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<The Mathematics Behind Quantum Computing: Part I ⁄ >

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Communication security today is almost universally ensured by the use of RSA encryption. This method relies on the inaccessibility of large prime factors of a large composite number. The problem is an artificial one: The encrypter takes two (or more) large primes and multiplies them. The decrypter tries to work backwards from the product to the factors. It is hard work. The largest number factored so far ("RSA-640") had 193 decimal digits and took "approximately 30 2.2GHz-Opteron-CPU years," over five months of calendar time. At that rate a 1024-bit number, the size currently recommended by a commercial cryptology site, would take on the order of 10145 years ("bit" is short for "binary digit;" each additional bit contributes a factor of 2 to the size of the calculation). The site adds: "For more security or if you are paranoid, use 2048 or even 4096 bits."

A quantum computer of suitable size could factor these large numbers in a much shorter time. For a 1024-bit number, Shor's Algorithm requires on the order of 10243, about one billion, operations. I do not have any information on how quickly quantum operations can be executed, but if each one took one second our factorization would last 34 years. If a quantum computer could run at the speed of today's electronic computers (100 million instructions per second and up) then factorization of the 1024-bit number would be a matter of seconds.

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