Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo $$F_n$$ F n , for every positive integer n not divisible by 3, where $$F_n$$ F n denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo $$F_n$$ F n , for every fixed integer $$a \ge 3$$ a ≥ 3 and for all positive integers n with $$\gcd (a, F_n) = 1$$ gcd ( a , F n ) = 1 . Our proof makes use of the so-called base-$$\varphi $$ φ expansion of real numbers.

We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group's structure over generalized Pell conics (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the Rédei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields.

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Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo Fn, for every positive integer n not divisible by 3, where Fn denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo Fn, for every fixed integer a ≥ 3 and for all positive integers n with gcd(a, Fn) = 1. Our proof makes use of the so-called base-ϕ expansion of real numbers.