A Public-Key Cryptography Algorithm which uses Prime Factorization as the Trapdoor Function. Define

(1) |

(2) |

(3) |

Let the message be converted to a number . The sender then makes and public and sends

(4) |

(5) |

(6) |

It is possible to break the cryptosystem by repeated encryption if a unit of
has small
Order (Simmons and Norris 1977, Meijer 1996), where
is the Ring of
Integers between 0 and under addition and multiplication (mod ). Meijer (1996) shows that
``almost'' every encryption exponent is safe from breaking using repeated encryption for factors of the form

(7) | |||

(8) |

where

(9) | |||

(10) |

and , , , , , and are all Primes. In this case,

(11) | |||

(12) |

Meijer (1996) also suggests that and should be of order 10

Using the RSA system, the identity of the sender can be identified as genuine without revealing his private code.

**References**

Honsberger, R. *Mathematical Gems III.* Washington, DC: Math. Assoc. Amer., pp. 166-173, 1985.

Meijer, A. R. ``Groups, Factoring, and Cryptography.'' *Math. Mag.* **69**, 103-109, 1996.

Rivest, R. L. ``Remarks on a Proposed Cryptanalytic Attack on the MIT Public-Key Cryptosystem.'' *Cryptologia* **2**, 62-65, 1978.

Rivest, R.; Shamir, A.; and Adleman, L. ``A Method for Obtaining Digital Signatures and Public Key Cryptosystems.''
*Comm. ACM* **21**, 120-126, 1978.

RSA Data Security. A Security Dynamics Company. http://www.rsa.com.

Simmons, G. J. and Norris, M. J. ``Preliminary Comments on the MIT Public-Key Cryptosystem.'' *Cryptologia* **1**, 406-414, 1977.

© 1996-9

1999-05-25