Primes and Codes. Dr Bill Lionheart, Department of Mathematics, UMIST. October 4, 2001

Transcription

1 Primes and Codes Dr Bill Lionheart, Department of Mathematics, UMIST October 4, Introduction Codes have been used since ancient times to send secret messages. Today they are even more important. They are used in military communication and increasingly in commerce. Electronic methods of communications have made it easier to intercept messages. One no longer needs to shoot the carrier pigeon!). Digital computers enable us to use more sophisticated codes, but also give code crackers faster means of breaking them. With the rise in demand for codes created by the communications industry and the need for secure transactions on-line, mathematicians specializing in the theory of codes are in ever increasing demand. Most codes start by translating the message in to a (possibly very long) whole number. Of course letters are already represented by number in a computer so this step is fairly easy. Number theory is the branch of mathematics which is concerned with properties of whole numbers (integers). The main idea of coding a message is to get the number representing the message and do some arithmetic operation to it which can easily be undone, if you know the secret, but very hard to undo if you don t. Many of the most useful codes use operations on integers such as factorizing numbers as the product of prime numbers and finding the remainder when one integer is divided by another. So its not surprising that number theorists are involved in creating, and cracking codes. In this talk I will tell you a little bit of number theory, and also I will explain the theory behind a very important type of code which is widely used for sending messages over the internet and also for digitally signing messages. 1

2 2 Prime numbers and factorization A positive integer a is said to divide an other b if b ka for some other integer k. We also say that a is a divisor, or factor of b. Every integer has itself and one as a factor. A positive whole number other than one, which only itself and one as a factor is called a prime number. Here are the first few primes There are infinitely many prime numbers (Euler gave a clever and easy proof of this you can look up), but they get rarer as you go to bigger numbers. In fact the number of primes less than or equal to n gets closer to n logn as n gets bigger. This is called the Prime Number Theorem. (What kind of logarithms do mathematicians mean when they just write log? Hint: not necessarily the same as your calculator s designer!) Positive whole numbers can be factorized as a product of powers of primes, such as in exactly one way (given you write the primes in increasing order). It will be important to us to find large primes. It will also be important to us (somewhat surprisingly perhaps) that it can be very hard to factorize large numbers, even if they are just the product of two large primes. The point is there are better ways of testing to see if a number is prime than looking for factors. You already know more about this than the richest man in the world, who wrote The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers. (Bill Gates, The Road Ahead, Viking Penguin (1995), page 265). There were some early successes in finding large prime numbers by hand. For example in 1588 Pietro Cataldi found that and were prime. These are examples of what are called Mersenne Numbers, they are of the form 2 p 1 for a smaller prime number p. Not all Mersenne numbers are prime, but even when they are not they tend to have large prime factors, for example which was found to be prime by Landry in With the advent of digital computers, as well as clever new methods for finding primes, bigger ones were found. In 1949 Newman (a topologist) used the prototype Manchester computer (with a massive 1024 bits of memory!) to make the first computerized attempt to find Mersenne primes. In 1951 the record was 79 decimal digits. By the 1980s digit primes had been found, and today primes with over digits are known. 2

3 3 Modulo (or clock) arithmetic In a clock, (or other instrument with a dial where the hand goes around more than one revolution) we have a strange kind of arithmetic. For a clock and Imaging a rather more mathematical clock which has zero instead of 12 at the top. In the arithmetic of this clock face we remove any multiples of 12, counting them as zero. So 38 for example is the same as 2. If we divide a whole number by another, say 38 divided by 12 find it goes 3 times remainder 2, or We also say that 38 is congruent to 2 modulo 12, written 38 2 mod 12. More generally a b mod n when a b kn for some integer k. There is a slightly different notation where we denote by amod n the number in the range 0 to n 1 which is congruent to a modulo n. This operation is often implemented in computer languages as a function called also called mod. Notations in various computer languages include a modb mod a b and a%b. The usual rules of arithmetic (commutative laws and distributive laws) work just as well in modulo arithmetic. One strange and different effect is that we can t always cancel. Notice that mod 12 and we can t say that mod 12 implies 8 4 mod 12, because it does n t! This troublesome behaviour is because 12 is not prime. Modulo arithmetic with a prime modulus is much better behaved. (Try writing out a multiplication table for modulo five for example, can you always solve ax b for x, where a is not congruent to 0?) 4 Highest Common Factor If a number c is a factor of both a and b we say c is a common factor of a and b. For example 5 is a common factor of 100 and 30. The highest common factor (hcf) of two numbers, also called the greatest common divisor, is just the biggest number which goes exactly into both. For example hcf If the highest common factor is one, hcf a b 1 we say the numbers are coprime. They don t have to be prime numbers, just have no factors in common. For example and are coprime. Euclid s algorithm gives us a way of working out the highest common factor of two numbers and it is easy to program on a computer. Here is an example of how it works for finding hcf (which is 4)

4 What we are doing here is first take the remainder when the larger number is divides by the smaller one, then the remainder when the smaller one is divided by the remainder, and so on. Eventually we always get to zero and stop and the previous remainder is always the hcf. The method works as whenever c is a factor of a and b it is also a factor of a b a 2b As a spin-off of this method, we can see that it is always possible to find integers x and y such that xa yb hcf a b. For example so hcf Public key codes In a traditional code both parties need to know some secret information (the key) which allows them to encrypt (put in to code) and decrypt (translate back to plain text) the message. If you have heard about the World War II German Enigma coding machine, this secret key would be the setting of the wheels on the machine, determined by a secret book which listed the settings for each day. This sort of code suits diplomatic and military uses reasonably well, as the diplomats or soldiers can set off knowing the secret key to send messages back home, and to decrypt messages sent to them. In civilian uses, the need often arises to send coded messages to people you may not have met before (such as at your bank or an online vendor). Even in military situations there might be a number of allies who would need different keys for each pair of allies (in case they later fell out). The disadvantage is that you need to send the key by some secure courier (a pigeon perhaps?). With a public key code, the key has two parts, a public key and a private 4

5 2a 3a key. If you want to receive coded messages you publicize your public key (you can put it on the internet for example). The people sending you the message use the public key to encode their message, but it cant be decoded without the private key, which you keep secret. With this system it has to be very hard to find your private key from the public key. One traditional way to crack a code is to arrange for the a particular message to be sent, perhaps you sink The Bismark and wait for the message BISMARK SUNK to be sent (well probably in German). With a public key system everyone knows how to encrypt the message using the public key so this method of attack does n t work. One thing which does work, is if you know one of a small number of very short messages was sent such as YES or NO then you can encrypt those messages and compare it with the messages you intercept. (The moral is simply not to send very short predictable messages). 6 Fermat s Little Theorem You may well have heard of the famous Fermat s Last Theorem, which says that x n y n z n has no positive integer solutions for n a whole number bigger than 2. (Actually it was n t Fermat s theorem but conjecture, it was only recently proved by Wiles and Taylor). Well there is a connection between the proof of this and codes, but we wont go into that now. We re talking about a little result Fermat proved, which we need for our public key codes. Fermat s Little Theorem If hcf a p 1then a p 1 mod p 1 Proof Consider the list of p 1 numbers a 2a 3a p 1 a if ra sa mod p then r s a 0 mod p. Now as hcf a p 1 we can cancel giving r s mod p and we see that all the numbers in the list are different the list is just 1 2 p 1 is some order. Now consider the product of the numbers in the list a p 1 a 1 2 p 1 p 1! mod p Factorizing the left hand side we see p 1!a p 1 p 1! mod p 5

6 which means that p 1! a p mod p. Now as p 1! and p are coprime we can cancel giving the result we desire a p 1 mod p 1. Its not hard to generalize this theorem (to a special case of Euler s Theorem) to show that if N pq is a product of two distinct primes and hcf a N 1 then a p 1 q 1 1 mod N. The number φ N p 1 q 1 will be important. More generally φ n (Euler s phi function) is the number of positive integers less than n which are coprime to n. For a product of two distinct primes it happens to be p 1 q 1. Check this for a few small numbers if you like, or try to find a general proof. 7 The RSA public key code We are now ready to describe the public key cryptosystem (that is code) first published by Adleman Rivest and Shamir in First we choose two large prime numbers p and q and a positive whole number e 1 such that hcf e φ N 1. We publish N and e as our public keys (without saying what p and q are). If someone wants to send us a coded message, they take their message which is an integer m N and work out c m e mod N and this c is the coded message. The private key d 1 ed mod φ N φ N is chosen so that and we can always find such a d using Euclid s algorithm. Now here is the magic: we can decode by raising to the power d. I claim that m c d mod N In other words we need to prove bf RSA Theorem m. m ed mod N 6

7 Proof Now ed 1 kφ N for some k and if m is not divisible by p then m ed m 1 k p 1 q 1 m m p 1 q 1 k m1 k mod p and if m is divisible by p then mmod p m ed mod p 0 anyway. Similarly m m ed mod q 0. But as m ed m is divisible by distinct prime numbers p and q it must be divisible by pq so we have the desired result m m ed mod N. 8 A little example Here s an example taken from Koblitz book. Lets assume we have an alphabet of only the 26 uppercase letters and no space nor punctuation. We want to send the message YES to someone who publishes their public key as N and e Actually N but we don t know that. One neat way to convert the text message into a number is to treat it as a number base 26 with the digits A 0 B 1 Z 25. So just as base we have Y ES We now need to work out mod To do this it would not be possible to work out then find the remainder. How many decimal digits do you think that would be? Instead we use the repeated squaring method see the prize challenge page for details. The answer is 2116 which is our encoded text message. As text it is BFIC. The person we sent the message to has the private key d works out mod which was our original message YES. If we wanted to send a longer message, we would split it into blocks of three characters. 9 Digital Signatures Often it is important to provide the reader with some evidence that some text was indeed written by the claimed author. When sending electronic mail it is very easy to pretend to be someone else. An important example is software, which you would like to be sure is as it was written by the trusted author, and has n t been tampered with. For example to check it has no computer viruses added. One way to digitally sign a piece of text is to send both the plain text and a coded 7

8 version coded using your private key. That you raise it to the power d and take the remainder modulo N. The recipient then does the same using your public key, and compares this with the plain text. If it is the same then the sender must have known the private key. In practice this is a bit awkward for long messages, so one applies a function to the message and encrypts that. The reader applies the same function to your plain text message. It is important that the function is what is called a trap door function. That a function f which is easy to apply but very hard to find a x such that f x is a given y. That way it is hard to alter the message without changing the signature. One popular trapdoor function is called the discrete logarithm. 10 Further reading One of our first year courses at UMIST, 117, covers this material in more detail with computer examples. You can look at all the handouts and example sheets on Dr Richard Booth s web site There is a nice little article in the magazine Mathematical Spectrum 1998/99 Volume 331 Number 1. The RSA algorithm: a public-key cryptosystem by Michael Williams and Linda Allen. Ask you teacher if they have a copy. Much more detail, some of it accessible to you and some perhaps too advanced for you at the moment, can be found in the book A course in number theory and cryptography by Neal Koblitz, published by Springer Verlag, second edition, You can find some interesting information about the history of primes at the web site The Largest Known Prime by Year: A Brief History, by Chris Caldwell: year.html. 8

9 Prize Challenge We put out this problem as a prize challenge on a visit to Schools and colleges in the Chesterfield area earlier in 2001, the first student to crack it took only two days to the correct answer and won the prize. Please feel free to have a go yourself, but sorry the prize has already gone (maybe I should have made it harder). If my public key is N 899 e 143 can you find my private key d? (The numbers are small so its not very secure). I ve been sent a message with 8 letters. Using the system A 0 B 1 Z 25 it has been coded using RSA in blocks of two letters. Suppose you have intercepted this message Can you crack the code? For the first part, finding the private key, you can do by hand. To do decryption you will almost certainly need a calculator (!) and it will be much easier if you have a computer and can use a programming language or a spread-sheet. Even if your computer system can handle long integers, raising numbers to very large powers then doing mod to them won t work. If you want to calculate x ab mod c you can calculate x a mod c b mod c instead. If the power you want only has large factors you are still stuck. Here is the trick. Find the binary expansion of the power first. For example so the binary expansion of 9 is If we want to work out mod 17 first we calculate 105mod 17 3, then mod mod 17 2 mod 17 9mod Now mod mod 17 2 mod 17 81mod and finally mod mod And so mod mod mod mod