Polynomial Approximation and Arithmetic Complexity of the Diffie-Hellman Secret Key

Abstract

Let g be a primitive root of a finite field ${\mathbb{F}_q}$ of q elements. One of the most popular public key cryptosystems, the Diffie- Hellman key exchange protocol, is based on the still unproved assumption that recovering the value of the Diffie-Hellman secret key

$$K(x,y) = {g^{xy}}$$

from the known values of g^Xand g^yis essentially equivalent to the discrete logarithm problem and therefore is hard. Here we show that even computation of ${g^{{x^2}}}$ from g^xcannot be realized by a polynomial of low degree.