Modular Exponentiation (a^b mod m) & Modular Inverse Calculator

Evaluate cryptographic modular powers and multiplicative inverses mod m.

Modular Power (a^b mod m)
7
Simple Remainder (a mod m) 7
Multiplicative Inverse a⁻¹ mod m 15
Coprime Check w6gcd(a, m) 1
Inputs
Base integer a.
Non-negative integer exponent b.
Positive integer modulus m (m ≥ 2).
Results & Graph
Modular Power (a^b mod m)
7
Simple Remainder (a mod m)
7
Multiplicative Inverse a⁻¹ mod m
15
Coprime Check w6gcd(a, m)
1

Modular Power (a^b mod m)

Graph 1 • Base A Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 5.4 10.8 50%: 3 3 50% 75%: 5 5 75% 100%: 7 7 100% 125%: 8 8 125% 150%: 10 10 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Modular Power (a^b mod m) 7 Simple Remainder (a mod m) 7 Multiplicative Inverse a⁻¹ mod m 15 Coprime Check w6gcd(a, m) 1
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Modular Power (a^b mod m)7Simple Remainder (a mod m)7Multiplicative Inverse a⁻¹ mod m15Coprime Check w6gcd(a, m)1
Summary: 7^13 ≡ 7 (mod 26). Multiplicative inverse a⁻¹ ≡ 15 (mod 26).
Summary: 7^13 ≡ 7 (mod 26). Multiplicative inverse a⁻¹ ≡ 15 (mod 26).

Formula & Mathematical Methodology: Modular Math Mathematical Identity

R ≡ ab (mod m)
a × a−1 ≡ 1 (mod m)
R ≡ a^b (mod m); a × a^(−1) ≡ 1 (mod m)
  • baseA: Base A (numeric parameter)
  • exponentB: Exponent B (numeric parameter)
  • modulusM: Modulus M (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Modular Math

Evaluate cryptographic modular powers and multiplicative inverses mod m.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: R ≡ a^b (mod m); a × a^(−1) ≡ 1 (mod m).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Modular Math via R ≡ a^b (mod m); a × a^(−1) ≡ 1 (mod m).

Frequently Asked Questions

What mathematical formula does the Modular Exponentiation (a^b mod m) & Modular Inverse Calculator use?

It directly evaluates R ≡ a^b (mod m); a × a^(−1) ≡ 1 (mod m) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Number Theory Guide

How to Use the Modular Exponentiation (a^b mod m) & Modular Inverse Calculator: Practical Guide & Visual Analysis

Compute fast binary modular exponentiation (a^b mod m), simple congruence (a mod m), and extended Euclidean modular multiplicative inverse. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching modular exponentiation (a^b mod m) & modular inverse calculator, modular math, modular arithmetic exponentiation inverse, number theory calculator, this interactive tool helps you compute fast binary modular exponentiation (a^b mod m), simple congruence (a mod m), and extended Euclidean modular multiplicative inverse.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust Base A, Exponent B, Modulus M, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Modular Math), Evaluates Modular Math via R ≡ a^b (mod m); a × a^(−1) ≡ 1 (mod m). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).