Projects

Publications

• Dickson permutation polynomials that decompose in cycles of the same length. (2008). Ivelisse M. Rubio, Gary L. Mullen, Carlos Corrada, & Francis N. Castro.
  
  Abstract: In this paper we study permutations of finite fields  F_q given by Dickson polynomials. For certain families of Dickson permutation polynomials we give the necessary and sufficient conditions on the degree of the polynomial in order to obtain a permutation that decomposes in cycles of the same length.
  
  Final version published in Finite Fields: Theory and Applications, Contemporary Mathematics, vol. 461 (2008), Amer. Math. Soc., Providence, RI, pp. 229-239.

• Cyclic Decomposition of Permutations of Finite Fields Obtained Using Monomials. (2003). Ivelisse M. Rubio & Carlos J. Corrada-Bravo.
  
  Abstract: In this paper we study permutations of finite fields F_q that decompose as products of cycles of the same length, and are obtained using monomials  xⁱ ∈ F_q[x]. We give the necessary and sufficient conditions on the exponent i to obtain such permutations. We also present formulas for counting the number of this type of permutations. An application to the construction of encoders for turbo codes is also discussed.
  
  Final version published in Finite Fields and Applications, Lecture Notes in Computer Science 2948, Springer, 2003.

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Publications

• Explicit Formulas for Monomial Involutions over Finite Fields. (2017). Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj & Ivelisse Rubio.
  
  Abstract: Permutations of finite fields have important applications in cryptography and coding theory. Involutions are permutations that are its own inverse and are of particular interest because the implementation used for coding can also be used for decoding. We present explicit formulas for all the involutions of Fq that are given by monomials and for their fixed points.
  
  Final version published in Advances in Mathematics of Communications (AMC) (pp. 301-306), Volume: 11, Number: 2, doi:10.3934/amc.2017022, 2017.

Student Projects

Master Students

• Involutions of Finite Fields Obtained From Binomials of the Form $x^m(x^{\frac{q−1}{2}} + a)$. Thesis. (2022). Lillian González.
  
  Abstract: Permutations of finite fields Fq have many applications ranging from cryptography and combinatorics to the theory of computation. In many of these applications, a permutation and its inverse are stored in memory. A good option to reduce the memory footprint is to generate the permutation with a polynomial at the time of implementation. A better option is to use a permutation polynomial that is its own inverse; in this case, the permutation is called an involution. In applications to cryptography, the number of fixed points is correlated with its cryptographic properties. In 2018, Zheng et al. characterized involutions of the form $x^mh(x^s)$ over $F_q$, and, in 2017, Castro et al. gave explicit formulas for monomial involutions of $F_q$ and their fixed points. The next simplest polynomials to study would be the binomials. In this work we characterize involutions of $F_q$ of the form $x^m(x^{\frac{q−1}{2}} + a)$ in terms of the number of fixed points. We present explicit formulas for obtaining these involutions and formulas for their fixed points. Additionally, we give an improvement of Zheng et al.’s result for polynomial involutions of $F_q$ of the form $x^mh(x^{\frac{q−1}{2}})$.

Undergraduate Students

• Involuciones de Cuerpos Finitos Obtenidos por Binomios. Technical Report. (2019). Dylan Cruz & Andrés Ramos.
  
  Abstract: Las permutaciones de cuerpos finitos tienen aplicaciones en varios campos de las Matemáticas y Ciencias de Cómputos, como lo es la Criptografía. En específico, las permutaciones que son su propia inversa (involuciones) son de interés porque ofrecen una ventaja al implementarlas ya que requieren menos memoria. Estudiamos binomios de la forma $x^m(x \frac{q−3}{2} +A)$ sobre $F_q$ con el propósito de encontrar las condiciones en $A$ y $q$ tal que el binomio sea una involución. Aquí probaremos que, si $m = 1$, esta familia de binomios nunca produce permutaciones cuando de $F_q$ donde $q$ es potencia de un primo impar. Conjeturamos que esto es cierto para toda $m$.

• Involuciones de Cuerpos Finitos obtenidos por Binomios . Technical Report. (2018). Lillian González.
  
  Abstract: Las permutaciones sobre cuerpos finitos son importantes ya que tienen aplicaciones desde cifrados de voz hasta teoría de computabilidad y criptografía. Polinomios que generan permutaciones son llamados polinomios de permutación; si estos polinomios son su propia inversa, son llamados involuciones. En esta investigación, queremos caracterizar binomios de involución la forma $x^m(x^{\frac{q−1}{2}} + a)$ sobre cuerpos finitos de característica $p$.

• Explicit Formulas for Monomial Involutions over Finite Fields. (2017). Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj & Ivelisse Rubio.
  
  Abstract: Permutations of finite fields have important applications in cryptography and coding theory. Involutions are permutations that are its own inverse and are of particular interest because the implementation used for coding can also be used for decoding. We present explicit formulas for all the involutions of Fq that are given by monomials and for their fixed points.
  
  Final version published in Advances in Mathematics of Communications (AMC) (pp. 301-306), Volume: 11, Number: 2, doi:10.3934/amc.2017022, 2017.

• Dickson Polynomials that Generate Involutions with More Than 3 Fixed Points. Technical Report. (2016). Natalia Pacheco.
  
  Abstract: Let $F_q$ be a finite field of prime order $q$. This research studies the Dickson Polynomials $D_i(x, a)$ that generate a permutation of $F_q$ such that $D_i$ decomposes into cycles of uniform length $2$, i.e. $D_i$ is an involution. Our goal is to provide a necessary and sufficient formula to construct $I$ such that the Dickson Polynomial $D_i(x, a)$ has a particular amount d of fixed points. We wish to find a formula for $I$ in terms of $d$ and $q$, similarly to what we were able to achieve with Permutation Monomials in [1].

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Publications

• Permutation binomials of the form x^r(x^{q – 1} + a) over F_q^e. (2022). Ariane Masuda, Ivelisse Rubio, & Javier Santiago.
  
  Abstract: We present several existence and nonexistence results for permutation binomials of the form x^r(x^{q – 1} + a), where e ≥ 2 and a ∈ F^*_q^e. As a consequence, we obtain a complete characterization of such permutation binomials over F_q² , F_q³ , F_q⁴ , F_p⁵ , and F_p⁶ , where p is an odd prime.
  
  Final version published in Finite Fields and Their Applications, Vol. 79, March 2022, https://doi.org/10.1016/j.ffa.2022.102003.

Student Projects

Undergraduate Students

• Permutation binomials of the form x^r(x^{q – 1} + a) over F_q^e. (2022). Ariane Masuda, Ivelisse Rubio, & Javier Santiago.
  
  Abstract: We present several existence and nonexistence results for permutation binomials of the form x^r(x^{q – 1} + a), where e ≥ 2 and a ∈ F^*_q^e. As a consequence, we obtain a complete characterization of such permutation binomials over F_q² , F_q³ , F_q⁴ , F_p⁵ , and F_p⁶ , where p is an odd prime.
  
  Final version published in Finite Fields and Their Applications, Vol. 79, March 2022, https://doi.org/10.1016/j.ffa.2022.102003.

• Binomials that do not produce permutations. Poster. (2019). Dylan Cruz & Andrés Ramos.
  
  Abstract: Permutations have important applications in fields such as coding theory and cryptography. We study binomials of the form $x^m(x^{\frac{q-3}{2} }+B)$ over finite fields $F_q$. In this research we prove that this class of binomials never permutes a finite fields when: (a) $A = \alpha^k$, where $k$ is odd and $alpha$ is a primitive root of the field, (b) $A = \alpha^k$, where $k$ is even, $\alpha$ is a primitive root of the field and $q = 4h+1$, $h ∈ N$. We are left with the case $A = \alpha^k$ where $k$ is even and $q = 4h + 3$, $h ∈ N$. We conjecture that the remaining class of polynomials do not permute $F_q$.

• Sumas de Monomios de Permutación. Technical Report. (2019). Karla Borges & Javier Santiago.
  
  Abstract: Las permutaciones juegan un papel importante en las comunicaciones, ya que se utilizan en una variedad de aplicaciones, desde la teoría de códigos hasta la criptografía. Los polinomios que generan permutaciones se llaman polinomios de permutación. Nosotros estudiamos binomios de la forma $f(x) = x^m +Ax^n = x^n(x^{m−n} +A)$ sobre cuerpos finitos $F_p$, donde $x^m$ y $Ax^n$ son también monomios de permutación. El objetivo es encontrar las condiciones para que $f(x)$ permute $F_p$. Aquí demostramos que $f(x)$ nunca es polinomio de permutación para $F_p$, donde $q = 2p+1$, $p$ primo, y conjeturamos que si $x^n(x^{m−n} +A)$ es un polinomio de permutación, donde $m − n = \frac{p−1}{d}$ , entonces $A^d + (−1)^{d+1}$ es un residuo edésimo.

• Generating Permutation Trinomials over Finite Fields. Technical Report. (2014). Christian A. Rodríguez & Alex D. Santos.
  
  Abstract: Permutation polynomials over finite fields have many applications in areas such as coding theory and cryptography. We consider polynomials of the form $F_{a,b} (X) = X(X^{\frac{q−1}{d_1}} + aX^{\frac{q−1}{d_2}} + b)$, where $a, b \in {F_q}^*$ and $d_1 < d_2$. We construct partitions of these polynomials where polynomials in the same partition have value sets of equal cardinality. As a consequence we provide families of permutation polynomials.

• Number of Permutation Polynomials. Poster. (2014). Christian A. Rodríguez & Alex D. Santos.
  
  Abstract: Given $q = p^r$, $d_1$ and $d_2$, we construct partitions of polynomials of the form $F_{a,b} (X) = X(X^{\frac{q−1}{d_1}} + aX^{\frac{q−1}{d_2}} + b)$, where $a, b \in {F_q}^*$, that have value sets of the same cardinality. As a consequence we provide families of permutation polynomials and of polynomials with small value sets.