from sage.crypto.boolean_function import BooleanFunction
from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
def check_bent_isomorphism(dim, k, certify=False):
r"""
Given the non-negative numbers $dim$ and $k$, `check_bent_isomorphism`
checks that all of the bent functions whose truth table has $k$ bits
set out of the $2^{dim}-1$ non-zero bits, have isomorphic Cayley graphs.
It does so by enumerating all combinations `a` with $k$ non-zero bits set,
testing if `a` yields a bent function, then testing the resulting
Cayley graph for isomorphism to the first graph so obtained.
If a non isomorphic Cayley graph is found, `check_bent_isomorphism` prints
the combination `a`, and returns the number of bent functions found so far.
Otherwise `check_bent_isomorphism` just returns the total number of
bent functions found.
The optional parameter `certify`, which defaults to `False`, prints
the truth table `v` and Cayley graph isomorphism `iso` corresponding
to each bent function.
"""
v = 2 ** dim
nbent = 0
for a in Combinations(range(1, v), k):
t = [1 if x in a else 0 for x in range(v)]
f = BooleanFunctionImproved(t)
f_is_bent = f.is_bent()
if f_is_bent:
nbent += 1
g = f.cayley_graph()
if nbent == 1:
g0 = g
print(g.is_strongly_regular(parameters=True))
if certify:
print(t)
else:
if certify:
g_is_iso, iso = g.is_isomorphic(g0, certify=True)
else:
g_is_iso = g.is_isomorphic(g0)
if not g_is_iso:
print(a)
return nbent
elif certify:
print(t, iso)
return nbent
