# A New Family of Boolean Functions with Good Cryptographic Properties

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**truth table**of a Boolean function f is the vector, indexed by the elements of ${\mathbb{F}}_{2}^{n}$ (in lexicographical order),

**polar truth table**of f is the $(1,-1)$ sequence defined by

**support**f, denoted by $Supp\left(f\right)$, is the set of vectors in ${\mathbb{F}}_{2}^{n}$ in which the image under f is 1, i.e.,

**balanced**if $w\left(f\right)={2}^{n-1}$, i.e., if the truth table of f contains the same number of 0 and 1. This property is desirable in a Boolean function to resist differential attacks, such as those introduced by Shamir against the DES algorithm [7]. A Boolean function $f\in {\mathcal{B}}_{n}$ is called

**affine**if we can write it as

**linear function**. The set of affine functions will be denoted by ${\mathcal{A}}_{n}$. Let $f,g\in {\mathcal{B}}_{n}$. The

**distance**, $d(f,g)$, between f and g, is the weight of the function $f\oplus g$, i.e.,

**non-linearity**of a Boolean function $f\in {\mathcal{B}}_{n}$, denoted by ${\mathcal{N}}_{f}$, is the minimum distance between f and the set of affine functions ${\mathcal{A}}_{n}$, i.e.,

**Algebraic Normal Form (ANF)**

**Möbius Transform**of f, denoted by $g=\mu \left(f\right)$. The

**Algebraic Degree**of a Boolean function f is the degree of its ANF. It follows that the algebraic degree of $f\in {\mathcal{B}}_{n}$ does not exceed n, that is, is the number of variables in the highest order term with non-zero coefficient.

**Walsh-Hadamard Transform**of a function f in ${\mathbb{F}}_{2}^{n}$ is the mapping $H\left(f\right):{\mathbb{F}}_{2}^{n}\to \mathbb{R}$, defined by

**Theorem**

**1.**

**Proof.**

**correlation immunity**of order m if and only if $H(\widehat{f})\left(u\right)=0$, with $1\le w\left(u\right)\le m$. A Boolean function with correlation immunity of order m and balanced is called

**m-resilient**. The fundamental relationship between the number of variables n, the algebraic degree d, and the order of correlation immunity m of a Boolean function is $m+d\le n$; see Reference [11].

**autocorrelation function**${r}_{\widehat{f}}\left(s\right)$ of a Boolean function f is defined from its polar representation as

**propagation criteria**of order l, denoted by $PC\left(l\right)$ if $f\left(x\right)\oplus f(x\oplus u)$ is balanced for all u with $1\le w\left(u\right)\le l$.

**Strict Avalanche Criterion (SAC)**[12], refers to the effect of changing all input bits. A Boolean function f is said to satisfy SAC if $f\left(x\right)\oplus f(x\oplus u)$ is balanced for all u with $w\left(u\right)=1$.

**error correcting code**C of length n is an ${\mathbb{F}}_{q}\u2014$linear subspace of ${\mathbb{F}}_{q}^{n}$. The elements of C are called words. The weight $wt\left(x\right)$ of a word x in C is the number of its non-zero coordinates. The minimum weight d of the code C is defined as the minimum of the weights among all non-zero words occurring in C. For $x,y\in C$, we define the Hamming distance $d(x,y)$ between x and y as $wt(x-y)$. The

**minimum distance**of a code C is defined as

## 3. Maiorana-McFarland-Guillot’s Construction

## 4. Construction of $\mathbf{\pi}$ and $\mathit{h}$

#### 4.1. $\pi $ One-to-One

#### 4.2. $\pi $ Two-to-One

## 5. Construction of $\mathit{f}$

## 6. Reed-Solomon Codes

## 7. Boolean Functions from $\mathit{RS}(\mathit{r},{\mathbf{2}}^{\mathit{m}})$

## 8. On the Number of Distinct Boolean Functions

## 9. Examples

#### 9.1. Example # 1. $\pi $ One-to-One

#### 9.2. Example # 2. $\pi $ Two to One

## 10. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

## References

- Lachowicz, P. Walsh–Hadamard Transform and Tests for Randomness of Financial Return- Series. Presented at Quant at Risk (Online), 7 April 2015. Available online: https://quantatrisk.com/2015/04/07/walsh-hadamard-transform-python-tests-for-randomness-of-financial-return-series/ (accessed on 15 March 2021).
- Menezes, A.J.; Van Oorschot, P.C.; Vanstone, S.A. Handbook of Applied Cryptography; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Pasalic, E. On Boolean Functions in Symmetric-Key Ciphers; Lund University: Lund, Sweden, 2003. [Google Scholar]
- Zeebaree, S.R. DES encryption and decryption algorithm implementation based on FPGA. Indones. J. Electr. Eng. Comput. Sci.
**2020**, 18, 774–781. [Google Scholar] [CrossRef] - Jiao, L.; Hao, Y.; Feng, D. Stream cipher designs: A review. Sci. China Inf. Sci.
**2020**, 63, 131101. [Google Scholar] [CrossRef][Green Version] - Cusick, T.W.; Stanica, P. Cryptographic Boolean Functions and Applications; Academic Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Biham, E.; Shamir, A. Differential cryptanalysis of DES-like cryptosystems. J. Cryptol.
**1991**, 4, 3–72. [Google Scholar] [CrossRef] - Langford, S.K.; Hellman, M.E. Differential-linear cryptanalysis. In Annual International Cryptology Conference; Springer: Berlin/Heidelberg, Germany, 1994; pp. 17–25. [Google Scholar]
- Chuan-Kun, W.; Dengguo, F. Boolean Functions and Their Applications in Cryptography; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar] [CrossRef]
- Henríquez, F.R. De la búsqueda de funciones booleanas con buenas propiedades criptográficas. Cinvestav
**2007**, 26, 50–65. [Google Scholar] - Chee, S.; Lee, S.; Lee, D.; Sung, S.H. On the correlation immune functions and their nonlinearity. In International Conference on the Theory and Application of Cryptology and Information Security; AsiaCrypt; Springer: Berlin/Heidelberg, Germany, 1996; pp. 232–243. [Google Scholar] [CrossRef]
- Forrié, R. The strict avalanche criterion: Spectral properties of Boolean functions and an extended definition. In Conference on the Theory and Application of Cryptography; Springer: Berlin/Heidelberg, Germany, 1988; pp. 450–468. [Google Scholar]
- Stichtenoth, H. A Note on Hermitian Codes Over GF(q2). IEEE Trans. Inf. Theory
**1988**, 34, 1345–1348. [Google Scholar] [CrossRef] - Forney, G.D. Concatenated Codes; Citeseer: Princeton, NJ, USA, 1966; Volume 11. [Google Scholar]
- Carlet, C. A Larger Class of Cryptographic Boolean Functions via a Study of the Maiorana-McFarland Construction. In Advances in Cryptology; Springer: Berlin/Heidelberg, Germany, 2002; Volume 2442, pp. 549–564. [Google Scholar] [CrossRef][Green Version]
- Guillot, P. Cryptographical boolean functions construction from linear codes. Boolean Funct. Cryptogr. Appl.
**2005**, 387, 141. [Google Scholar] - Wicker, S.B.; Bhargava, V.K. Reed-Solomon Codes and Their Applications; John Wiley & Sons: Hoboken, NJ, USA, 1999. [Google Scholar]
- Van Lint, J.H. Introduction to Coding Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012; Volume 86. [Google Scholar]

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**MDPI and ACS Style**

Sosa-Gómez, G.; Paez-Osuna, O.; Rojas, O.; Madarro-Capó, E.J. A New Family of Boolean Functions with Good Cryptographic Properties. *Axioms* **2021**, *10*, 42.
https://doi.org/10.3390/axioms10020042

**AMA Style**

Sosa-Gómez G, Paez-Osuna O, Rojas O, Madarro-Capó EJ. A New Family of Boolean Functions with Good Cryptographic Properties. *Axioms*. 2021; 10(2):42.
https://doi.org/10.3390/axioms10020042

**Chicago/Turabian Style**

Sosa-Gómez, Guillermo, Octavio Paez-Osuna, Omar Rojas, and Evaristo José Madarro-Capó. 2021. "A New Family of Boolean Functions with Good Cryptographic Properties" *Axioms* 10, no. 2: 42.
https://doi.org/10.3390/axioms10020042