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# How do asymetrical cyphers work mathmatically speaking. Can...

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 Question by Flunky Submitted on 11/14/2003 Related FAQ: Cryptography FAQ (01/10: Overview) Rating: Not yet rated Rate this question: N/A Worst Weak OK Good Great How do asymetrical cyphers work mathmatically speaking.  Can someone please show me an example of an equation that has a public and private key.  I don't understand how it all works

 Answer by Regie Submitted on 12/16/2003 Rating: Rate this answer: N/A Worst Weak OK Good Great Mathematically, the public and private keys are simply two factors of a prime number.  The prime number should be rather large to be practical. The “secrecy” is due only to the exponential nature of current factoring algorithms and the speed of currently architected computers. This is what keeps the private key “secret”, yet with the knowledge of the private keys, the public key can be computed in seconds.  Take the following examples: 1.  What is the number whose prime factors are 77237 and 3217? 2.  What are the prime factors of the number 77219? In example 1, the answer can be arrived at quickly by simply multiplying the numbers. In example 2, we can derive a simple brute force algorithm that looks like this: Start at the number 2, divide it into 77219, if there is a remainder, increment 2 to 3 and continue until the division has no remainder. This takes 36 steps for this example, and if implemented on a slow (human) computer, will take a number of minutes longer than the first example. The number of steps to compute the factor increases exponentially, based on the length of the number.  For a number 100 digits long, this becomes billions of years, even for a fast computer that can complete 10,000,000,000 divisions per second (see http://qubit.org/intros/cyptana.html). It is this degree of difficulty, from a time perspective, that makes current public key cryptography “secure”.

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