Multiplicative Order Calculator mod n (ordₙ(a))

Multiplicative Order Calculator mod n (ordₙ(a)) | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Multiplicative Order Calculator finds the smallest positive integer k such that a^k ≡ 1 (mod n) for any coprime integer a and modulus n. It computes values using Euler's totient function φ(n) and Carmichael's function λ(n), providing a step-by-step modular power table, minimality checks, the generated cyclic subgroup ⟨a⟩, and primitive root identification.

When to use

Determining if an integer a is a primitive root modulo n by checking if its order equals φ(n).
Analyzing cyclic subgroups and subgroup sizes within multiplicative groups (ℤ/nℤ)* for abstract algebra and number theory problems.
Selecting generator candidates and verifying cycle lengths for modular arithmetic, discrete logarithm protocols, or cryptographic algorithms.

How it works

1Checks coprimality by evaluating gcd(a, n); if gcd(a, n) = 1, it computes φ(n) and λ(n) using the prime factorization of n.
2Reduces potential order candidates from divisors of φ(n) by iteratively testing prime factors via modular exponentiation.
3Generates the sequence of modular powers a^1, a^2, ..., a^k (mod n) to list the elements in the cyclic subgroup ⟨a⟩.
4Verifies minimality by ensuring a^(k/p) ≢ 1 (mod n) for all prime divisors p of k, then flags whether a achieves maximal order or is a primitive root.

Use cases

Verifying generator elements for Diffie-Hellman key exchanges and discrete log constructions.
Solving number theory assignments involving subgroup orders, order dividing φ(n), and modular exponentiation cycles.
Determining whether a composite modulus n admits a primitive root by evaluating maximal orders against λ(n).

Examples

1. Finding a Primitive Root Modulo 7

Mathematics Student

Background

Working on modular arithmetic problem sets requiring identification of generators for the multiplicative group modulo 7.

Problem

Determine if 3 is a primitive root modulo 7 and list its generated cyclic subgroup.

How to use

Enter 3 for Element a and 7 for Modulus n, then compute the order.

a = 3, n = 7

Outcome

Returns ord₇(3) = 6, matches φ(7) = 6, lists the 6-element subgroup {3, 2, 6, 4, 5, 1}, and confirms 3 is a primitive root.

2. Checking Subgroup Order Modulo 15

Cryptography Researcher

Background

Analyzing unit group structures and element periods in composite moduli where primitive roots do not exist.

Problem

Find the order of 2 modulo 15 and compare it against φ(15) and λ(15).

How to use

Enter 2 for Element a and 15 for Modulus n, then run the calculation.

FAQ

What happens if a and n are not coprime?

The multiplicative order is undefined when gcd(a, n) > 1 because no positive integer power of a can be congruent to 1 modulo n.

What is the difference between φ(n) and λ(n)?

Euler's totient function φ(n) counts integers coprime to n, while Carmichael's function λ(n) represents the smallest universal exponent such that a^λ(n) ≡ 1 (mod n) for all coprime a.

How does the tool prove minimality?

It tests a^(k/p) mod n for each prime factor p dividing the computed order k to demonstrate that no smaller proper divisor yields 1.

What are the input limits for a and n?

Element a accepts values up to 10⁵¹² (which are reduced modulo n), while modulus n supports values between 2 and 10¹².

What does a primitive root flag indicate?

It indicates that ordₙ(a) = φ(n), meaning the powers of a generate every coprime residue in the multiplicative group (ℤ/nℤ)*.

a = 2, n = 15

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Finding a Primitive Root Modulo 7

2. Checking Subgroup Order Modulo 15

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