PDF 417 for visual Factorisation of large integers in .NET Maker ANSI/AIM Code 128 in .NET Factorisation of large integers

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Factorisation of large integers using none touse none in web,windows applicationpdf417 code examples It is a relatively ea none for none sy task to multiply two numbers together, particularly if we have a calculator available, provided that they are not too large. If the numbers are both less than 10 a child should be able to do it unaided. If they are both less than 100 most people, hopefully, could get the answer with paper and pencil.

If both numbers are bigger than 10 000 it is likely that a calculator would be employed.. Microsoft Office Excel Website chapter 13 The opposite problem, none for none nding two or more numbers which when multiplied together produce a given number, is called factorisation. This is a much harder task than multiplying numbers together, as anyone who has tried it knows. For example: if we are asked to multiply 89 by 103 we should be able to get the answer, 9167, in less than a minute.

If, however, we were asked to nd two numbers whose product is 9167 it would probably take us considerably longer. How would we do it . The standard method of factorisation If we are asked to fa none for none ctorise a large number, N, we should use the fact that unless N is a prime number it must have at least two factors and the smaller of these cannot exceed the square root of N. This means that in the case where N 9167 we need only test for divisibility by the primes less than 9167, which is nearly 96. The largest prime below 96 is 89, so in this case we would succeed on the very last test, by which time we would have carried out more than 20 divisions.

Had we carried out the tests on N 9161 we would have failed to nd a divisor, since 9161 is a prime. As N increases so does the number of tests that we have to make. Thus if N 988 027 N either is a prime or is divisible by some number less than 988 027, which is a little less than 994.

We would then try dividing 988 027 by each prime less than 994. If we nd a prime that divides 988 027 exactly, i.e.

without leaving a remainder, we have solved the problem. If no such number is found we would know that 988 027 is prime. In fact 988 027 991 997.

and since both 991 an d 997 are prime numbers the factorisation is complete. It would have taken us quite a lot of effort to do this because there are more than 160 primes less than 991, and we would have had to try them all before we were successful. Even with a calculator this would be a time-consuming and tedious job.

Someone who has a computer and can program could, of course, get the computer to do the work. Irrespective of how it was done, by increasing N from 9167 to 988 027, a factor of about 108, we, or the computer, were faced with an increase in the number of divisions from about 20 to over 160. Note that although N increased by a factor of over 100, so that N increased by a factor of more than 10, the.

Encipherment and the internet number of tests incre ased by a factor of only about 8. The explanation for this can be found in M21. This method of nding the prime factors of a number by dividing the number by each prime less than its square root is, essentially, due to Eratosthenes and is the standard method both for factorising (if it works) and for showing that a number is a prime (if it fails).

This is not the only method that might be used, sometimes a short-cut can be found; for example, someone might notice that 9167 9216 49 (96)2 (7)2 (96 7)(96 7) 89 103. and, even better, tha none none t 988 027 988 036 9 (994)2 (3)2 (994 991 997 3)(994 3). but, in general, we a re not so lucky. Sometimes, such as when the number that we are trying to factorise has a particular form such as 2p 1 where p is a prime number, there are special techniques which reduce the number of possibilities, but in the type of number which is relevant to the RSA system these special techniques are not applicable. The RSA system of encryption, which is described below, relies for its security upon this fact: that it is very time-consuming to factorise a large number even if we are told that it is the product of two large primes.

As for the encryption process in the RSA system the basis of this is an elegant and powerful theorem stated, without proof, by the French mathematician Pierre Fermat early in the seventeenth century. This is often referred to as Fermat s Little Theorem and is not to be confused with the notorious Fermat s Last Theorem , which he also stated without proof, and which was not proved until 1993 [13.2].

Fermat may have had a proof of his Little Theorem ; it is extremely unlikely that he had a proof of his Last Theorem . The Swiss mathematician Leonhard Euler gave a proof of Fermat s Little Theorem in 1760 and also generalised it, so giving us what is known as the Fermat Euler Theorem and it is this that is used in the RSA encryption/decryption process. As a preliminary it is instructive to look at some examples of.

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