Calculator Input
Example Data Table
| Base a | Target b | Modulus m | Discrete log x | Check |
|---|---|---|---|---|
| 5 | 8 | 23 | 6 | 56 mod 23 = 8 |
| 3 | 13 | 17 | 4 | 34 mod 17 = 13 |
| 7 | 5 | 13 | 3 | 73 mod 13 = 5 |
| 10 | 15 | 17 | 2 | 102 mod 17 = 15 |
Formula Used
The calculator solves the modular equation
a^x ≡ b (mod m).
The unknown value is x. The values a, b, and
m are supplied by the user.
In baby-step giant-step search, the exponent is split as
x = in + j. Baby steps store smaller powers. Giant steps jump by
n powers. A matching residue gives the exponent.
For non-coprime cases, the calculator first reduces common factors. It then solves the reduced congruence and verifies the final answer against the original equation.
How to Use This Calculator
- Enter the base value in the first field.
- Enter the target residue you want the power to match.
- Enter the modulus greater than one.
- Set a maximum exponent, or leave it blank.
- Choose automatic search for the best general option.
- Press the calculate button and review the result above the form.
- Download the result as a CSV or PDF file when needed.
Understanding Discrete Logarithms
What the Problem Means
A discrete logarithm asks for an exponent inside modular arithmetic. Instead of solving a normal logarithm, it searches for a whole number power. The expression has the form a raised to x, reduced by a modulus. The answer is the exponent that creates the target residue. This makes the problem useful in number theory, coding, and cryptography.
Why Modular Powers Behave Differently
Modular powers repeat. A large power can return a small residue after division by the modulus. These repeated residues form cycles. Some bases reach every nonzero residue when the modulus is prime. Other bases reach only part of the residue set. This is why some inputs have no solution. The calculator checks the cycle and verifies every answer.
Search Methods
Direct search is simple. It tests powers one by one. It is easy to understand, but it can be slow. Baby-step giant-step is faster for many practical inputs. It stores a table of smaller powers. Then it jumps through larger blocks. A collision between both tables gives the answer. The generalized version also handles many cases where the base and modulus share a factor.
Practical Limits
Discrete logarithms can become very hard for large cryptographic numbers. This page is designed for learning, homework checks, demonstrations, and moderate integer experiments. It reports the method, the normalized residues, the gcd condition, and a verification line. These details help you spot mistakes in signs, modulus choice, and exponent bounds.
Interpreting the Answer
The smallest displayed exponent is the first verified match inside the selected search range. More answers may exist because residues can repeat. Use the order value and preview table to study that pattern. Always read the verification line. It confirms the final congruence using the original values.
FAQs
What is a discrete logarithm?
It is the exponent x that satisfies a modular power equation, such as ax ≡ b mod m.
Does every input have a solution?
No. Some bases never produce the target residue under the chosen modulus. The calculator reports no solution when none is verified.
Which method should I choose?
Use automatic search for most cases. It uses faster logic and handles many non-coprime inputs safely.
What does the maximum exponent mean?
It limits the search range. The calculator only accepts answers from x = 0 through your selected bound.
Can I use a negative base?
Yes. The calculator normalizes the base into the selected modulus before performing the search.
Why is the gcd shown?
The gcd helps explain whether standard group methods apply. Non-coprime values may need reduction before searching.
Why are very large numbers restricted?
Large discrete log problems can require huge memory or time. Limits protect the page from slow calculations.
Is this suitable for secure cryptography?
It is best for education and moderate experiments. Real cryptographic analysis needs specialized tools and careful security review.