Structure and Randomness of the Discrete Lambert Map

We investigate the structure and cryptographic applications of the Discrete Lambert Map (DLM), the mapping x 7→ xgxmod p, for p a prime and some fixed g ∈ (Z/pZ)∗. The mapping is closely related to the Discrete Log Problem, but has received far less attention since it is considered to be a more complicated map that is likely even harder to invert. However, this mapping is quite important because it underlies the security of the ElGamal Digital Signature Scheme. Using functional graphs induced by this mapping, we were able to find non-random properties that could potentially be used to exploit the ElGamal DSS. Acknowledgements: This research was performed at the Rose-Hulman Institute of Technology Mathematics REU 2011, which was sponsored by NSF grant DMS-1003924. We would like to thank Joshua Holden for his indispensable role in advising and guiding us on this project. We also recognize previous REU participants Friedrichsen, Larson, and McDowell, whose paper served as inspiration for our own work, and Cloutier, Lindle, and Hoffman, whose code helped us to collect the relevant statistical data for our investigations. Finally, we thank the NSF for the grant that made this REU program possible. Page 64 RHIT Undergrad. Math. J., Vol. 13, No. 1