Bent Boolean Functions: A Better Procedure to Generate Non-crypto 4-bit S-boxes

Dey, Sankhanil; Ghosh, Ranjan

doi:10.1007/s40031-021-00653-y

Bent Boolean Functions: A Better Procedure to Generate Non-crypto 4-bit S-boxes

Original Contribution
Published: 22 August 2021

Volume 103, pages 385–393, (2022)
Cite this article

Journal of The Institution of Engineers (India): Series B Aims and scope Submit manuscript

73 Accesses
Explore all metrics

Abstract

Crypto 4-bit substitution boxes or crypto 4-bit S-boxes are used in block ciphers for nonlinear substitution very frequently. If the 16 elements of a 4-bit S-box are unique, distinct and vary between 0 and f in hex then the said 4-bit S-box is called as a crypto 4-bit S-box. There are 16! crypto 4-bit S-boxes available in crypto literature. Other than crypto 4-bit S-boxes, there are another type of 4-bit S-boxes exist. In such 4-bit S-boxes, 16 elements of the 4-bit S-box are not unique and distinct i.e., at least one element must repeat more than one time. They are called as non-crypto 4-bit S-boxes. There are 16¹⁶–16! Numbers of non-crypto 4-bit S-boxes can be found in crypto literature. The non-crypto 4-bit S-boxes can be generated from 4-bit Boolean Functions (BFs) in the same manner as that crypto 4-bit S-boxes are generated in [C. Adams, S. Tavares, “The structured design of cryptographically good S-boxes”, J. Cryptol. (1990) 344 vol. 3, pp : 27–41]. But to generate crypto 4-bit S-boxes the security of the generated 4-bit S-boxes is sacrificed into some extend. Since 12,870 4-bit balanced BFs are responsible for 16! crypto 4-bit S-boxes and the nonlinearity of the balanced 4-bit BFs are at most 4. So, the 4-bit BFs with highest nonlinearity 6 are left abandoned. These 4-bit BFs are called as 4-bit Bent BFs. Here in this paper, we generate non-crypto 4-bit S-boxes from 4-bit Bent BFs. The generated non-crypto 4-bit S-boxes are analyzed with the existing cryptanalysis techniques to prove them much secure 4-bit S-boxes from crypto angle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Construction of nonlinear component based on bent Boolean functions

Article 06 January 2024

Exploring Semi-bent Boolean Functions Arising from Cellular Automata

A Note on Semi-bent and Hyper-bent Boolean Functions

References

S. Dey , A. Chakrabarti, R. Ghosh, 4-bit Boolean functions in generation and cryptanalysis of secure 4-bit crypto S-boxes. Security and Privacy. 2019; e90. Willey Periodicals Inc. https://doi.org/10.1002/spy2.90.
S. Dey, R. Ghosh, (2018) A Review of Existing 4-Bit Crypto S-Box Cryptanalysis Techniques and Two New Techniques with 4-Bit Boolean Functions for Cryptanalysis of 4-Bit Crypto S-Boxes*. Advances in Pure Mathematics, 8, 272–306. ISSN Online: 2160–0384 ISSN Print: 2160–0368. https://doi.org/10.4236/apm.2018.83015.
S. Dey, R. Ghosh, “A smart review and two new techniques using 4-bit Boolean functions for cryptanalysis of 4-bit crypto S-boxes.”, Vol.40, issue.3, pp.1–19, International Journal of Computers and Applications, Taylor and Francis publishers, Year: 2018. ISSN. 1206–212X. https://doi.org/10.1080/1206212X.2018.1504459.
S. Dey, R. Ghosh, “Crypto-Archaeology: Unearthing Design Methodology of DES S-Boxes”, Circulation in Computer Science, Vol-2, Number -9, pp-30–34, CSL Press, NY, Oct-2017, ISSN. 2456–3692. https://doi.org/10.22632/ccs-2017-252-57.
D. Joan, R. Vincent (2000), AES Proposal: Rijndael. http://csrc.nist.gov/encryption/aes/ Last Visited: 7th February 2001.
S. Dey, A. Chakrabarti, R. Ghosh, (2019) 4-bit crypto S-boxes: Generation with irreducible polynomials over Galois field GF(2^4) and cryptanalysis., International Journal of Tomography and Simulation, ISSN: 2319–3336, Vol. 32, Issue No. 3, Centre for Environment, Social & Economic Research.
H. Feistel, "Block Cipher Cryptographic System", US Patent 3798359 (Filed June 30, 1971).
A. Sorkin, (1984). LUCIFER: a cryptographic algorithm. Cryptologia, 8(1), 22–35, 1984.
Data Encryption Standard, Federal Information Processing Standards Publication (FIPS PUB) 46 (National Bureau of Standards, Washington, DC, 1977)
Google Scholar
Data Encryption Standard (DES), Federal Information Processing Standards Publication (FIPS PUB) 46–3, National Institute of Standards and Technology, Gaithersburg, MD (1999).
C. Adams, S. Tavares “The structured design of cryptographically good S-boxes”, J. Cryptology (1990) 344 vol. 3, pp : 27–41.
H.M. Heys, A tutorial on linear and differential cryptanlysis.cryptologia,26(2002),189–221.
H.M. Heys, S.E. Tavares, Substitution-permutation networks resistant to differential and linear cryptanalysis. J. Cryptol. 9, 1–19 (1996)
Article MathSciNet Google Scholar

Download references

Acknowledgements

For this exhaustive work, the authors would like to acknowledge the Institute of Radio Physics and Electronics, University of Calcutta and A K Choudhury School of Information Technology, University of Calcutta for their continuous encouragement and help.

Funding

No funding was received to perform this research.

Author information

Authors and Affiliations

Department of Radio Physics and Electronics, University of Calcutta, 92 A P C Road, Kolkata, 700009, India
Sankhanil Dey & Ranjan Ghosh

Authors

Sankhanil Dey
View author publications
You can also search for this author in PubMed Google Scholar
Ranjan Ghosh
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sankhanil Dey.

Ethics declarations

Conflict of interest

The authors have no conflict of interest related to the content of this study.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Properties of 10 4-bit Bent BFs with their Complement 4-bit Bent BFs given below:-

BF		BF	CBF			CBF				FO-SAC		SFO-SAC	SFO			MHO-SAC	MHO
Sl.No	(Dec)	(BInary)	10	L	(Dec)	(BInary)	10	L	ANF Coefficients	Mm 8 4 21		3569AC	7BDE	F	SUM	3569AC	7BDE	F	SUM
00,001	00,854	0,000,001,101,010,110	6a	2	64,681	1,111,110,010,101,001	a6	4	C-0000–001,100-0000–0	A6	1111	111,111	1111	1	011	111,111	1111	1	011
00,002	00,857	0,000,001,101,011,001	6a	2	64,678	1,111,110,010,100,110	a6	4	C-0000–001,101-0000–0	A6	1111	111,111	1111	1	011	111,111	1011	1	010
00,003	00,869	0,000,001,101,100,101	6a	2	64,666	1,111,110,010,011,010	a6	4	C-0000–001,110-0000–0	A6	1111	111,111	1111	1	011	111,111	1011	1	010
00,004	00,874	0,000,001,101,101,010	6a	2	64,661	1,111,110,010,010,101	a6	4	C-0000–001,111-0000–0	A6	1111	111,111	1111	1	011	111,111	1111	1	011
00,005	00,917	0,000,001,110,010,101	6a	2	64,618	1,111,110,001,101,010	a6	4	C-0001–001,111-0000–0	A6	1111	111,111	1111	1	011	111,111	1111	1	011
00,006	00,922	0,000,001,110,011,010	6a	2	64,613	1,111,110,001,100,101	a6	4	C-0001–001,110-0000–0	A6	1111	111,111	1111	1	011	111,111	1011	1	010
00,007	00,934	0,000,001,110,100,110	6a	2	64,601	1,111,110,001,011,001	a6	4	C-0001–001,101-0000–0	A6	1111	111,111	1111	1	011	111,111	1011	1	010
00,008	00,937	0,000,001,110,101,001	6a	2	64,598	1,111,110,001,010,110	a6	4	C-0001–001,100-0000–0	A6	1111	111,111	1111	1	011	111,111	1111	1	011
00,009	01,334	0,000,010,100,110,110	6a	2	64,201	1,111,101,011,001,001	a6	4	C-0000–010,010-0000–0	A6	1111	111,111	1111	1	011	111,111	1111	1	011
00,010	01,337	0,000,010,100,111,001	6a	2	64,198	1,111,101,011,000,110	a6	4	C-0000–010,011-0000–0	A6	1111	111,111	1111	1	011	111,111	1011	1	010

BIC, LC and DC analysis of 4! or 24 permuted non-crypto 4-bit S-boxes given below:-

DEs of BFs-crypto S-box		DEs of BFs-complement crypto S-box		Complement	Balancedness of the six xored 4-bit BFs									Total
Sl No	SB(DBF1,DBF2,DBF3,DBF4)	S-box in Hex	CSB(DBF1,DBF2,DBF3,DBF4)	S-box in Hex	(1,2)		10 (1,3)		10 (1,4)		10 (2,3)	10 (2,4)	10 (3,4)	10	88	97	A6	L	0	D	0
001.01	SB(00,854,00,857,00,869,00,874)	000000ff0f3c5a96	CSB(64,681,64,678,64,666,64,661)	ffffff00f0c3a569	00,015	4c	00,051	4c	00,060	4c	00,060	4c	00,051	4c	00,015	4c	00	00	00	102	165
001.02	SB(00,857,00,854,00,869,00,874)	000000ff0f3c965a	CSB(64,678,64,681,64,666,64,661)	ffffff00f0c369a5	00,015	4c	00,060	4c	00,051	4c	00,051	4c	00,060	4c	00,015	4c	00	00	00	102	165
001.03	SB(00,854,00,869,00,857,00,874)	000000ff0f5a3c96	CSB(64,681,64,666,64,678,64,661)	ffffff00f0a5c369	00,051	4c	00,015	4c	00,060	4c	00,060	4c	00,015	4c	00,051	4c	00	00	00	102	165
001.04	SB(00,857,00,869,00,854,00,874)	000000ff0f5a963c	CSB(64,678,64,666,64,681,64,661)	ffffff00f0a569c3	00,060	4c	00,015	4c	00,051	4c	00,051	4c	00,015	4c	00,060	4c	00	00	00	102	165
001.05	SB(00,869,00,854,00,857,00,874)	000000ff0f963c5a	CSB(64,666,64,681,64,678,64,661)	ffffff00f069c3a5	00,051	4c	00,060	4c	00,015	4c	00,015	4c	00,060	4c	00,051	4c	00	00	00	102	165
001.06	SB(00,869,00,857,00,854,00,874)	000000ff0f965a3c	CSB(64,666,64,678,64,681,64,661)	ffffff00f069a5c3	00,060	4c	00,051	4c	00,015	4c	00,015	4c	00,051	4c	00,060	4c	00	00	00	102	165
001.07	SB(00,854,00,857,00,874,00,869)	000000ff0f3c69a5	CSB(64,681,64,678,64,661,64,666)	ffffff00f0c3965a	00,015	4c	00,060	4c	00,051	4c	00,051	4c	00,060	4c	00,015	4c	00	00	00	102	165
001.08	SB(00,857,00,854,00,874,00,869)	000000ff0f3ca569	CSB(64,678,64,681,64,661,64,666)	ffffff00f0c35a96	00,015	4c	00,051	4c	00,060	4c	00,060	4c	00,051	4c	00,015	4c	00	00	00	102	165
001.09	SB(00,854,00,869,00,874,00,857)	000000ff0f693ca5	CSB(64,681,64,666,64,661,64,678)	ffffff00f096c35a	00,051	4c	00,060	4c	00,015	4c	00,015	4c	00,060	4c	00,051	4c	00	00	00	102	165
001.10	SB(00,857,00,869,00,874,00,854)	000000ff0f69a53c	CSB(64,678,64,666,64,661,64,681)	ffffff00f0965ac3	00,060	4c	00,051	4c	00,015	4c	00,015	4c	00,051	4c	00,060	4c	00	00	00	102	165
001.11	SB(00,869,00,854,00,874,00,857)	000000ff0fa53c69	CSB(64,666,64,681,64,661,64,678)	ffffff00f05ac396	00,051	4c	00,015	4c	00,060	4c	00,060	4c	00,015	4c	00,051	4c	00	00	00	102	165
001.12	SB(00,869,00,857,00,874,00,854)	000000ff0fa5693c	CSB(64,666,64,678,64,661,64,681)	ffffff00f05a96c3	00,060	4c	00,015	4c	00,051	4c	00,051	4c	00,015	4c	00,060	4c	00	00	00	102	165
001.13	SB(00,854,00,874,00,857,00,869)	000000ff0f5a69c3	CSB(64,681,64,661,64,678,64,666)	ffffff00f0a5963c	00,060	4c	00,015	4c	00,051	4c	00,051	4c	00,015	4c	00,060	4c	00	00	00	102	165
001.14	SB(00,857,00,874,00,854,00,869)	000000ff0f5ac369	CSB(64,678,64,661,64,681,64,666)	ffffff00f0a53c96	00,051	4c	00,015	4c	00,060	4c	00,060	4c	00,015	4c	00,051	4c	00	00	00	102	165
001.15	SB(00,854,00,874,00,869,00,857)	000000ff0f695ac3	CSB(64,681,64,661,64,666,64,678)	ffffff00f096a53c	00,060	4c	00,051	4c	00,015	4c	00,015	4c	00,051	4c	00,060	4c	00	00	00	102	165
001.16	SB(00,857,00,874,00,869,00,854)	000000ff0f69c35a	CSB(64,678,64,661,64,666,64,681)	ffffff00f0963ca5	00,051	4c	00,060	4c	00,015	4c	00,015	4c	00,060	4c	00,051	4c	00	00	00	102	165
001.17	SB(00,869,00,874,00,854,00,857)	000000ff0fc35a69	CSB(64,666,64,661,64,681,64,678)	ffffff00f03ca596	00,015	4c	00,051	4c	00,060	4c	00,060	4c	00,051	4c	00,015	4c	00	00	00	102	165
001.18	SB(00,869,00,874,00,857,00,854)	000000ff0fc3695a	CSB(64,666,64,661,64,678,64,681)	ffffff00f03c96a5	00,015	4c	00,060	4c	00,051	4c	00,051	4c	00,060	4c	00,015	4c	00	00	00	102	165
001.19	SB(00,874,00,854,00,857,00,869)	000000ff0f96a5c3	CSB(64,661,64,681,64,678,64,666)	ffffff00f0695a3c	00,060	4c	00,051	4c	00,015	4c	00,015	4c	00,051	4c	00,060	4c	00	00	00	102	165
001.20	SB(00,874,00,857,00,854,00,869)	000000ff0f96c3a5	CSB(64,661,64,678,64,681,64,666)	ffffff00f0693c5a	00,051	4c	00,060	4c	00,015	4c	00,015	4c	00,060	4c	00,051	4c	00	00	00	102	165
001.21	SB(00,874,00,854,00,869,00,857)	000000ff0fa596c3	CSB(64,661,64,681,64,666,64,678)	ffffff00f05a693c	00,060	4c	00,015	4c	00,051	4c	00,051	4c	00,015	4c	00,060	4c	00	00	00	102	165
001.22	SB(00,874,00,857,00,869,00,854)	000000ff0fa5c396	CSB(64,661,64,678,64,666,64,681)	ffffff00f05a3c69	00,051	4c	00,015	4c	00,060	4c	00,060	4c	00,015	4c	00,051	4c	00	00	00	102	165
001.23	SB(00,874,00,869,00,854,00,857)	000000ff0fc396a5	CSB(64,661,64,666,64,681,64,678)	ffffff00f03c695a	00,015	4c	00,060	4c	00,051	4c	00,051	4c	00,060	4c	00,015	4c	00	00	00	102	165
001.24	SB(00,874,00,869,00,857,00,854)	000000ff0fc3a596	CSB(64,661,64,666,64,678,64,681)	ffffff00f03c5a69	00,015	4c	00,051	4c	00,060	4c	00,060	4c	00,051	4c	00,015	4c	00	00	00	102	165

.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dey, S., Ghosh, R. Bent Boolean Functions: A Better Procedure to Generate Non-crypto 4-bit S-boxes. J. Inst. Eng. India Ser. B 103, 385–393 (2022). https://doi.org/10.1007/s40031-021-00653-y

Download citation

Received: 11 June 2020
Accepted: 14 July 2021
Published: 22 August 2021
Issue Date: April 2022
DOI: https://doi.org/10.1007/s40031-021-00653-y

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Bent Boolean Functions: A Better Procedure to Generate Non-crypto 4-bit S-boxes

Abstract

Access this article

Similar content being viewed by others

Construction of nonlinear component based on bent Boolean functions

Exploring Semi-bent Boolean Functions Arising from Cellular Automata

A Note on Semi-bent and Hyper-bent Boolean Functions

References

Acknowledgements

Funding

Author information

Authors and Affiliations

Corresponding author

Ethics declarations

Conflict of interest

Additional information

Publisher's Note

Appendix

Properties of 10 4-bit Bent BFs with their Complement 4-bit Bent BFs given below:-

BIC, LC and DC analysis of 4! or 24 permuted non-crypto 4-bit S-boxes given below:-

Rights and permissions

About this article

Cite this article

Navigation

Bent Boolean Functions: A Better Procedure to Generate Non-crypto 4-bit S-boxes

Abstract

Access this article

Similar content being viewed by others

Construction of nonlinear component based on bent Boolean functions

Exploring Semi-bent Boolean Functions Arising from Cellular Automata

A Note on Semi-bent and Hyper-bent Boolean Functions

References

Acknowledgements

Funding

Author information

Authors and Affiliations

Corresponding author

Ethics declarations

Conflict of interest

Additional information

Publisher's Note

Appendix

Appendix

Properties of 10 4-bit Bent BFs with their Complement 4-bit Bent BFs given below:-

BIC, LC and DC analysis of 4! or 24 permuted non-crypto 4-bit S-boxes given below:-

Rights and permissions

About this article

Cite this article

Share this article

Search

Navigation