The reader will now appreciate that the collection of counting numbers has, from the earliest times, been recognized as the repository of riddles and secrets, many of which have never been revealed to this day. For many of us, this is enough to justify the continued serious study of numbers but others may take a different attitude. Intriguing and difficult as these conundrums may be, it might be imagined that they have little bearing on the rest of human wisdom. But that would be a mistake.
Over the last few decades it has emerged that ordinary secrets, of thekindsweallindulgeinfromtimetotime,canbecodedas secrets about numbers. This has now all been put into practice and our most precious secrets, whether they be commercial or military, personal orfinancial, political or downright scandalous,can all be protected on the Internet by masking them using secrets about ordinary counting numbers.
Secrets turned into numbers
How is all this possible?Any information, whether it be a poem or a bank statement, a blueprint for a weapon or a computer program, can be described in words. We may, however, need to augment the alphabet that is used to make up our words beyond the ordinary letters of the alphabet. We may include number symbols, punctuation symbols including special symbols for space between ordinary words, but it is nonetheless the case that all the information we wish to transfer, including instructions for producing pictures and diagrams, can be expressed using words from an alphabet of, let us say, no more than one thousand symbols. We can count these symbols and so represent each symbol uniquely as a number. Since numbers are cheap and inexhaustible, it may be convenient to use numbers all with the same number of digits for this purpose(so, for example, every symbol was represented uniquely by its own four-digit PIN). We could string the symbols together as required to give one big long numberthattoldtheentirestory.Wecanevenworkinbinaryif we wish and so devise a way of translating any information into one long string of 0s and 1s. Every message we might ever want to send could then be coded as a binary string and then decoded at the other end by a suitably programmed computer, to be compiled in ordinary language that we can all comprehend. This then is the first realization:in order to send messages between one person and another, it is enough, both in theory and in practice, to be able to send numbers from one person to another.