Doi: http://dx.doi.org/10.15446/recolma.v49n1.54163

¹Universidad Nacional de Colombia, Medellín, Colombia. Email: jporras@unal.edu.co
²Universidad Nacional de Colombia, Medellín, Colombia. Email: jbbaena@unal.edu.co
³University of Cincinnati, Cincinnati, OH, USA. Email: jintai.ding@uc.edu

We present a new method for building pairs of HFE polynomials of high degree, such that the map constructed with one of these pairs is easy to invert. The inversion is accomplished using a low degree polynomial of Hamming weight three, which is derived from a special reduction via Hamming weight three polynomials produced by these two HFE polynomials. This allows us to build new candidates for multivariate trapdoor functions in which we use the pair of HFE polynomials to fabricate the core map. We performed the security analysis for the case where the base field is GF(2) and showed that these new trapdoor functions have high degrees of regularity, and therefore they are secure against the direct algebraic attack. We also give theoretical arguments to show that these new trapdoor functions over GF(2) are secure against the MinRank attack as well.

Key words: Multivariate cryptography, HFE polynomials, HFE cryptosystem, Trapdoor functions, Zhuang-zi algorithm.

Presentamos un nuevo método de reducción que permite construir parejas de polinomios HFE de grado alto, tal que la función construida con cada una de estas parejas de polinomios es fácil de invertir. Para invertir la pareja de polinomios usamos un polinomio de grado bajo y de peso de Hamming tres, el cual es derivado mediante un método especial de reducción que involucra polinomios de peso de Hamming tres producidos a partir de los dos polinomios HFE. Esto nos permite construir nuevas candidatas para funciones trampa multivariadas usando la pareja de polinomios HFE para construir la función central. Realizamos un análisis de seguridad cuando el campo base es GF(2) y mostramos que estas nuevas funciones trampa multivariadas tienen grado de regularidad alto, y por lo tanto resisten el ataque algebraico. Además damos argumentos teóricos para mostrar que estas nuevas funciones trampa sobre GF(2) también resisten el ataque MinRank.

Palabras clave: Criptografía multivariada, polinomios HFE, criptosistema HFE, funciones trampa, algoritmo Zhuang-zi.

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

Doi: http://dx.doi.org/ @ARTICLE{RCMv49n1a03,
    AUTHOR  = {Porras, Jaiberth and Baena, John B. and Ding, Jintai},
    TITLE   = {{New Candidates for Multivariate Trapdoor Functions}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {57--76}
}