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…

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Up one level

In my last post (well, last excepting Wednesday’s little topical deviation), I talked about the real nuts and bolts of a computer, detailing the function of the transistors that are so vital to the workings of a computer. Today, I’m going to take one step up and study a slightly broader picture, this time concerned with the integrated circuits that utilise such components to do the real grunt work of computing.

An integrated circuit is simply a circuit that is not comprised of multiple, separate, electronic components- in effect, whilst a standard circuit might consist of a few bits of metal and plastic connected to one another by wires, in an IC they are all stuck in the same place and all assembled as one. The main advantage of this is that since all the components don’t have to be manually stuck to one another, but are built in circuit form from the start, there is no worrying about the fiddliness of assembly and they can be mass-produced quickly and cheaply with components on a truly microscopic scale. They generally consist of several layers on top of the silicon itself, simply to allow space for all of the metal connecting tracks and insulating materials to run over one another (this pattern is usually, perhaps ironically, worked out on a computer), and the sheer detail required of their manufacture surely makes it one of the marvels of the engineering world.

But… how do they make a computer work? Well, let’s start by looking at a computer’s memory, which in all modern computers takes the form of semiconductor memory. Memory takes the form of millions upon millions of microscopically small circuits known as memory circuits, each of which consists of one or more transistors. Computers are electronic, meaning to only thing they understand is electricity- for the sake of simplicity and reliability, this takes the form of whether the current flowing in a given memory circuit is ‘on’ or ‘off’. If the switch is on, then the circuit is represented as a 1, or a 0 if it is switched off. These memory circuits are generally grouped together, and so each group will consist of an ordered pattern of ones and zeroes, of which there are many different permutations. This method of counting in ones and zeroes is known as binary arithmetic, and is sometimes thought of as the simplest form of counting. On a hard disk, patches of magnetically charged material represent binary information rather than memory circuits.

Each little memory circuit, with its simple on/off value, represents one bit of information. 8 bits grouped together forms a byte, and there may be billions of bytes in a computer’s memory. The key task of a computer programmer is, therefore, to ensure that all the data that a computer needs to process is written in binary form- but this pattern of 1s and 0s might be needed to represent any information from the content of an email to the colour of one pixel of a video. Clearly, memory on its own is not enough, and the computer needs some way of translating the information stored into the appropriate form.

A computer’s tool for doing this is known as a logic gate, a simple electronic device consisting of (you guessed it) yet more transistor switches. This takes one or two inputs, either ‘on’ or ‘off’ binary ones, and translates them into another value. There are three basic types:  AND gates (if both inputs equal 1, output equals 1- otherwise, output equals 0), OR gates (if either input equals 1, output equals 1- if both inputs equal 0, output equals 0), and NOT gates (if input equals 1, output equals 0, if input equals 0, output equals 1). The NOT gate is the only one of these with a single input, and combinations of these gates can perform other functions too, such as NAND (not-and) or XOR (exclusive OR; if either input equals 1, output equals 1, but if both inputs equal 1 or 0, output equals 0) gates. A computer’s CPU (central processing unit) will contain hundreds of these, connected up in such a way as to link various parts of the computer together appropriately, translate the instructions of the memory into what function a given program should be performing, and thus cause the relevant bit (if you’ll pardon the pun) of information to translate into the correct process for the computer to perform.

For example, if you click on an icon on your desktop, your computer will put the position of your mouse and the input of the clicking action through an AND gate to determine that it should first highlight that icon. To do this, it orders the three different parts of each of the many pixels of that symbol to change their shade by a certain degree, and the the part of the computer responsible for the monitor’s colour sends a message to the Arithmetic Logic Unit (ALU), the computer’s counting department, to ask what the numerical values of the old shades plus the highlighting is, to give it the new shades of colour for the various pictures. Oh, and the CPU should also open the program. To do this, its connections send a signal off to the memory to say that program X should open now. Another bit of the computer then searches through the memory to find program X, giving it the master ‘1’ signal that causes it to open. Now that it is open, this program routes a huge amount of data back through the CPU to tell it to change the pattern of pretty colours on the screen again, requiring another slue of data to go through the ALU, and that areas of the screen A, B and C are now all buttons, so if you click there then we’re going to have to go through this business all over again. Basically the CPU’s logical function consists of ‘IF this AND/OR this happens, which signal do I send off to ask the right part of the memory what to do next?’. And it will do all this in a miniscule fraction of a second. Computers are amazing.

Obviously, nobody in their right mind is going to go through the whole business of telling the computer exactly what to do with each individual piece of binary data manually, because if they did nothing would ever get done. For this purpose, therefore, programmers have invented programming languages to translate their wishes into binary, and for a little more detail about them, tune in to my final post on the subject…

What we know and what we understand are two very different things…

If the whole Y2K debacle over a decade ago taught us anything, it was that the vast majority of the population did not understand the little plastic boxes known as computers that were rapidly filling up their homes. Nothing especially wrong or unusual about this- there’s a lot of things that only a few nerds understand properly, an awful lot of other stuff in our life to understand, and in any case the personal computer had only just started to become commonplace. However, over 12 and a half years later, the general understanding of a lot of us does not appear to have increased to any significant degree, and we still remain largely ignorant of these little feats of electronic witchcraft. Oh sure, we can work and operate them (most of us anyway), and we know roughly what they do, but as to exactly how they operate, precisely how they carry out their tasks? Sorry, not a clue.

This is largely understandable, particularly given the value of ‘understand’ that is applicable in computer-based situations. Computers are a rare example of a complex system that an expert is genuinely capable of understanding, in minute detail, every single aspect of the system’s working, both what it does, why it is there, and why it is (or, in some cases, shouldn’t be) constructed to that particular specification. To understand a computer in its entirety, therefore, is an equally complex job, and this is one very good reason why computer nerds tend to be a quite solitary bunch, with quite few links to the rest of us and, indeed, the outside world at large.

One person who does not understand computers very well is me, despite the fact that I have been using them, in one form or another, for as long as I can comfortably remember. Over this summer, however, I had quite a lot of free time on my hands, and part of that time was spent finally relenting to the badgering of a friend and having a go with Linux (Ubuntu if you really want to know) for the first time. Since I like to do my background research before getting stuck into any project, this necessitated quite some research into the hows and whys of its installation, along with which came quite a lot of info as to the hows and practicalities of my computer generally. I thought, then, that I might spend the next couple of posts or so detailing some of what I learned, building up a picture of a computer’s functioning from the ground up, and starting with a bit of a history lesson…

‘Computer’ was originally a job title, the job itself being akin to accountancy without the imagination. A computer was a number-cruncher, a supposedly infallible data processing machine employed to perform a range of jobs ranging from astronomical prediction to calculating interest. The job was a fairly good one, anyone clever enough to land it probably doing well by the standards of his age, but the output wasn’t. The human brain is not built for infallibility and, not infrequently, would make mistakes. Most of these undoubtedly went unnoticed or at least rarely caused significant harm, but the system was nonetheless inefficient. Abacuses, log tables and slide rules all aided arithmetic manipulation to a great degree in their respective fields, but true infallibility was unachievable whilst still reliant on the human mind.

Enter Blaise Pascal, 17th century mathematician and pioneer of probability theory (among other things), who invented the mechanical calculator aged just 19, in 1642. His original design wasn’t much more than a counting machine, a sequence of cogs and wheels so constructed as to able to count and convert between units, tens, hundreds and so on (ie a turn of 4 spaces on the ‘units’ cog whilst a seven was already counted would bring up eleven), as well as being able to work with currency denominations and distances as well. However, it could also subtract, multiply and divide (with some difficulty), and moreover proved an important point- that a mechanical machine could cut out the human error factor and reduce any inaccuracy to one of simply entering the wrong number.

Pascal’s machine was both expensive and complicated, meaning only twenty were ever made, but his was the only working mechanical calculator of the 17th century. Several, of a range of designs, were built during the 18th century as show pieces, but by the 19th the release of Thomas de Colmar’s Arithmometer, after 30 years of development, signified the birth of an industry. It wasn’t a large one, since the machines were still expensive and only of limited use, but de Colmar’s machine was the simplest and most reliable model yet. Around 3,000 mechanical calculators, of various designs and manufacturers, were sold by 1890, but by then the field had been given an unexpected shuffling.

Just two years after de Colmar had first patented his pre-development Arithmometer, an Englishmen by the name of Charles Babbage showed an interesting-looking pile of brass to a few friends and associates- a small assembly of cogs and wheels that he said was merely a precursor to the design of a far larger machine: his difference engine. The mathematical workings of his design were based on Newton polynomials, a fiddly bit of maths that I won’t even pretend to understand, but that could be used to closely approximate logarithmic and trigonometric functions. However, what made the difference engine special was that the original setup of the device, the positions of the various columns and so forth, determined what function the machine performed. This was more than just a simple device for adding up, this was beginning to look like a programmable computer.

Babbage’s machine was not the all-conquering revolutionary design the hype about it might have you believe. Babbage was commissioned to build one by the British government for military purposes, but since Babbage was often brash, once claiming that he could not fathom the idiocy of the mind that would think up a question an MP had just asked him, and prized academia above fiscal matters & practicality, the idea fell through. After investing £17,000 in his machine before realising that he had switched to working on a new and improved design known as the analytical engine, they pulled the plug and the machine never got made. Neither did the analytical engine, which is a crying shame; this was the first true computer design, with two separate inputs for both data and the required program, which could be a lot more complicated than just adding or subtracting, and an integrated memory system. It could even print results on one of three printers, in what could be considered the first human interfacing system (akin to a modern-day monitor), and had ‘control flow systems’ incorporated to ensure the performing of programs occurred in the correct order. We may never know, since it has never been built, whether Babbage’s analytical engine would have worked, but a later model of his difference engine was built for the London Science Museum in 1991, yielding accurate results to 31 decimal places.

…and I appear to have run on a bit further than intended. No matter- my next post will continue this journey down the history of the computer, and we’ll see if I can get onto any actual explanation of how the things work.

The Problems of the Real World

My last post on the subject of artificial intelligence was something of a philosophical argument on its nature- today I am going to take on a more practical perspective, and have a go at just scratching the surface of the monumental challenges that the real world poses to the development of AI- and, indeed, how they are (broadly speaking) solved.

To understand the issues surrounding the AI problem, we must first consider what, in the strictest sense of the matter, a computer is. To quote… someone, I can’t quite remember who: “A computer is basically just a dumb adding machine that counts on its fingers- except that it has an awful lot of fingers and counts terribly fast”. This, rather simplistic model, is in fact rather good for explaining exactly what it is that computers are good and bad at- they are very good at numbers, data crunching, the processing of information. Information is the key thing here- if something can be inputted into a computer purely in terms of information, then the computer is perfectly capable of modelling and processing it with ease- which is why a computer is very good at playing games. Even real-world problems that can be expressed in terms of rules and numbers can be converted into computer-recognisable format and mastered with ease, which is why computers make short work of things like ballistics modelling (such as gunnery tables, the US’s first usage of them), and logical games like chess.

However, where a computer develops problems is in the barrier between the real world and the virtual. One must remember that the actual ‘mind’ of a computer itself is confined exclusively to the virtual world- the processing within a robot has no actual concept of the world surrounding it, and as such is notoriously poor at interacting with it. The problem is twofold- firstly, the real world is not a mere simulation, where rules are constant and predictable; rather, it is an incredibly complicated, constantly changing environment where there are a thousand different things that we living humans keep track of without even thinking. As such, there are a LOT of very complicated inputs and outputs for a computer to keep track of in the real world, which makes it very hard to deal with. But this is merely a matter of grumbling over the engineering specifications and trying to meet the design brief of the programmers- it is the second problem which is the real stumbling block for the development of AI.

The second issue is related to the way a computer processes information- bit by bit, without any real grasp of the big picture. Take, for example, the computer monitor in front of you. To you, it is quite clearly a screen- the most notable clue being the pretty pattern of lights in front of you. Now, turn your screen slightly so that you are looking at it from an angle. It’s still got a pattern of lights coming out of it, it’s still the same colours- it’s still a screen. To a computer however, if you were to line up two pictures of your monitor from two different angles, it would be completely unable to realise that they were the same screen, or even that they were the same kind of objects. Because the pixels are in a different order, and as such the data’s different, the two pictures are completely different- the computer has no concept of the idea that the two patterns of lights are the same basic shape, just from different angles.

There are two potential solutions to this problem. Firstly, the computer can look at the monitor and store an image of it from every conceivable angle with every conceivable background, so that it would be able to recognise it anywhere, from any viewpoint- this would however take up a library’s worth of memory space and be stupidly wasteful. The alternative requires some cleverer programming- by training the computer to spot patterns of pixels that look roughly similar (either shifted along by a few bytes, or missing a few here and there), they can be ‘trained’ to pick out basic shapes- by using an algorithm to pick out changes in colour (an old trick that’s been used for years to clean up photos), the edges of objects can be identified and separate objects themselves picked out. I am not by any stretch of the imagination an expert in this field so won’t go into details, but by this basic method a computer can begin to step back and begin to look at the pattern of a picture as a whole.

But all that information inputting, all that work…  so your computer can identify just a monitor? What about all the myriad of other things our brains can recognise with such ease- animals, buildings, cars? And we haven’t even got on to differentiating between different types of things yet… how will we ever match the human brain?

This idea presented a big setback for the development of modern AI- so far we have been able to develop AI that allows one computer to handle a few real-world tasks or applications very well (and in some cases, depending on the task’s suitability to the computational mind, better than humans), but scientists and engineers were presented with a monumental challenge when faced with the prospect of trying to come close to the human mind (let alone its body) in anything like the breadth of tasks it is able to perform. So they went back to basics, and began to think of exactly how humans are able to do so much stuff.

Some of it can be put down to instinct, but then came the idea of learning. The human mind is especially remarkable in its ability to take in new information and learn new things about the world around it- and then take this new-found information and try to apply it to our own bodies. Not only can we do this, but we can also do it remarkably quickly- it is one of the main traits which has pushed us forward as a race.

So this is what inspires the current generation of AI programmers and robotocists- the idea of building into the robot’s design a capacity for learning. The latest generation of the Japanese ‘Asimo’ robots can learn what various objects presented to it are, and is then able to recognise them when shown them again- as well as having the best-functioning humanoid chassis of any existing robot, being able to run and climb stairs. Perhaps more excitingly are a pair of robots currently under development that start pretty much from first principles, just like babies do- first they are presented with a mirror and learn to manipulate their leg motors in such a way that allows them to stand up straight and walk (although they aren’t quite so good at picking themselves up if they fail in this endeavour). They then face one another and begin to demonstrate and repeat actions to one another, giving each action a name as they do so.  In doing this they build up an entirely new, if unsophisticated, language with which to make sense of the world around them- currently, this is just actions, but who knows what lies around the corner…