Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 13, 30 November 2023
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Discrete logarithms and primitive roots: Algorithms, properties, and typical solution methods
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Abstract
In mathematics, the logarithm, log_a〖b,〗 where a∈(0,1)∪(1,∞) and b>0, is always defined as the real number x, such that a^x=b. Moreover, in the field of number theory, a similar concept called the discrete logarithm can be defined as follows: For a given positive integer m(m≥2), let a∈N^(+ ) with (a,m)=1, and r is the primitive root of m, x=〖ind〗_r a if r^x≡a (mod m). Here, x is the discrete logarithm. The Discrete Logarithm Problem, which is a famous problem in number theory, is formulized as: For a positive integer b and a prime number p, and a is the primitive root of p, the goal is to find the exact value of i, such that a^i≡b (mod p), in other words, it is targeted at finding the exact value of 〖ind〗_a b. The goal of this research is to give several solutions to the Discrete Logarithm Problem, so firstly, some background concept like order and primitive root will be introduced with the proof of some foundational theories of these two concepts, then this essay will give two methods that can solve the Discrete Logarithm Problem called Shanks' Babystep-Giantstep Algorithm and Pohlig-Hellman Discrete Logarithm Algorithm.
Keywords
Discrete Logarithm, The Discrete Logarithm Problem, Order, Primitive Root
References
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8. Stromquist, W. (2017). What are Primitive Roots? Mathematics. Bryn Mawr College.
9. Gauss & Clarke. (1986). Arts 92.
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11. Mollin, R. (2006). An Introduction to Cryptography. Chapman and Hall/CRC.p.344.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
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- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-189-6
- ISBN (Online)
- 978-1-83558-190-2
- Published Date
- 30 November 2023
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/13/20240801
- Copyright
- 30 November 2023
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited