Wave-shaped round functions and primitive groups

Riccardo Aragona; Marco Calderini; Roberto Civino; Massimiliano Sala; Ilaria Zappatore

doi:10.3934/amc.2019004

Article Contents

2019, Volume 13, Issue 1: 67-88. Doi: 10.3934/amc.2019004

This issue Previous Article Connecting Legendre with Kummer and Edwards Next Article Optimal information ratio of secret sharing schemes on Dutch windmill graphs

Wave-shaped round functions and primitive groups

1.
DISIM, University of L'Aquila, Italy
2.
Department of Informatics, University of Bergen, Norway
3.
Department of Mathematics, University of Trento, Italy
4.
LIRMM of Montpellier, France

* Corresponding author: Roberto Civino
* Corresponding author: Roberto Civino

Received: February 28, 2018

Revised: July 31, 2018

Published: December 2018

R. Aragona is member of INdAM-GNSAGA (Italy). R. Civino thankfully acknowledges support by the Department of Mathematics of the University of Trento. R. Aragona, R.Civino, and M. Sala thankfully acknowledge support by MIUR-Italy via PRIN 2015TW9LSR "Group theory and applications".

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Abstract

Round functions used as building blocks for iterated block ciphers, both in the case of Substitution-Permutation Networks (SPN) and Feistel Networks (FN), are often obtained as the composition of different layers. The bijectivity of any encryption function is guaranteed by the use of invertible layers or by the Feistel structure. In this work a new family of ciphers, called wave ciphers, is introduced. In wave ciphers, round functions feature wave functions, which are vectorial Boolean functions obtained as the composition of non-invertible layers, where the confusion layer enlarges the message which returns to its original size after the diffusion layer is applied. Efficient decryption is guaranteed by the use of wave functions in FNs. It is shown how to avoid that the group generated by the round functions acts imprimitively, a serious flaw for the cipher. The primitivity is a consequence of a more general result, which reduce the problem of proving that a given FN generates a primitive group to proving that an SPN, directly related to the given FN, generates a primitive group. Finally, a concrete instance of real-world size wave cipher is proposed as an example, and its resistance against differential and linear cryptanalyses is also established.

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Mathematics Subject Classification: 20B15, 20B35, 94A60.

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Figure 1. Round function of an SPN and of an FN

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Figure 2. Wave functions

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Figure 3. A 4x5 APN S-box

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Figure 4. Feistel structure of wave ciphers

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Figure 5. Feistel to SPN reduction

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Figure 6. An example of 40 × 32 proper diffusion layer with parallel kernel, where each "·" represents 0

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Figure 7. Diffusion properties of the matrix λ of Fig. 6.

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Table 1. Difference distribution table of the S-box $\gamma_1$ defined in Section 3.1

	0_x	1_x	2_x	3_x	4_x	5_x	6_x	7_x	8_x	9_x	A_x	B_x	C_x	D_x	E_x	F_x	10_x	11_x	12_x	13_x	14_x	15_x	16_x	17_x	18_x	19_x	1A_x	1B_x	1C_x	1D_x	1E_x	1F_x
0_x	16	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $
1_x	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2
2_x	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $
3_x	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2
4_x	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $
5_x	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2
6_x	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2
7_x	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $
8_x	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	2
9_x	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2
A_x	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $
B_x	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	2	$\cdot $	2	$\cdot $
C_x	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $
D_x	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $
E_x	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	2	$\cdot $	2	$\cdot $	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2
F_x	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	2	2	2	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	2	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $	$\cdot $

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