"""
Bent functions related to amicability relationships in the real representations of Clifford algebras.
Paul Leopardi.
"""
from boolean_cayley_graphs.bent_function import BentFunction
from sage.rings.integer import Integer
base4 = lambda length, num: num.digits(4, padto=length)
def sigma_tau(which, m, x):
r"""
Given the numbers `which`, $m$ and $x$,
`sigma_tau(0,m,x)` returns $\sigma_m(x)$, and
`sigma_tau(1,m,x)` returns $\tau_m(x)$,
where $\sigma_m$ and $\tau_m$ are
two specific bent functions with disjoint support,
defined via the properties of real monomial representation matrices
of real Clifford algebras.
"""
xbase4 = base4(Integer(m), Integer(x))
nbr1 = xbase4.count(1)
nbr2 = xbase4.count(2)
return 0 if nbr1 + nbr2 == 0 else (nbr1 + which) % 2
clifford_sign_of_square_sigma = lambda m, x: sigma_tau(0, m, x)
clifford_non_diag_symmetry_tau = lambda m, x: sigma_tau(1, m, x)
power_4_truth_table = lambda m, f: [
f(m, x)
for x in range(4 ** m)]
sigma_list = lambda n: [
BentFunction(power_4_truth_table(m, clifford_sign_of_square_sigma))
for m in range(n)]
r"""
The list `sigma_list(n)` contains each `BentFunction`
corresponding to $\sigma_m$ for $m$ from 0 to n-1.
"""
tau_list = lambda n: [
BentFunction(power_4_truth_table(m, clifford_non_diag_symmetry_tau))
for m in range(n)]
r"""
The list `tau_list(n)` contains each `BentFunction`
corresponding to $\tau_m$ for $m$ from 0 to n-1.
"""
