Read Online Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart - Free Book Online
Authors: Ian Stewart
Tags: General, Mathematics
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big number is extremely hard - so hard that in practice it can’t be done, no matter how big and fast your computer might be. Finding big primes is much easier, and so is multiplying them together.
Of course, in my example, with impractically small numbers, finding the decoding key 37 is easy. Alice could work it out, and so could any eavesdropper. But with 100-digit primes, say, calculating the decoding key seems to be impossible if all you know is the product of the two primes. On the other hand, if you do know the primes, then it is relatively straightforward to find the decoding key. That’s why it’s possible to set up the system to begin with.
Systems like RSA are very suitable for the internet, where every user has to ‘know’ how to send an encrypted message (such as a credit card number). That is, the method for encrypting this message has to be stored on their computer. So a skilled programmer could find it. But only the bank needs to know the decryption key. So until criminals discover efficient ways to factorise big numbers into primes, your money is safe. Assuming it’s safe in the hands of the banks, which has suddenly become questionable.
In practical applications, some care has to be taken and the method isn’t quite this simple. See, for example: en.wikipedia.org/wiki/RSA
It is also worth remarking that, in practice, RSA is mainly used for sending encrypted versions of keys to other, simpler cryptosystems, which can then be used to send messages, rather than using RSA for the messages themselves. RSA involves a bit too much computational time to be used routinely for messages.
There is a curious historical postscript to this story. In 1973, the same method was invented by Clifford Cocks, a mathematician working for British Intelligence, but it was considered impractical at the time. Because his work was
classified Top Secret, no one knew about his anticipation of the RSA system until 1997.
Calendar Magic
‘My beautiful assistant,’ stated the Great Whodunni, ‘will now hand me a perfectly ordinary calendar.’
Grumpelina smiled sweetly and did as instructed. It was indeed an ordinary calendar, with seven columns per month, headed by the days Sunday-Saturday, and the numbers of the days written in order.
Whodunni then called for a volunteer from the audience, while Grumpelina blindfolded him (Whodunni, that is).
‘I want you to choose any month from the calendar, and then draw a 3×3 square round nine dates. Don’t include any blank spaces. I will then ask you to tell me the smallest number from those dates, and I will instantly tell you what the nine numbers add up to.’
The volunteer did so, and, as it happened, he chose a square of dates for which the smallest number was 11. As soon as he told the magician this number, Whodunni immediately replied ‘171’.
Whodunni’s method works whichever 3×3 square is chosen. How does he do it?
Answer on page 289
The volunteer’s choice.
Mathematical Cats
Isaac Newton, it is said, 15 had a cat. He cut a hole in the bottom of his study door so that puss could get in and out. So we must add to Newton’s achievements the invention of the catflap, except that his version lacked the flap. Anyway, the cat had kittens. So Newton cut a small hole in the door next to the bigger one.
I don’t know whether Lewis Carroll - pen-name of the mathematician Charles Lutwidge Dodgson - had a cat, but he created one of the most memorable fictional cats: the Cheshire Cat, which slowly faded away until only its grin remained. The Cheshire isn’t a breed of cat: it is an English county where cheese was (and still is) made. Possibly Carroll was referring to the British shorthair, a breed of cat that appeared on Cheshire Cheese labels.
The Cheshire Cat.
Problem 79 of the ancient Egyptian Rhind Papyrus (pages 77-8) poses the sum
houses
7
cats
49
mice
343
wheat seed
2,401
hekat
16,807
(a hekat is a measure of volume)
TOTAL
19,607
where each number is
Of course, in my example, with impractically small numbers, finding the decoding key 37 is easy. Alice could work it out, and so could any eavesdropper. But with 100-digit primes, say, calculating the decoding key seems to be impossible if all you know is the product of the two primes. On the other hand, if you do know the primes, then it is relatively straightforward to find the decoding key. That’s why it’s possible to set up the system to begin with.
Systems like RSA are very suitable for the internet, where every user has to ‘know’ how to send an encrypted message (such as a credit card number). That is, the method for encrypting this message has to be stored on their computer. So a skilled programmer could find it. But only the bank needs to know the decryption key. So until criminals discover efficient ways to factorise big numbers into primes, your money is safe. Assuming it’s safe in the hands of the banks, which has suddenly become questionable.
In practical applications, some care has to be taken and the method isn’t quite this simple. See, for example: en.wikipedia.org/wiki/RSA
It is also worth remarking that, in practice, RSA is mainly used for sending encrypted versions of keys to other, simpler cryptosystems, which can then be used to send messages, rather than using RSA for the messages themselves. RSA involves a bit too much computational time to be used routinely for messages.
There is a curious historical postscript to this story. In 1973, the same method was invented by Clifford Cocks, a mathematician working for British Intelligence, but it was considered impractical at the time. Because his work was
classified Top Secret, no one knew about his anticipation of the RSA system until 1997.
Calendar Magic
‘My beautiful assistant,’ stated the Great Whodunni, ‘will now hand me a perfectly ordinary calendar.’
Grumpelina smiled sweetly and did as instructed. It was indeed an ordinary calendar, with seven columns per month, headed by the days Sunday-Saturday, and the numbers of the days written in order.
Whodunni then called for a volunteer from the audience, while Grumpelina blindfolded him (Whodunni, that is).
‘I want you to choose any month from the calendar, and then draw a 3×3 square round nine dates. Don’t include any blank spaces. I will then ask you to tell me the smallest number from those dates, and I will instantly tell you what the nine numbers add up to.’
The volunteer did so, and, as it happened, he chose a square of dates for which the smallest number was 11. As soon as he told the magician this number, Whodunni immediately replied ‘171’.
Whodunni’s method works whichever 3×3 square is chosen. How does he do it?
Answer on page 289
The volunteer’s choice.
Mathematical Cats
Isaac Newton, it is said, 15 had a cat. He cut a hole in the bottom of his study door so that puss could get in and out. So we must add to Newton’s achievements the invention of the catflap, except that his version lacked the flap. Anyway, the cat had kittens. So Newton cut a small hole in the door next to the bigger one.
I don’t know whether Lewis Carroll - pen-name of the mathematician Charles Lutwidge Dodgson - had a cat, but he created one of the most memorable fictional cats: the Cheshire Cat, which slowly faded away until only its grin remained. The Cheshire isn’t a breed of cat: it is an English county where cheese was (and still is) made. Possibly Carroll was referring to the British shorthair, a breed of cat that appeared on Cheshire Cheese labels.
The Cheshire Cat.
Problem 79 of the ancient Egyptian Rhind Papyrus (pages 77-8) poses the sum
houses
7
cats
49
mice
343
wheat seed
2,401
hekat
16,807
(a hekat is a measure of volume)
TOTAL
19,607
where each number is
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