15 results sorted by ID
Possible spell-corrected query: Boolean derivatives
Construction of quadratic APN functions with coefficients in $\mathbb{F}_2$ in dimensions $10$ and $11$
Yuyin Yu, Jingchen Li, Nadiia Ichanska, Nikolay Kaleyski
Foundations
Yu et al. described an algorithm for conducting computational searches for quadratic APN functions over the finite field $\mathbb{F}_{2^n}$, and used this algorithm to give a classification of all quadratic APN functions with coefficients in $\mathbb{F}_{2}$ for dimensions $n$ up to 9. In this paper, we speed up the running time of that algorithm by a factor of approximately $\frac{n \times 2^n}{n^3}$. Based on this result, we give a complete classification of all quadratic APN functions...
Some Classes of Cubic Monomial Boolean Functions with Good Second-Order Nonlinearity
RUCHI TELANG GODE
Secret-key cryptography
It is well known that estimating a sharp lower bound on the second-order nonlinearity of a general class of cubic Booleanfunction is a difficult task. In this paper for a given integer $n \geq 4$, some values of $s$ and $t$ are determined for which cubic monomial Boolean functions of the form $h_{\mu}(x)=Tr( \mu x^{2^s+2^t+1})$ $(n>s>t \geq 1)$ possess good lower bounds on their second-order nonlinearity. The obtained functions are worth considering for securing symmetric...
Quadratic-like balanced functions and permutations
Claude Carlet, Irene Villa
Secret-key cryptography
We study those $(n,n)$-permutations, and more generally those balanced $(n,m)$-functions, whose component functions all admit a derivative equal to constant function 1 (this property itself implies balancedness). We call these functions quadratic-like permutations (resp. quadratic-like balanced functions) since all quadratic balanced functions have this property. We show that all Feistel permutations, all crooked permutations and (more generally) all balanced strongly plateaued functions...
Simple Vs Vectorial: Exploiting Structural Symmetry to Beat the ZeroSum Distinguisher Applications to SHA3, Xoodyak and Bash
SAHIBA SURYAWANSHI, Shibam Ghosh, Dhiman Saha, Prathamesh Ram
Attacks and cryptanalysis
Higher order differential properties constitute a very insightful tool at the hands
of a cryptanalyst allowing for probing a cryptographic primitive from an algebraic perspective. In FSE 2017, Saha et al. reported SymSum (referred to as
SymSum_Vec in this paper), a new distinguisher based on higher order vectorial
Boolean derivatives of SHA-3, constituting one of the best distinguishers on the
latest cryptographic hash standard. SymSum_Vec exploits the difference in the
algebraic degree...
Improving logarithmic derivative lookups using GKR
Shahar Papini, Ulrich Haböck
Cryptographic protocols
In this informal note, we instantiate the Goldwasser-Kalai-Rothblum (GKR) protocol to prove fractional sumchecks as present in lookup arguments based on logarithmic derivatives, with the following impact on the prover cost of logUp (IACR eprint 2022/1530):
When looking up $M\geq 1$ columns in a (for the sake of simplicity) single column table, the prover has to commit only to a single extra column, i.e. the multiplicities of the table entries.
In order to carry over the GKR fractional...
On cubic-like bent Boolean functions
Claude Carlet, Irene Villa
Secret-key cryptography
Cubic bent Boolean functions (i.e. bent functions of algebraic degree at most 3) have the property that, for every nonzero element $a$ of $\mathbb{F}_2^n$, the derivative $D_af(x)=f(x)+f(x+a)$ of $f$ admits at least one derivative $D_bD_af(x)=f(x)+f(x+a)+f(x+b)+f(x+a+b)$ that is equal to constant function 1. We study the general class of those Boolean functions having this property, which we call cubic-like bent. We study the properties of such functions and the structure of their...
Multivariate lookups based on logarithmic derivatives
Ulrich Haböck
Cryptographic protocols
Logarithmic derivatives translate products of linear factors into sums of their reciprocals, turning zeroes into simple poles of same multiplicity. Based on this simple fact, we construct an interactive oracle proof for multi-column lookups over the boolean hypercube, which makes use of a single multiplicity function instead of working with a rearranged union of table and witnesses. For single-column lookups the performance is comparable to the well-known Plookup strategy used by...
Classification of all DO planar polynomials with prime field coefficients over GF(3^n) for n up to 7
Diana Davidova, Nikolay Kaleyski
Foundations
We describe how any function over a finite field $\mathbb{F}_{p^n}$ can be represented in terms of the values of its derivatives. In particular, we observe that a function of algebraic degree $d$ can be represented uniquely through the values of its derivatives of order $(d-1)$ up to the addition of terms of algebraic degree strictly less than $d$. We identify a set of elements of the finite field, which we call the degree $d$ extension of the basis, which has the property that for any...
Recovering or Testing Extended-Affine Equivalence
Anne Canteaut, Alain Couvreur, Léo Perrin
Secret-key cryptography
Extended Affine (EA) equivalence is the equivalence relation between two vectorial Boolean functions $F$ and $G$ such that there exist two affine permutations $A$, $B$, and an affine function $C$ satisfying $G = A \circ F \circ B + C$. While a priori simple, it is very difficult in practice to test whether two functions are EA-equivalent. This problem has two variants: EA-testing deals with figuring out whether the two functions can be EA-equivalent, and EA-recovery is about recovering the...
On derivatives of polynomials over finite fields through integration
Enes Pasalic, Amela Muratovic-Ribic, Samir Hodzic, Sugata Gangopadhyay
Secret-key cryptography
In this note, using rather elementary technique and the derived formula that relates the coefficients of a polynomial over a finite field and its derivative, we deduce many interesting results related to derivatives of Boolean functions and derivatives of mappings over finite fields. For instance, we easily identify several infinite classes of polynomials which cannot possess linear structures. The same technique can be applied for deducing a nontrivial upper bound on the degree of...
Some properties of q-ary functions based on spectral analysis
Deep Singh, Maheshanand Bhaintwal
Secret-key cryptography
In this paper, we generalize some existing results on Boolean
functions to the $q$-ary functions defined over $\BBZ_q$, where
$q\geq 2$ is an integer, and obtain some new characterization of
$q$-ary functions based on spectral analysis. We provide a
relationship between Walsh-Hadamard spectra of two $p$-ary functions
$f$ and $g$ (for $p$ a prime) and their derivative $D_{f, g}$. We
provide a relationship between the Walsh-Hadamard spectra and the
decompositions of any two $p$-ary functions....
Additive autocorrelation of some classes of cubic semi-bent Boolean functions
Deep Singh, Maheshanand Bhaintwal
Secret-key cryptography
In this paper, we investigate the relation between the autocorrelation of a cubic Boolean function $f\in \cB_n$ at $a \in \BBF_{2^n}$ and the kernel of the bilinear form associated with $D_{a}f$, the derivative of $f$ at $a$. Further, we apply this technique to obtain the tight upper bounds of absolute indicator and sum-of-squares indicator for avalanche characteristics of various classes of highly nonlinear non-bent cubic Boolean functions.
Distinguishing Properties of Higher Order Derivatives of Boolean Functions
Ming Duan, Xuejia Lai, Mohan Yang, Xiaorui Sun, Bo Zhu
Foundations
Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define \emph{fast point} as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a $n$-variable Boolean function form a linear subspace and its dimension plus the algebraic degree of the function is at...
The Lower Bounds on the Second Order Nonlinearity of Cubic Boolean Functions
Xuelian Li, Yupu Hu, Juntao Gao
Secret-key cryptography
It is a difficult task to compute the $r$-th order nonlinearity of a
given function with algebraic degree strictly greater than $r>1$.
Even the lower bounds on the second order nonlinearity is known only
for a few particular functions. We investigate the lower bounds on
the second order nonlinearity of cubic Boolean functions
$F_u(x)=Tr(\sum_{l=1}^{m}\mu_{l}x^{d_{l}})$, where $u_{l} \in
F_{2^n}^{*}$, $d_{l}=2^{i_{l}}+2^{j_{l}}+1$, $i_{l}$ and $j_{l}$ are
positive integers, $n>i_{l}> j_{l}$....
On second order nonlinearities of cubic monomial Boolean functions
Ruchi Gode, Sugata Gangopadhyay
Secret-key cryptography
We study cubic monomial Boolean functions of the
form $Tr_1^n(\mu x^{2^i+2^j+1})$ where
$\mu \in \mathbb{F}_{2^n}$. We prove that the functions
of this form do not have any affine derivative. A lower
bound on the second order nonlinearities of these
functions is also derived.
Yu et al. described an algorithm for conducting computational searches for quadratic APN functions over the finite field $\mathbb{F}_{2^n}$, and used this algorithm to give a classification of all quadratic APN functions with coefficients in $\mathbb{F}_{2}$ for dimensions $n$ up to 9. In this paper, we speed up the running time of that algorithm by a factor of approximately $\frac{n \times 2^n}{n^3}$. Based on this result, we give a complete classification of all quadratic APN functions...
It is well known that estimating a sharp lower bound on the second-order nonlinearity of a general class of cubic Booleanfunction is a difficult task. In this paper for a given integer $n \geq 4$, some values of $s$ and $t$ are determined for which cubic monomial Boolean functions of the form $h_{\mu}(x)=Tr( \mu x^{2^s+2^t+1})$ $(n>s>t \geq 1)$ possess good lower bounds on their second-order nonlinearity. The obtained functions are worth considering for securing symmetric...
We study those $(n,n)$-permutations, and more generally those balanced $(n,m)$-functions, whose component functions all admit a derivative equal to constant function 1 (this property itself implies balancedness). We call these functions quadratic-like permutations (resp. quadratic-like balanced functions) since all quadratic balanced functions have this property. We show that all Feistel permutations, all crooked permutations and (more generally) all balanced strongly plateaued functions...
Higher order differential properties constitute a very insightful tool at the hands of a cryptanalyst allowing for probing a cryptographic primitive from an algebraic perspective. In FSE 2017, Saha et al. reported SymSum (referred to as SymSum_Vec in this paper), a new distinguisher based on higher order vectorial Boolean derivatives of SHA-3, constituting one of the best distinguishers on the latest cryptographic hash standard. SymSum_Vec exploits the difference in the algebraic degree...
In this informal note, we instantiate the Goldwasser-Kalai-Rothblum (GKR) protocol to prove fractional sumchecks as present in lookup arguments based on logarithmic derivatives, with the following impact on the prover cost of logUp (IACR eprint 2022/1530): When looking up $M\geq 1$ columns in a (for the sake of simplicity) single column table, the prover has to commit only to a single extra column, i.e. the multiplicities of the table entries. In order to carry over the GKR fractional...
Cubic bent Boolean functions (i.e. bent functions of algebraic degree at most 3) have the property that, for every nonzero element $a$ of $\mathbb{F}_2^n$, the derivative $D_af(x)=f(x)+f(x+a)$ of $f$ admits at least one derivative $D_bD_af(x)=f(x)+f(x+a)+f(x+b)+f(x+a+b)$ that is equal to constant function 1. We study the general class of those Boolean functions having this property, which we call cubic-like bent. We study the properties of such functions and the structure of their...
Logarithmic derivatives translate products of linear factors into sums of their reciprocals, turning zeroes into simple poles of same multiplicity. Based on this simple fact, we construct an interactive oracle proof for multi-column lookups over the boolean hypercube, which makes use of a single multiplicity function instead of working with a rearranged union of table and witnesses. For single-column lookups the performance is comparable to the well-known Plookup strategy used by...
We describe how any function over a finite field $\mathbb{F}_{p^n}$ can be represented in terms of the values of its derivatives. In particular, we observe that a function of algebraic degree $d$ can be represented uniquely through the values of its derivatives of order $(d-1)$ up to the addition of terms of algebraic degree strictly less than $d$. We identify a set of elements of the finite field, which we call the degree $d$ extension of the basis, which has the property that for any...
Extended Affine (EA) equivalence is the equivalence relation between two vectorial Boolean functions $F$ and $G$ such that there exist two affine permutations $A$, $B$, and an affine function $C$ satisfying $G = A \circ F \circ B + C$. While a priori simple, it is very difficult in practice to test whether two functions are EA-equivalent. This problem has two variants: EA-testing deals with figuring out whether the two functions can be EA-equivalent, and EA-recovery is about recovering the...
In this note, using rather elementary technique and the derived formula that relates the coefficients of a polynomial over a finite field and its derivative, we deduce many interesting results related to derivatives of Boolean functions and derivatives of mappings over finite fields. For instance, we easily identify several infinite classes of polynomials which cannot possess linear structures. The same technique can be applied for deducing a nontrivial upper bound on the degree of...
In this paper, we generalize some existing results on Boolean functions to the $q$-ary functions defined over $\BBZ_q$, where $q\geq 2$ is an integer, and obtain some new characterization of $q$-ary functions based on spectral analysis. We provide a relationship between Walsh-Hadamard spectra of two $p$-ary functions $f$ and $g$ (for $p$ a prime) and their derivative $D_{f, g}$. We provide a relationship between the Walsh-Hadamard spectra and the decompositions of any two $p$-ary functions....
In this paper, we investigate the relation between the autocorrelation of a cubic Boolean function $f\in \cB_n$ at $a \in \BBF_{2^n}$ and the kernel of the bilinear form associated with $D_{a}f$, the derivative of $f$ at $a$. Further, we apply this technique to obtain the tight upper bounds of absolute indicator and sum-of-squares indicator for avalanche characteristics of various classes of highly nonlinear non-bent cubic Boolean functions.
Higher order differential cryptanalysis is based on the property of higher order derivatives of Boolean functions that the degree of a Boolean function can be reduced by at least 1 by taking a derivative on the function at any point. We define \emph{fast point} as the point at which the degree can be reduced by at least 2. In this paper, we show that the fast points of a $n$-variable Boolean function form a linear subspace and its dimension plus the algebraic degree of the function is at...
It is a difficult task to compute the $r$-th order nonlinearity of a given function with algebraic degree strictly greater than $r>1$. Even the lower bounds on the second order nonlinearity is known only for a few particular functions. We investigate the lower bounds on the second order nonlinearity of cubic Boolean functions $F_u(x)=Tr(\sum_{l=1}^{m}\mu_{l}x^{d_{l}})$, where $u_{l} \in F_{2^n}^{*}$, $d_{l}=2^{i_{l}}+2^{j_{l}}+1$, $i_{l}$ and $j_{l}$ are positive integers, $n>i_{l}> j_{l}$....
We study cubic monomial Boolean functions of the form $Tr_1^n(\mu x^{2^i+2^j+1})$ where $\mu \in \mathbb{F}_{2^n}$. We prove that the functions of this form do not have any affine derivative. A lower bound on the second order nonlinearities of these functions is also derived.