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.
Encryption Algorithm Punctuation Mark Small Prime Factor Carmichael Number Multiple Precision Arithmetic
This is a preview of subscription content, log in to check access
Claude Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal27 (1948) pp. 379–423 and 623–656.MathSciNetMATHGoogle Scholar
. Martin Gardner, “A New Kind of Cipher that Would Take Millions of Years to Break;” Scientific Am.237 (Aug. 1977) pp. 120–124.CrossRefGoogle Scholar
R. L. Rivest, A. Shamir, and L. Adleman, “A Method For Obtaining Digital Signatures and Public-Key Cryptosystems,” Comm. ACM21 (1978) pp. 120–126.MathSciNetMATHCrossRefGoogle Scholar
Alan G. Konheim, Cryptography: A Primer, Wiley-Interscience, New York, 1981.MATHGoogle Scholar
. János Pintz, William L. Steiger, and Endre Szemerédi, “Infinite Sets of Primes with Fast Primality Tests and Quick Generation of Large Primes,” Math. Comp.53 (1989) pp. 399–406.MathSciNetMATHCrossRefGoogle Scholar