Shaping Postquantum Cryptography: The Hidden Subgroup and Shift Problems

The security of popular public key-cryptographic protocols, such as RSA, Diffie–Hellman key exchange and the digital signature algorithm (DSA), is endangered by the advent of quantum computers. Shor brought a big breakthrough with his quantum algorithm that can be used to factor an arbitrarily large integer into the product of its prime factors, hence jeopardizing the security of RSA, and that at the same time also solves the Discrete Logarithm Problem, which raises issues for certain Diffie–Hellman-based cryptosystems and digital signatures. It is hence crucial to upgrade our current tools for postquantum cryptography: it should be infeasible, even using quantum algorithms, to break the new cryptosystems. Popular candidates include for example elliptic curve or lattice-based cryptography, but they share something in common: they are specific cases of the more general Hidden Subgroup and connected Hidden Shift Problem.