Symmetric and asymmetric cryptographic key exchange protocols in the octonion algebra
The design of cryptosystems and cryptographic protocols using as primitives some hard problems in non-commutative group theory is one of the ways to construct schemes secure against the (supposed) next arrival of the quantum computation. Following this way the present paper proposes three cryptographic key exchange protocols based on the non-associative octonion algebra \(\mathcal{O}\),\, see [\textit{J. H. Conway} and \textit{D. A. Smith}, On quaternions and octonions: their geometry, arithmetic, and symmetry. Natick, MA: A K Peters (2003; Zbl 1098.17001)]. The properties and the arithmetic of \(\mathcal{O}\)\, are summarized in Section 1.\par The first proposed scheme (Section 2) is the translation of the public-key cryptosystem RSA to the octonion setting. Giving the totient function \(\lambda(k,m)\),\, with \(k\in \mathcal{O}\)\, an integral octonion and \(m\)\, a natural number product of two primes \(p\), \(q\)\, one randomly takes \(e\)\, coprime to \(\lambda(k,m)\)\, (the public key) and \(d= e^{-1} \bmod \lambda(k,m)\)\, (the private key). The security of the scheme is based both on the difficulty of the factorization of \(m\)\, and on the difficulty of the computation of \(\lambda(k,p)\)\, and \(\lambda(k,q)\).\par The two other proposals are symmetric key exchange protocols. Section 3 first provides a key exchange algorithm in a quaternion subalgebra \(D\subset \mathcal{O}\)\, and then with two such keys generates an octonion key, which is an automorphism of \(\mathcal{O}\).\par Finally Section 4 shows the second symmetric key exchange algorithm based on a derivation of \(\mathcal{O}\).
scientific article; zbMATH DE number 7357743
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