galois.primitive_root(n: int, start: int = 1, stop: int | None = None, method: 'min' | 'max' | 'random' = 'min') → int

Finds a primitive root modulo \(n\) in the range \([\texttt{start}, \texttt{stop})\).

Parameters:
n: int

A positive integer.

start: int = 1

Starting value (inclusive) in the search for a primitive root. The default is 1.

stop: int | None = None

Stopping value (exclusive) in the search for a primitive root. The default is None, which corresponds to n.

method: 'min' | 'max' | 'random' = 'min'

The search method for finding the primitive root. Choices are:

  • "min": Returns the smallest primitive root in the range.

  • "max": Returns the largest primitive root in the range.

  • "random": Returns a random primitive root in the range.

Returns:

A primitive root modulo \(n\) in the specified range.

See also

primitive_roots, is_primitive_root, is_cyclic, totatives, euler_phi, carmichael_lambda

Notes

An integer \(g\) is a primitive root modulo \(n\) if and only if \(g\) generates the multiplicative group of units \((\mathbb{Z}/n\mathbb{Z})^\times\):

\[ (\mathbb{Z}/n\mathbb{Z})^\times = \{ [g^0]_n, [g^1]_n, \dots, [g^{\varphi(n)-1}]_n \}, \]

where \(\varphi(n)\) is Euler’s totient function and equals the order of this group.

Existence of primitive roots depends only on \(n\). By a classical theorem, the group \((\mathbb{Z}/n\mathbb{Z})^\times\) is cyclic (and hence has primitive roots) if and only if

\[ n \in \{1, 2, 4\} \quad \text{or} \quad n = p^k \text{ or } n = 2p^k \]

for some odd prime \(p\) and integer \(k \ge 1\). See is_cyclic() for a convenience test.

  • If \((\mathbb{Z}/n\mathbb{Z})^\times\) is not cyclic, no primitive roots modulo \(n\) exist and this function raises a RuntimeError.

  • If \((\mathbb{Z}/n\mathbb{Z})^\times\) is cyclic, the number of primitive roots modulo \(n\) is \(\varphi(\varphi(n))\), and this function finds one in the specified range.

The edge cases n = 1 and n = 2 are handled by convention:

  • For n = 1, the trivial group is considered cyclic and this function returns 0.

  • For n = 2, the unique primitive root modulo 2 is 1.

References

Examples

The elements of \((\mathbb{Z}/14\mathbb{Z})^\times = \{1, 3, 5, 9, 11, 13\}\) are the totatives of 14.

In [1]: n = 14

In [2]: Znx = galois.totatives(n); Znx
Out[2]: [1, 3, 5, 9, 11, 13]

The Euler totient \(\varphi(n)\) function counts the totatives of \(n\), which is equivalent to the order of \((\mathbb{Z}/n\mathbb{Z})^\times\).

In [3]: phi = galois.euler_phi(n); phi
Out[3]: 6

In [4]: assert len(Znx) == phi

Since 14 is of the form \(2p^k\), the multiplicative group \((\mathbb{Z}/14\mathbb{Z})^\times\) is cyclic, meaning there exists at least one element that generates the group by its powers.

In [5]: assert galois.is_cyclic(n)

Find the smallest primitive root modulo 14. Observe that the powers of g uniquely represent each element in \((\mathbb{Z}/14\mathbb{Z})^\times\).

In [6]: g = galois.primitive_root(n); g
Out[6]: 3

In [7]: [pow(g, i, n) for i in range(0, phi)]
Out[7]: [1, 3, 9, 13, 11, 5]

Find the largest primitive root modulo 14. Observe that the powers of g also uniquely represent each element in \((\mathbb{Z}/14\mathbb{Z})^\times\), although in a different order.

In [8]: g = galois.primitive_root(n, method="max"); g
Out[8]: 5

In [9]: [pow(g, i, n) for i in range(0, phi)]
Out[9]: [1, 5, 11, 13, 9, 3]

A non-cyclic group is \((\mathbb{Z}/15\mathbb{Z})^\times = \{1, 2, 4, 7, 8, 11, 13, 14\}\).

In [10]: n = 15

In [11]: Znx = galois.totatives(n); Znx
Out[11]: [1, 2, 4, 7, 8, 11, 13, 14]

In [12]: phi = galois.euler_phi(n); phi
Out[12]: 8

Since 15 is not of the form \(2\), \(4\), \(p^k\), or \(2p^k\), the multiplicative group \((\mathbb{Z}/15\mathbb{Z})^\times\) is not cyclic, meaning no elements exist whose powers generate the group.

In [13]: assert not galois.is_cyclic(n)

Below, every element is tested to see if it spans the group.

In [14]: for a in Znx:
   ....:     span = set([pow(a, i, n) for i in range(0, phi)])
   ....:     primitive_root = span == set(Znx)
   ....:     print("Element: {:2d}, Span: {:<13}, Primitive root: {}".format(a, str(span), primitive_root))
   ....: 
Element:  1, Span: {1}          , Primitive root: False
Element:  2, Span: {8, 1, 2, 4} , Primitive root: False
Element:  4, Span: {1, 4}       , Primitive root: False
Element:  7, Span: {1, 4, 13, 7}, Primitive root: False
Element:  8, Span: {8, 1, 2, 4} , Primitive root: False
Element: 11, Span: {1, 11}      , Primitive root: False
Element: 13, Span: {1, 4, 13, 7}, Primitive root: False
Element: 14, Span: {1, 14}      , Primitive root: False

The Carmichael \(\lambda(n)\) function finds the maximum multiplicative order of any element, which is 4 and not 8.

In [15]: galois.carmichael_lambda(n)
Out[15]: 4

Observe that no primitive roots modulo 15 exist and a RuntimeError is raised.

In [16]: galois.primitive_root(n)
---------------------------------------------------------------------------
StopIteration                             Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.13.11/x64/lib/python3.13/site-packages/galois/_primitive_root.py:307, in primitive_root(n, start, stop, method)
    306 if method == "min":
--> 307     root = next(primitive_roots(n, start, stop=stop))
    308 elif method == "max":

StopIteration: 

The above exception was the direct cause of the following exception:

RuntimeError                              Traceback (most recent call last)
Cell In[16], line 1
----> 1 galois.primitive_root(n)

File /opt/hostedtoolcache/Python/3.13.11/x64/lib/python3.13/site-packages/galois/_primitive_root.py:315, in primitive_root(n, start, stop, method)
    312     return root
    313 except StopIteration as e:
    314     # No primitive root found in the requested range (either non-cyclic group or range too small).
--> 315     raise RuntimeError(f"No primitive roots modulo {n} exist in the range [{start}, {stop}).") from e

RuntimeError: No primitive roots modulo 15 exist in the range [1, 15).

The algorithm is also efficient for very large \(n\).

In [17]: n = 1000000000000000035000061

In [18]: phi = galois.euler_phi(n); phi
Out[18]: 1000000000000000035000060

Find the smallest, the largest, and a random primitive root modulo \(n\).

In [19]: galois.primitive_root(n)
Out[19]: 7

In [20]: galois.primitive_root(n, method="max")
Out[20]: 1000000000000000035000054

In [21]: galois.primitive_root(n, method="random")
Out[21]: 883323915705962523632990