#### Abstract Title

Characterizing Finite Fields With Anomalous Elliptic Curves

#### Acknowledgement of Funding Sources

The project described was supported by the National Science Foundation REU Program at Boise State University under Award No. DMS-1359425, Department of Mathematics, College of Arts and Sciences, and Division of Research and Economic Development at Boise State University.

#### Disciplines

Number Theory

#### Abstract

With the dramatically increasing volume of sensitive information being transmitted wirelessly via the Internet, information security is a critical component of numerous applications. Cryptography, in turn, has become a fundamental component of secure applications. The strength and performance advantages of Elliptic Curve Cryptography make it an effective solution to secure data transmission. The security of Elliptic Curve Cryptography relies on the difficulty of solving the discrete logarithm problem. Depending on the curve chosen, solving the discrete logarithm problem ranges from nearly impossible to extremely easy. In particular, anomalous curves are weak for cryptographic purposes. Closely related are anomalous primes, which are orders of anomalous curves. We have shown that anomalous primes are precisely those primes which can be written as a difference of consecutive cubes, and have generalized this result to finite fields of non-prime order. These results efficiently identify the anomalous primes.

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Characterizing Finite Fields With Anomalous Elliptic Curves

With the dramatically increasing volume of sensitive information being transmitted wirelessly via the Internet, information security is a critical component of numerous applications. Cryptography, in turn, has become a fundamental component of secure applications. The strength and performance advantages of Elliptic Curve Cryptography make it an effective solution to secure data transmission. The security of Elliptic Curve Cryptography relies on the difficulty of solving the discrete logarithm problem. Depending on the curve chosen, solving the discrete logarithm problem ranges from nearly impossible to extremely easy. In particular, anomalous curves are weak for cryptographic purposes. Closely related are anomalous primes, which are orders of anomalous curves. We have shown that anomalous primes are precisely those primes which can be written as a difference of consecutive cubes, and have generalized this result to finite fields of non-prime order. These results efficiently identify the anomalous primes.

## Comments

Poster #W69