Узагальнені послідовності Фібоначчі як основа побудови порогових схем розподілу секрету

Abstract

The theses substantiate the use of generalized Fibonacci sequences (Stakhov’s p-numbers) as a mathematical foundation for constructing threshold secret sharing schemes. The limitations of classical approaches (Shamir’s scheme) are demonstrated, and a scheme structure is proposed in which recurrent relations of the form F p(n) = Fp(n − 1) + Fp(n − p − 1) define the method of share generation. It is established that varying the generalization order p provides a mechanism for flexible construction of (t, n)-threshold schemes with additional properties: weighted authority distribution and share compactness. The results open prospects for developing a new class of cryptographic secret sharing protocols.