Primitive Root Finder

Primitive Root Finder | 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, select
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Primitive Root Finder calculates the smallest primitive root modulo n, validates cyclic group structures, and verifies generator candidates for moduli up to 10¹². It provides complete certificates based on prime factors of Euler's totient φ(n), determines total primitive root counts with φ(φ(n)), and can list up to 50 valid primitive roots.

When to use

When selecting cyclic group generators for cryptographic algorithms such as Diffie-Hellman key exchange or ElGamal encryption.
When verifying whether a specific candidate integer g has full multiplicative order modulo n.
When solving number theory assignments or analyzing multiplicative group structures (ℤ/nℤ)*.

How it works

1Evaluates whether the modulus n admits primitive roots by verifying the form n ∈ {2, 4, p^k, 2p^k} for odd primes p.
2Computes Euler's totient φ(n), Carmichael's function λ(n), and factors φ(n) into its distinct prime divisors q.
3Tests candidates g using modular exponentiation to confirm that g^(φ(n)/q) ≢ 1 (mod n) for every prime factor q.
4Outputs the minimal root, total count φ(φ(n)), up to 50 primitive roots, or detailed verification results.

Use cases

Generating public base parameters for discrete logarithm cryptosystems.
Checking group cyclicity and finding generators in abstract algebra coursework.
Auditing whether proposed protocol constants are valid primitive roots modulo prime p.

Examples

1. Finding and listing primitive roots modulo 23

Cryptography student

Background

Studying discrete logarithms in cyclic prime fields and needs all valid generators modulo 23.

Problem

Find the smallest primitive root and enumerate all generators for modulus 23.

How to use

Enter 23 into Modulus n and select the 'Also list primitive roots (up to 50)' output mode.

n = 23, mode = list

Outcome

Returns smallest generator g = 5, certification steps 5^11 ≡ 22 and 5^2 ≡ 2 (mod 23), total count φ(22) = 10, and lists all 10 roots: 5, 7, 10, 11, 14, 15, 17, 19, 20, 21.

2. Verifying generator candidate modulo 41

Security researcher

Background

Evaluating whether g = 2 can serve as a full-order generator modulo the prime 41.

Problem

Determine whether candidate g = 2 generates the entire group (ℤ/41ℤ)*.

How to use

Enter 41 for Modulus n, select 'Verify a candidate g' mode, and set Candidate g to 2.

FAQ

Which moduli n have primitive roots?

A primitive root exists modulo n if and only if n is 2, 4, p^k, or 2·p^k, where p is an odd prime and k ≥ 1.

What is the maximum modulus n supported?

The tool supports integers n up to 10¹² using trial-division factorization.

How is a primitive root mathematically certified?

For gcd(g, n) = 1, g is certified if g^(φ(n)/q) ≢ 1 (mod n) for every prime factor q of φ(n).

How many primitive roots exist modulo n?

If the multiplicative group is cyclic, the total number of primitive roots is exactly φ(φ(n)).

What happens if candidate g fails verification?

The tool identifies the prime factor witness q where g^(φ(n)/q) ≡ 1 (mod n) and reports the actual order of g.

n = 41, mode = verify, g = 2

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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 and listing primitive roots modulo 23

2. Verifying generator candidate modulo 41

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