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Full-Text Articles in Discrete Mathematics and Combinatorics
Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh
Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh
Mathematical Sciences Technical Reports (MSTR)
The Welch map x -> gx-1+c is similar to the discrete exponential map x -> gx, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: gx-1+c = x (mod pe) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo pe, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo p …
Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh
Deconstructing The Welch Equation Using P-Adic Methods, Abigail Mann, Adelyn Yeoh
Rose-Hulman Undergraduate Research Publications
The Welch map x -> gx-1+c is similar to the discrete exponential map x -> gx, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: gx-1+c = x (mod pe) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo pe, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo p …
Mapping The Discrete Logarithm, Daniel R. Cloutier
Mapping The Discrete Logarithm, Daniel R. Cloutier
Mathematical Sciences Technical Reports (MSTR)
The discrete logarithm is a problem that surfaces frequently in the field of cryptog- raphy as a result of using the transformation ga mod n. This paper focuses on a prime modulus, p, for which it is shown that the basic structure of the functional graph is largely dependent on an interaction between g and p-1. In fact, there are precisely as many different functional graph structures as there are divisors of p-1. This paper extracts two of these structures, permutations and binary functional graphs. Estimates exist for the shape of a random permutation, but …
Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden
Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden
Mathematical Sciences Technical Reports (MSTR)
We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …