Discrete logarithms and primitive roots: Algorithms, properties, and typical solution methods
Bibliographic record
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.010 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".