galois.is_cyclic(n: int) → bool

Determines whether the multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic.

Parameters:
n: int

A positive integer.

Returns:

True if the multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic.

See also

euler_phi, carmichael_lambda, totatives

Notes

The multiplicative group \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is the set of positive integers \(1 \le a < n\) that are coprime with \(n\). \((\mathbb{Z}/n\mathbb{Z}){^\times}\) being cyclic means that some primitive root of \(n\), or generator, \(g\) can generate the group \(\{1, g, g^2, \dots, g^{\phi(n)-1}\}\), where \(\phi(n)\) is Euler’s totient function and calculates the order of the group. If \((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic, the number of primitive roots is found by \(\phi(\phi(n))\).

\((\mathbb{Z}/n\mathbb{Z}){^\times}\) is cyclic if and only if \(n\) is 2, 4, \(p^k\), or \(2p^k\), where \(p\) is an odd prime and \(k\) is a positive integer.

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 \(\phi(n)\) 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]: len(Znx) == phi
Out[4]: True

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]: galois.is_cyclic(n)
Out[5]: True

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]: galois.is_cyclic(n)
Out[13]: False

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/3.11.9/x64/lib/python3.11/site-packages/galois/_modular.py:560, in primitive_root(n, start, stop, method)
    559 if method == "min":
--> 560     root = next(primitive_roots(n, start, stop=stop))
    561 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/3.11.9/x64/lib/python3.11/site-packages/galois/_modular.py:567, in primitive_root(n, start, stop, method)
    565     return root
    566 except StopIteration as e:
--> 567     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).

For prime \(n\), a primitive root modulo \(n\) is also a primitive element of the Galois field \(\mathrm{GF}(n)\).

In [17]: n = 31

In [18]: galois.is_cyclic(n)
Out[18]: True

A primitive element is a generator of the multiplicative group \(\mathrm{GF}(p)^{\times} = \{1, 2, \dots, p-1\} = \{1, g, g^2, \dots, g^{\phi(n)-1}\}\).

In [19]: GF = galois.GF(n)

In [20]: galois.primitive_root(n)
Out[20]: 3

In [21]: GF.primitive_element
Out[21]: GF(3, order=31)

The number of primitive roots/elements is \(\phi(\phi(n))\).

In [22]: list(galois.primitive_roots(n))
Out[22]: [3, 11, 12, 13, 17, 21, 22, 24]

In [23]: GF.primitive_elements
Out[23]: GF([ 3, 11, 12, 13, 17, 21, 22, 24], order=31)

In [24]: galois.euler_phi(galois.euler_phi(n))
Out[24]: 8