Solving $x+x^{2^{l}}+\cdots +x^{2^{ml}}=a$ over $\mathbb {F}_{2^{n}}$
Tóm tắt
This paper presents an explicit representation for the solutions of the equation
${\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}$
for any given positive integers k, l with l|k and n, in the closed field
${\overline {\mathbb {F}_{2}}}$
and in the finite field
$\mathbb {F}_{2^{n}}$
. As a by-product of our study, we are able to completely characterize the a’s for which this equation has solutions in
$\mathbb {F}_{2^{n}}$
.
Tài liệu tham khảo
Blake, I., Seroussi, G., Smart, N.: Elliptic curves in cryptography. Number 265 in London mathematical society lecture note series. Cambridge University Press, Cambridge (1999)
Carlet, C.: Boolean functions for cryptography and error correcting codes. Chapter of the monography. In: Crama, Y., Hammer, P. (eds.) Boolean models and methods in mathematics, computer science, and engineering, pp 257–397. Cambridge University Press, Cambridge (2010)
Carlet, C.: Vectorial Boolean functions for cryptography. Chapter of the monography. In: Crama, Y., Hammer, P. (eds.) Boolean models and methods in mathematics, computer science, and engineering, pp 398–469. Cambridge University Press, Cambridge (2010)
Mullen, G.L., Panario, D.: Handbook of finite fields. Discrete mathematics and its applications. CRC Press, Boca Raton (2013)