Finding exceptional bent functions in the CAST128 S-boxes

Only a small number of the $8 \times 32 = 256$ bent functions in the CAST128 S-boxes have fewer than $2^{17} = 131072$ Cayley classes in the union of their Extended Translation class and its dual. The code in this notebook finds these bent functions.

Import the required modules

In [1]:

import datetime
import os
import sqlite3

from boolean_cayley_graphs.bent_function import BentFunction
from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification

Using saved CAST128 classifcations, find the exceptional bent functions.

In [2]:

sobj_dirname=os.path.join("..","sobj")
exceptions = []
max_len_cayley_graph_class_list = 2 ** 17
then = datetime.datetime.now()
for i in range(1,9):
    print(i)
    stri = "%01d" % i
    for j in range(32):
        strj = "%02d" % j
        sobj_name = "cast128_" + stri + "_" + str(j) + ".sobj"
        cgc = BentFunctionCayleyGraphClassification.load_mangled(
            sobj_name,
            dir=sobj_dirname)
        len_cayley_graph_class_list = len(cgc.cayley_graph_class_list)
        if len_cayley_graph_class_list < max_len_cayley_graph_class_list:
            exceptions.append((i,j))
            now = datetime.datetime.now()
            print(stri, strj, now, now - then, len_cayley_graph_class_list)
            then = now
print(exceptions)

1
2
2 01 2023-01-22 17:35:59.209772 0:04:58.706344 8256
2 16 2023-01-22 17:38:01.334311 0:02:02.124539 65536
3
4
4 27 2023-01-22 17:48:44.514829 0:10:43.180518 65536
5
5 16 2023-01-22 17:51:34.786826 0:02:50.271997 66560
5 27 2023-01-22 17:53:04.571871 0:01:29.785045 6144
6
6 17 2023-01-22 17:56:23.275130 0:03:18.703259 65536
7
7 15 2023-01-22 18:00:32.579023 0:04:09.303893 65536
7 21 2023-01-22 18:01:17.083889 0:00:44.504866 65536
8
[(2, 1), (2, 16), (4, 27), (5, 16), (5, 27), (6, 17), (7, 15), (7, 21)]

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