Orthogonal-state-based cryptography in quantum mechanics and local post-quantum theories

TL;DR

This paper introduces a framework for cryptographic reduction between quantum key distribution and quantum secure direct communication, demonstrating how security in one implies security in the other, and proposes a new orthogonal-state-based key distribution protocol secure in post-quantum theories.

Contribution

It establishes a formal reduction framework linking QKD and QSDC security, and introduces a novel orthogonal-state-based protocol secure beyond standard quantum mechanics.

Findings

Block encoding transforms QKD into QSDC protocols.

Security of QKD and QSDC are interconnected through reductions.

Proposed orthogonal-state-based protocol is secure in local post-quantum theories.

Abstract

We introduce the concept of cryptographic reduction, in analogy with a similar concept in computational complexity theory. In this framework, class $A$ of crypto-protocols reduces to protocol class $B$ in a scenario $X$, if for every instance $a$ of $A$, there is an instance $b$ of $B$ and a secure transformation $X$ that reproduces $a$ given $b$, such that the security of $b$ guarantees the security of $a$. Here we employ this reductive framework to study the relationship between security in quantum key distribution (QKD) and quantum secure direct communication (QSDC). We show that replacing the streaming of independent qubits in a QKD scheme by block encoding and transmission (permuting the order of particles block by block) of qubits, we can construct a QSDC scheme. This forms the basis for the \textit{block reduction} from a QSDC class of protocols to a QKD class of protocols, whereby if the latter is secure, then so is the former. Conversely, given a secure QSDC protocol, we can of course construct a secure QKD scheme by transmitting a random key as the direct message. Then the QKD class of protocols is secure, assuming the security of the QSDC class which it is built from. We refer to this method of deduction of security for this class of QKD protocols, as \textit{key reduction}. Finally, we propose an orthogonal-state-based deterministic key distribution (KD) protocol which is secure in some local post-quantum theories. Its security arises neither from geographic splitting of a code state nor from Heisenberg uncertainty, but from post-measurement disturbance.