# Prime Numbers and Cryptography

## Abstract

There is a remarkable disparity between the degree of difficulty of the task of multiplication and that of factorization. Multiplying integers together is a reasonable exercise for a young child if the integers are small, and it remains a very straightforward task even when the integers are very large. The reverse operation, however, that of resolving a given integer into factors, is cumbersome except for the very smallest integers and becomes near to impossible for large numbers. This assymmetry is exploited in a new kind of cryptosystem, called RSA after its discoverers, Rivest, Shamir and Adleman. In the RSA system secrecy is provided by placing a would-be codebreaker in a situation where in principle he *commands all information necessary* for reading the protected message but is confronted with an arithmetic task which in practice is prohibitively time-consuming.

## Keywords

Encryption Algorithm Punctuation Mark Small Prime Factor Carmichael Number Multiple Precision Arithmetic## Bibliography

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