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RSA Another concept that made it into Mathematical Games shortly aer its discovery was public key cryptography. In mid-1977, mathematicians Ron Rivest, Adi Shamir and Leonard Adleman invented the method of encryption now known as RSA (the initials of their surnames). Here, messages are encoded using two publicly shared numbers, or keys. These numbers and the method used to encrypt messages can be publicly shared as knowing this information does not reveal how to decrypt the message. Rather, decryption of the message requires knowing the prime factors of one of the keys. If this key is the product of two very large prime numbers, then this is a very difficult task.

Something to think about

Encrypting with RSA To encode the message 809, we will use the public key: s = 19 and r = 1769 The encoded message is the remainder when the message to the power of s is divided by r : 80919 ≡ 388 mod 1769 Decrypting with RSA To decode the message, we need the two prime factors of r (29 and 61). We multiply one less than each of these together: a = (29 − 1) × (61 − 1) = 1680. We now need to find a number t such that st ≡ 1 mod a. Or in other words: 19t ≡ 1 mod 1680 One solution of this equation is t = 619 (calculated via the extended Euclidean algorithm). Then we calculate the remainder when the encoded message to the power of t is divided by r : 388619 ≡ 809 mod 1769

Gardner had no education in maths beyond high school, and at times had difficulty understanding the material he was writing about. He believed, however, that this was a strength and not a weakness: his struggle to understand led him to write in a way that other nonmathematicians could follow. This goes a long way to explaining the popularity of his column. Aer Gardner finished working on the column, it was continued by Douglas Hofstadter and then AK Dewney before being passed down to Ian Stewart (see pages 4–9). Gardner died in May 2010, leaving behind hundreds of books and articles. There could be no beer way to end than with something for you to go away and think about. These of course all come from Martin Gardner’s Mathematical Games: • Find a number base other than 10 in which 121 is a perfect square. • Why do mirrors reverse le and right, but not up and down? • Every square of a 5-by-5 chessboard is occupied by a knight. Is it possible for all 25 knights to move simultaneously in such a way that at the finish all cells are still occupied as before?

Mahew Scroggs came from Lower Brailes, and is now a PhD student at UCL working on finite and boundary element methods. His website,, and Twier, @mscroggs, contain lots of maths, almost all of which originally came from Martin Gardner’s articles.


Chalkdust, Issue 03  

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