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Discrete Mathematics and Combinatorics Commons™
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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 …
The Discrete Logarithm Problem And Ternary Functional Graphs, Max F. Brugger, Christina A. Frederick
The Discrete Logarithm Problem And Ternary Functional Graphs, Max F. Brugger, Christina A. Frederick
Mathematical Sciences Technical Reports (MSTR)
Encryption is essential to the security of transactions and communications, but the algorithms on which they rely might not be as secure as we all assume. In this paper, we investigate the randomness of the discrete exponentiation function used frequently in encryption. We show how we used exponential generating functions to gain theoretical data for mapping statistics in ternary functional graphs. Then, we compare mapping statistics of discrete exponentiation functional graphs, for a range of primes, with mapping statistics of the respective ternary functional graphs.
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 …