9
Deep Math // BY A DRIEN NE BER N H A R D //
40
November/December 2021
IBM says that a quantum system with just 50 qubits might overpower today’s top supercomputers.
E
NCRYPTION —TH E PROCESS OF SE N D I NG
a scra mbled messa ge that on ly the intended recipient’s device can decode— allows private and public sectors alike to safeguard information. Traditional encryption uses schemes based on complex mathematics such as factoring (breaking an integer down to its prime factors) or discrete logarithm. Classical computers would require billions of years to crack these codes. Quantum computers, however, won’t be stumped by such hard problems; their exponential leaps in processing power will render classical cyphers obsolete, potentially exposing troves of sensitive data across commercial entities, healthcare providers, government institutions, and billions of individual users. Experts are working to devise cryptographic schemes that can run on today’s computers, but that can also be used in ciphers to protect data against quantum attackers. Quantum computers can perform certain functions that ordinary computers simply can’t; in part, this is because qubits (the way information is encoded on quantum computers) adopt the properties of quantum mechanics, using individual atoms, ions, photons, or electrons to take on a combination of various states at once. This gives quantum computers—once little more than a laboratory curiosity—access to a larger space of values than conventional computers, with their binary 0’s and 1’s. Someday, quantum computers will be able to perform a vast number of calculations almost instantaneously, breaking the ciphers that protect personal data. “We need to identify alternative problems we can use as the foundation for quantum-secure cryptosystems,” says Chris Peikert, professor of computer science and engineering at the University of Michigan. “What hard problem are we going to use? How do we use that problem to encrypt messages securely?” Enter lattices. Abstract algebraic structures, lattices are enormous grids with many individual points across two, three, or potentially hundreds of dimensions. In a high enough dimension, for instance, a bounded distance decoding problem—a type of lattice-based conundrum—should be able to f lummox these machines (see sidebar). “The best algorithms we come up with still
CO U R T E S Y I B M
Quantum Cyberattacks Are Coming. This Math Can Stop Them
