publishes n and e, allowing Alice (or anyone) to encode a message and send it to him. Bob keeps to himself the key d needed to decode the message. If p and q are chosen sufficiently large it is remarkably difficult to crack the code with only knowledge of n and e; it would require finding the prime factors of a large number which currently takes approximately `the age of the universe' with the best CPU. RSA cryptography is extensively used over the internet.
However, it was then discussed how the parallel processing capabilities of a quantum computer may one day reduce this prime factoring time to seconds! In the example described the `0' and `1' of the `Qubits' of the quantum computer correspond to the ground and first excited metastable states of calcium atoms. After showing how a `one-time pad' can be used to encode and decode a string of bits (and why it should only be used once!) it was explained how quantum cryptography may in the future be used to securely transmit a one-time pad from Alice to Bob (without Eve eavesdropping). Quantum cryptography involves the transmission and detection of photons through polarizing filters, different orientations of which correspond to `0' and `1'. It was shown how the quantum nature of light prevents Eve from intercepting the transmission without Bob and Alice noticing.
There followed a general discussion about the status and future of cryptography. The general conclusion was that one can perhaps never be 100% sure of having a secure system in practice, however by utilizing the quantum properties of photons one could get pretty close.