General Linear Group Action on Tensors: A Candidate for PostQuantum Cryptography. Starting from the oneway group action framework of Brassard and Yung (Crypto u002790), we revisit building cryptography based on group actions. Several previous candidates for oneway group actions no longer stand, due to progress both on classical algorithms (e.g., graph isomorphism) and quantum algorithms (e.g., discrete logarithm).  :47,133,propose the general linear group action on tensors as a new candidate to build cryptography based on group actions. Recent works (FutornyGrochowSergeichuk, Lin. Alg. Appl., 2019) suggest that the underlying algorithmic problem, the tensor isomorphism problem, is the hardest one among several isomorphism testing problems arising from areas including coding theory, computational group theory, and multivariate cryptography. We present evidence to justify the viability of this proposal from comprehensive study of the stateofart heuristic algorithms, theoretical algorithms, and hardness results, as well as quantum algorithms.  then introduce a new notion called pseudorandom group actions to further develop groupaction based cryptography. Briefly speaking, given a group G acting on a set S, we assume that it is hard to distinguish two distributions of (s, t) either uniformly chosen from Stimes S, or where s is randomly chosen from S and t is the result of applying a random group action of gin G on s. This subsumes the classical decisional DiffieHellman assumption when specialized to a particular group action. We carefully analyze various attack strategies that support the general linear group action on tensors as a candidate for this assumption.  we :238,establish the quantum security of several cryptographic primitives based on the oneway group action assumption and the pseudorandom group action assumption.

Semiquantum cryptography Semiquantum communication, a model introduced by Boyer et al. (Phys Rev Lett 99:140501, 2007), involves the use of fully quantum users and semiquantum, or xe2x80x9cclassicalxe2x80x9d users. These restricted users are only allowed to interact with the quantum channel in a limited manner. Originally introduced to study the keydistribution problem, semiquantum research has since expanded, and continues to grow, with new protocols, security proof methods, experimental implementations, and new cryptographic applications beyond key distribution. Research in the field of semiquantum cryptography requires new insights into working with restricted protocols and, so, the tools and techniques derived in this field can translate to results in broader quantum information science. Furthermore, other questions such as the connection between quantum and classical processing, including how classical information processing can be used to counteract a quantum deficiency in a protocol, can shed light on important theoretical questions. This work surveys the history and current state of the art in semiquantum research. We discuss the model and several protocols offering the reader insight into how protocols are constructed in this realm. We discuss security proof methods and how classical postprocessing can be used to counteract usersxe2x80x99 inability to perform certain quantum operations. Moving beyond key distribution, we survey current work in other semiquantum cryptographic protocols and current trends. We also survey recent work done in attempting to construct practical semiquantum systems including recent experimental results in this field. Finally, as this is still a growing field, we highlight, throughout this survey, several open problems that we feel are important to investigate in the hopes that this will spur even more research in this topic.