galois.is_cyclic¶

galois.is_cyclic(n)¶

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

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

\(\mathbb{Z}{_n^\times}\) 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.

Parameters: n (int) – A positive integer.
Returns: True if the multiplicative group \(\mathbb{Z}{_n^\times}\) is cyclic.
Return type: bool

References

https://en.wikipedia.org/wiki/Primitive_root_modulo_n

Examples

The elements of \(\mathbb{Z}{_n^\times}\) are the positive integers less than \(n\) that are coprime with \(n\). For example when \(n = 14\), then \(\mathbb{Z}{_{14}^\times} = \{1, 3, 5, 9, 11, 13\}\).

# n is of type 2*p^k, which is cyclic
In [351]: n = 14

In [352]: galois.is_cyclic(n)
Out[352]: True

# The congruence class coprime with n
In [353]: Znx = set([a for a in range(1, n) if math.gcd(n, a) == 1]); Znx
Out[353]: {1, 3, 5, 9, 11, 13}

# Euler's totient function counts the "totatives", positive integers coprime with n
In [354]: phi = galois.euler_totient(n); phi
Out[354]: 6

In [355]: len(Znx) == phi
Out[355]: True

# The primitive roots are the elements in Znx that multiplicatively generate the group
In [356]: for a in Znx:
   .....:     span = set([pow(a, i, n) for i in range(1, phi + 1)])
   .....:     primitive_root = span == Znx
   .....:     print("Element: {:2d}, Span: {:<20}, Primitive root: {}".format(a, str(span), primitive_root))
   .....: 
Element:  1, Span: {1}                 , Primitive root: False
Element:  3, Span: {1, 3, 5, 9, 11, 13}, Primitive root: True
Element:  5, Span: {1, 3, 5, 9, 11, 13}, Primitive root: True
Element:  9, Span: {9, 11, 1}          , Primitive root: False
Element: 11, Span: {9, 11, 1}          , Primitive root: False
Element: 13, Span: {1, 13}             , Primitive root: False

In [357]: roots = galois.primitive_roots(n); roots
Out[357]: [3, 5]

# Euler's totient function phi(phi(n)) counts the primitive roots of n
In [358]: len(roots) == galois.euler_totient(phi)
Out[358]: True

A counterexample is \(n = 15 = 3*5\), which doesn’t fit the condition for cyclicness. \(\mathbb{Z}{_{15}^\times} = \{1, 2, 4, 7, 8, 11, 13, 14\}\).

# n is of type p1^k1 * p2^k2, which is not cyclic
In [359]: n = 15

In [360]: galois.is_cyclic(n)
Out[360]: False

# The congruence class coprime with n
In [361]: Znx = set([a for a in range(1, n) if math.gcd(n, a) == 1]); Znx
Out[361]: {1, 2, 4, 7, 8, 11, 13, 14}

# Euler's totient function counts the "totatives", positive integers coprime with n
In [362]: phi = galois.euler_totient(n); phi
Out[362]: 8

In [363]: len(Znx) == phi
Out[363]: True

# The primitive roots are the elements in Znx that multiplicatively generate the group
In [364]: for a in Znx:
   .....:     span = set([pow(a, i, n) for i in range(1, phi + 1)])
   .....:     primitive_root = span == 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

In [365]: roots = galois.primitive_roots(n); roots
Out[365]: []

# Note the max order of any element is 4, not 8, which is Carmichael's lambda function
In [366]: galois.carmichael(n)
Out[366]: 4