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Don't forget your roots: constant-time root finding over F2m
Side-channel Attack Post-quantum Cryptography Code-based Cryptography
2019/7/15
In the last few years, post-quantum cryptography has received much attention. NIST is running a competition to select some post-quantum schemes as standard. As a consequence, implementations of post-q...
Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences
Coppersmith's Algorithm Small Roots of Polynomial Equations LLL
2016/1/9
In a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysi...
Finding Roots in GF(p^n) with the Successive Resultant Algorithm
number theory Successive Resultant Algorithm
2016/1/8
The problem of solving polynomial equations over finite fields has many applications
in cryptography and coding theory. In this paper, we consider polynomial equations
over a “large” finite field wi...
Rounding LLL: Finding Faster Small Roots of Univariate Polynomial Congruences
Coppersmith's Algorithm Small Roots of Polynomial Equations
2014/3/10
In a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysi...
Roots of Square: Cryptanalysis of Double-Layer Square and Square+
Multivariate Cryptography Algebraic Cryptanalysis Square Double-Layer Square Square+ MinRank Key Recovery
2012/3/27
Square is a multivariate quadratic encryption scheme proposed in 2009. It is a specialization of Hidden Field Equations by using only odd characteristic fi elds and also X^2 as its central map. In add...
A remark on the computation of cube roots in finite fields
cube root finite field the Tonelli-Shanks method
2009/9/27
We consider the computation of cube roots in finite fields. For the computation
of square roots in finite fields, there are two typical methods; the
Tonelli-Shanks method [9, 12] and the Cipolla-Leh...
A remark on the computation of cube roots in finite fields
cube root finite field the Tonelli-Shanks method
2009/9/22
We consider the computation of cube roots in finite fields. For the computation
of square roots in finite fields, there are two typical methods; the
Tonelli-Shanks method [9, 12] and the Cipolla-Leh...
A strategy for recovering roots of bivariate polynomials modulo a prime
recovering roots bivariate polynomials modulo prime
2009/6/15
We show how, when given an irreducible bivariate polynomial with
coefficients in a finite prime field and an approximation to one of its
roots, one can recover that root efficiently, if the approxim...
A note on efficient computation of cube roots in characteristic 3
cube roots folklore algorithm
2009/4/1
The cost of the folklore algorithm for computing cube roots
in F3m in standard polynomial basis is less that one multiplication, but
still O(m2). Here we show that, if F3m is represented in trinomia...
An Algorithm for Finding Small Roots of Multivariate Polynomials over the Integers
Algorithm Finding Small Roots Multivariate Polynomials Integers
2008/9/10
In this paper we present a new algorithm for finding small
roots of multivariate polynomials over the integers based on lattice reduction
techniques. Our simpler heuristic method is inspired in algo...
Another Look at Square Roots and Traces (and Quadratic Equations) in Fields of Even Characteristic
Binary fields Polynomial basis Square root extraction
2008/9/8
We discuss irreducible polynomials that can be used to speed
up square root extraction in fields of characteristic two. We call such
polynomials square root friendly. The obvious applications are to...