Efficient Tate Pairing Computation for Supersingular Elliptic Curves over Binary Fields

We present a closed formula for the Tate pairing computation for supersingular elliptic curves defined over the binary field F_{2^m} of odd dimension. There are exactly three isomorphism classes of supersingular elliptic curves over F_{2^m} for odd m and our result is applicable to all these curves. Moreover we show that our algorithm and also the Duursma-Lee algorithm can be modified to another algorithm which does not need any inverse Frobenius operation (square root or cube root extractions) without sacrificing any of the computational merits of the original algorithm. Since the computation of the inverse Frobenius map is not at all trivial in a polynomial basis and since a polynomial basis is still a preferred choice for the Tate pairing computation in many situations, this new algorithm avoiding the inverse Frobenius operation has some advantage over the existing algorithms

Partially APN Boolean functions and classes of functions that are not APN infinitely often

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field \F_2, extending some earlier results of Leander and Rodier.Comment: 24 pages; to appear in Cryptography and Communication

cc-differential uniformity, (almost) perfect cc-nonlinearity, and equivalences

In this article, we introduce new notions $cc$-differential uniformity, $cc$-differential spectrum, PccN functions and APccN functions, and investigate their properties. We also introduce $c$-CCZ equivalence, $c$-EA equivalence, and $c1$-equivalence. We show that $c$-differential uniformity is invariant under $c1$-equivalence, and $cc$-differential uniformity and $cc$-differential spectrum are preserved under $c$-CCZ equivalence. We characterize $cc$-differential uniformity of vectorial Boolean functions in terms of the Walsh transformation. We investigate $cc$-differential uniformity of power functions $F(x)=x^d$. We also illustrate examples to prove that $c$-CCZ equivalence is strictly more general than $c$-EA equivalence.Comment: 18 pages. Comments welcom

On r-th Root Extraction Algorithm in F_q For q=lr^s+1 (mod r^(s+1)) with 0 < l < r and Small s

We present an r-th root extraction algorithm over a finite field F_q. Our algorithm precomputes a primitive r^s-th root of unity where s is the largest positive integer satisfying r^s| q-1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the r-th root computation and is favorably compared to the existing algorithms

Square Root Algorithm in F_q for q=2^s+1 (mod 2^(s+1))

We present a square root algorithm in F_q which generalizes Atkins\u27s square root algorithm for q=5(mod 8) and Kong et al.\u27s algorithm for q=9(mod 16) Our algorithm precomputes a primitive 2^s-th root of unity where s is the largest positive integer satisfying 2^s| q-1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favorably compared with the algorithms of Atkin, Muller and Kong et al

Trace Expression of r-th Root over Finite Field

Efficient computation of $r$-th root in $\mathbb F_q$ has many applications in computational number theory and many other related areas. We present a new $r$-th root formula which generalizes Müller\u27s result on square root, and which provides a possible improvement of the Cipolla-Lehmer algorithm for general case. More precisely, for given $r$-th power $c\in \mathbb F_q$, we show that there exists $\alpha \in \mathbb F_{q^r}$ such that $Tr\left(\alpha^\frac{(\sum_{i=0}^{r-1}q^i)-r}{r^2}\right)^r=c$ where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots +\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible polynomial of degree $r$ over $\mathbb F_q$

New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field

In this paper, we present a new cube root algorithm in finite field $\mathbb{F}_{q}$ with $q$ a power of prime, which extends the Cipolla-Lehmer type algorithms \cite{Cip,Leh}. Our cube root method is inspired by the work of Müller \cite{Muller} on quadratic case. For given cubic residue $c \in \mathbb{F}_{q}$ with $q \equiv 1 \pmod{9}$, we show that there is an irreducible polynomial $f(x)=x^{3}-ax^{2}+bx-1$ with root $\alpha \in \mathbb{F}_{q^{3}}$ such that $Tr(\alpha^{\frac{q^{2}+q-2}{9}})$ is a cube root of $c$. Consequently we find an efficient cube root algorithm based on third order linear recurrence sequence arising from $f(x)$. Complexity estimation shows that our algorithm is better than previously proposed Cipolla-Lehmer type algorithms