Crypto

Cryptography is a funny business; shady from the beginning, the whole business of codes and ciphers has been specifically designed to hide your intentions and move in the shadows, unnoticed. However, the art of cryptography has been changed almost beyond recognition in the last hundred years thanks to the invention of the computer, and what was once an art limited by the imagination of the nerd responsible has now turned into a question of sheer computing might. But, as always, the best way to start with this story is at the beginning…

There are two different methods of applying cryptography to a message; with a code or with a cipher. A code is a system involving replacing words with other words (‘Unleash a fox’ might mean ‘Send more ammunition’, for example), whilst a cipher involves changing individual letters and their ordering. Use of codes can generally only be limited to a few words that can be easily memorised, and/or requires endless cross-referencing with a book of known ‘translations’, as well as being relatively insecure when it comes to highly secretive information. Therefore, most modern encoding (yes, that word is still used; ‘enciphering’ sounds stupid) takes the form of employing ciphers, and has done for hundreds of years; they rely solely on the application of a simple rule, require far smaller reference manuals, and are more secure.

Early attempts at ciphers were charmingly simple; the ‘Caesar cipher’ is a classic example, famously invented and used by Julius Caesar, where each letter is replaced by the one three along from it in the alphabet (so A becomes D, B becomes E and so on). Augustus Caesar, who succeeded Julius, didn’t set much store by cryptography and used a similar system, although with only a one-place transposition (so A to B and such)- despite the fact that knowledge of the Caesar cipher was widespread, and his messages were hopelessly insecure. These ‘substitution ciphers’ suffered from a common problem; the relative frequency with which certain letters appear in the English language (E being the most common, followed by T) is well-known, so by analysing the frequency of occurring letters in a substitution-enciphered message one can work out fairly accurately what letter corresponds to which, and work out the rest from there. This problem can be partly overcome by careful phrasing of messages and using only short ones, but it’s nonetheless a problem.

Another classic method is to use a transposition cipher, which changes the order of letters- the trick lies in having a suitable ‘key’ with which to do the reordering. A classic example is to write the message in a rectangle of a size known to both encoder and recipient, writing in columns but ‘reading it off’ in rows. The recipient can then reverse the process to read the original message. This is a nice method, and it’s very hard to decipher a single message encoded this way, but if the ‘key’ (e.g. the size of the rectangle) is not changed regularly then one’s adversaries can figure it out after a while. The army of ancient Sparta used a kind of transposition cipher based on a tapered wooden rod called a skytale (pronounced skih-tah-ly), around which a strip of paper was wrapped and the message written down it, one on each turn of paper. The recipient then wrapped the paper around a skytale of identical girth and taper (the tapering prevented letters being evenly spaced, making it harder to decipher), and read the message off- again, a nice idea, but the need to make a new set of skytale’s for everyone every time the key needed changing rendered it impractical. Nonetheless, transposition ciphers are a nice idea, and the Union used them to great effect during the American Civil War.

In the last century, cryptography has developed into even more of an advanced science, and most modern ciphers are based on the concept of transposition ciphers- however, to avoid the problem of using letter frequencies to work out the key, modern ciphers use intricate and elaborate systems to change by how much the ‘value’ of the letter changes each time. The German Lorenz cipher machine used during the Second World War (and whose solving I have discussed in a previous post) involved putting the message through three wheels and electronic pickups to produce another letter; but the wheels moved on one click after each letter was typed, totally changing the internal mechanical arrangement. The only way the British cryptographers working against it could find to solve it was through brute force, designing a computer specifically to test every single possible starting position for the wheels against likely messages. This generally took them several hours to work out- but if they had had a computer as powerful as the one I am typing on, then provided it was set up in the correct manner it would have the raw power to ‘solve’ the day’s starting positions within a few minutes. Such is the power of modern computers, and against such opponents must modern cryptographers pit themselves.

One technique used nowadays presents a computer with a number that is simply too big for it to deal with; they are called ‘trapdoor ciphers’. The principle is relatively simple; it is far easier to find that 17 x 19 = 323 than it is to find the prime factors of 323, even with a computer, so if we upscale this business to start dealing with huge numbers a computer will whimper and hide in the corner just looking at them. If we take two prime numbers, each more than 100 digits long (this is, by the way, the source of the oft-quoted story that the CIA will pay $10,000 to anyone who finds a prime number of over 100 digits due to its intelligence value) and multiply them together, we get a vast number with only two prime factors which we shall, for now, call M. Then, we convert our message into number form (so A=01, B=02, I LIKE TRAINS=0912091105201801091419) and the resulting number is then raised to the power of a third (smaller, three digits will do) prime number. This will yield a number somewhat bigger than M, and successive lots of M are then subtracted from it until it reaches a number less than M (this is known as modulo arithmetic, and can be best visualised by example: so 19+16=35, but 19+16 (mod 24)=11, since 35-24=11). This number is then passed to the intended recipient, who can decode it relatively easily (well, so long as they have a correctly programmed computer) if they know the two prime factors of M (this business is actually known as the RSA problem, and for reasons I cannot hope to understand current mathematical thinking suggests that finding the prime factors of M is the easiest way of solving this; however, this has not yet been proven, and the matter is still open for debate). However, even if someone trying to decode the message knows M and has the most powerful computer on earth, it would take him thousands of years to find out what its prime factors are. To many, trapdoor ciphers have made cryptoanalysis (the art of breaking someone else’s codes), a dead art.

Man, there’s a ton of cool crypto stuff I haven’t even mentioned yet… screw it, this is going to be a two-parter. See you with it on Wednesday…

Determinism

In the early years of the 19th century, science was on a roll. The dark days of alchemy were beginning to give way to the modern science of chemistry as we know it today, the world of physics and the study of electromagnetism were starting to get going, and the world was on the brink of an industrial revolution that would be powered by scientists and engineers. Slowly, we were beginning to piece together exactly how our world works, and some dared to dream of a day where we might understand all of it. Yes, it would be a long way off, yes there would be stumbling blocks, but maybe, just maybe, so long as we don’t discover anything inconvenient like advanced cosmology, we might one day begin to see the light at the end of the long tunnel of science.

Most of this stuff was the preserve of hopeless dreamers, but in the year 1814 a brilliant mathematician and philosopher, responsible for underpinning vast quantities of modern mathematics and cosmology, called Pierre-Simon Laplace published a bold new article that took this concept to extremes. Laplace lived in the age of ‘the clockwork universe’, a theory that held Newton’s laws of motion to be sacrosanct truths and claimed that these laws of physics caused the universe to just keep on ticking over, just like the mechanical innards of a clock- and just like a clock, the universe was predictable. Just as one hour after five o clock will always be six, presuming a perfect clock, so every result in the world can be predicted from the results. Laplace’s arguments took such theory to its logical conclusion; if some vast intellect were able to know the precise positions of every particle in the universe, and all the forces and motions of them, at a single point in time, then using the laws of physics such an intellect would be able to know everything, see into the past, and predict the future.

Those who believed in this theory were generally disapproved of by the Church for devaluing the role of God and the unaccountable divine, whilst others thought it implied a lack of free will (although these issues are still considered somewhat up for debate to this day). However, among the scientific community Laplace’s ideas conjured up a flurry of debate; some entirely believed in the concept of a predictable universe, in the theory of scientific determinism (as it became known), whilst others pointed out the sheer difficulty in getting any ‘vast intellect’ to fully comprehend so much as a heap of sand as making Laplace’s arguments completely pointless. Other, far later, observers, would call into question some of the axiom’s upon which the model of the clockwork universe was based, such as Newton’s laws of motion (which collapse when one does not take into account relativity at very high velocities); but the majority of the scientific community was rather taken with the idea that they could know everything about something should they choose to. Perhaps the universe was a bit much, but being able to predict everything, to an infinitely precise degree, about a few atoms perhaps, seemed like a very tempting idea, offering a delightful sense of certainty. More than anything, to these scientists there work now had one overarching goal; to complete the laws necessary to provide a deterministic picture of the universe.

However, by the late 19th century scientific determinism was beginning to stand on rather shaky ground; although ¬†the attack against it came from the rather unexpected direction of science being used to support the religious viewpoint. By this time the laws of thermodynamics, detailing the behaviour of molecules in relation to the heat energy they have, had been formulated, and fundamental to the second law of thermodynamics (which is, to this day, one of the fundamental principles of physics) was the concept of entropy. ¬†Entropy (denoted in physics by the symbol S, for no obvious reason) is a measure of the degree of uncertainty or ‘randomness’ inherent in the universe; or, for want of a clearer explanation, consider a sandy beach. All of the grains of sand in the beach can be arranged in a vast number of different ways to form the shape of a disorganised heap, but if we make a giant, detailed sandcastle instead there are far fewer arrangements of the molecules of sand that will result in the same structure. Therefore, if we just consider the two situations separately, it is far, far more likely that we will end up with a disorganised ‘beach’ structure rather than a castle forming of its own accord (which is why sandcastles don’t spring fully formed from the sea), and we say that the beach has a higher degree of entropy than the castle. This increased likelihood of higher entropy situations, on an atomic scale, means that the universe tends to increase the overall level of entropy in it; if we attempt to impose order upon it (by making a sandcastle, rather than waiting for one to be formed purely by chance), we must input energy, which increases the entropy of the surrounding air and thus resulting in a net entropy increase. This is the second law of thermodynamics; entropy always increases, and this principle underlies vast quantities of modern physics and chemistry.

If we extrapolate this situation backwards, we realise that the universe must have had a definite beginning at some point; a starting point of order from which things get steadily more chaotic, for order cannot increase infinitely as we look backwards in time. This suggests some point at which our current universe sprang into being, including all the laws of physics that make it up; but this cannot have occurred under ‘our’ laws of physics that we experience in the everyday universe, as they could not kickstart their own existence. There must, therefore, have been some other, higher power to get the clockwork universe in motion, destroying the image of it as some eternal, unquestionable predictive cycle. At the time, this was seen as vindicating the idea of the existence of God to start everything off; it would be some years before Edwin Hubble would venture the Big Bang Theory, but even now we understand next to nothing about the moment of our creation.

However, this argument wasn’t exactly a death knell for determinism; after all, the laws of physics could still describe our existing universe as a ticking clock, surely? True; the killer blow for that idea would come from Werner Heisenburg in 1927.

Heisenburg was a particle physicist, often described as the person who invented quantum mechanics (a paper which won him a Nobel prize). The key feature of his work here was the concept of uncertainty on a subatomic level; that certain properties, such as the position and momentum of a particle, are impossible to know exactly at any one time. There is an incredibly complicated explanation for this concerning wave functions and matrix algebra, but a simpler way to explain part of the concept concerns how we examine something’s position (apologies in advance to all physics students I end up annoying). If we want to know where something is, then the tried and tested method is to look at the thing; this requires photons of light to bounce off the object and enter our eyes, or hypersensitive measuring equipment if we want to get really advanced. However, at a subatomic level a photon of light represents a sizeable chunk of energy, so when it bounces off an atom or subatomic particle, allowing us to know where it is, it so messes around with the atom’s energy that it changes its velocity and momentum, although we cannot predict how. Thus, the more precisely we try to measure the position of something, the less accurately we are able to know its velocity (and vice versa; I recognise this explanation is incomplete, but can we just take it as red that finer minds than mine agree on this point). Therefore, we cannot ever measure every property of every particle in a given space, never mind the engineering challenge; it’s simply not possible.

This idea did not enter the scientific consciousness comfortably; many scientists were incensed by the idea that they couldn’t know everything, that their goal of an entirely predictable, deterministic universe would forever remain unfulfilled. Einstein was a particularly vocal critic, dedicating the rest of his life’s work to attempting to disprove quantum mechanics and back up his famous statement that ‘God does not play dice with the universe’. But eventually the scientific world came to accept the truth; that determinism was dead. The universe would never seem so sure and predictable again.