We propose a fully homomorphic encryption scheme -- i.e., a scheme that allows one to evaluate circuits over encrypted data without being able to decrypt. Our solution comes in three steps. First, we provide a general result -- that, to construct an encryption scheme that permits evaluation of arbitrary circuits, it suffices to construct an encryption scheme that can evaluate (slightly augmented versions of) its own decryption circuit; we call a scheme that can evaluate its (augmented) decryption circuit bootstrappable.Next, we describe a public key encryption scheme using ideal lattices that is almost bootstrappable.Lattice-based cryptosystems typically have decryption algorithms with low circuit complexity, often dominated by an inner product computation that is in NC1. Also, ideal lattices provide both additive and multiplicative homomorphisms (modulo a public-key ideal in a polynomial ring that is represented as a lattice), as needed to evaluate general circuits.Unfortunately, our initial scheme is not quite bootstrappable -- i.e., the depth that the scheme can correctly evaluate can be logarithmic in the lattice dimension, just like the depth of the decryption circuit, but the latter is greater than the former. In the final step, we show how to modify the scheme to reduce the depth of the decryption circuit, and thereby obtain a bootstrappable encryption scheme, without reducing the depth that the scheme can evaluate. Abstractly, we accomplish this by enabling the encrypter to start the decryption process, leaving less work for the decrypter, much like the server leaves less work for the decrypter in a server-aided cryptosystem.

/pdf/fully-homomorphic-encryption-using-ideal-lattices-3oirjo0huq.pdf

Fully homomorphic encryption using ideal lattices

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away --- we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.

/pdf/quantum-computing-in-the-nisq-era-and-beyond-4jnwt61yqa.pdf

Quantum Computing in the NISQ era and beyond

Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansatze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry--calculating the ground-state molecular energy for He-H(+). The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future.

/pdf/a-variational-eigenvalue-solver-on-a-photonic-quantum-2nxrfjmr5w.pdf

A variational eigenvalue solver on a photonic quantum processor

Our main result is a reduction from worst-case lattice problems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the 'learning from parity with error' problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP and SIVP. A main open question is whether this reduction can be made classical.Using the main result, we obtain a public-key cryptosystem whose hardness is based on the worst-case quantum hardness of SVP and SIVP. Previous lattice-based public-key cryptosystems such as the one by Ajtai and Dwork were only based on unique-SVP, a special case of SVP. The new cryptosystem is much more efficient than previous cryptosystems: the public key is of size O(n2) and encrypting a message increases its size by O(n)(in previous cryptosystems these values are O(n4) and O(n2), respectively). In fact, under the assumption that all parties share a random bit string of length O(n2), the size of the public key can be reduced to O(n).

/pdf/on-lattices-learning-with-errors-random-linear-codes-and-2dvfl9suqh.pdf

On lattices, learning with errors, random linear codes, and cryptography

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the
near future. Quantum computers with 50-100 qubits may be able to perform tasks
which surpass the capabilities of today's classical digital computers, but
noise in quantum gates will limit the size of quantum circuits that can be
executed reliably. NISQ devices will be useful tools for exploring many-body
quantum physics, and may have other useful applications, but the 100-qubit
quantum computer will not change the world right away --- we should regard it
as a significant step toward the more powerful quantum technologies of the
future. Quantum technologists should continue to strive for more accurate
quantum gates and, eventually, fully fault-tolerant quantum computing.

/pdf/quantum-computing-in-the-nisq-era-and-beyond-3utyn832f7.pdf

Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the “learning from parity with error” problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., nonquantum).We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size O(n2) and encrypting a message increases its size by a factor of O(n) (in previous cryptosystems these values are O(n4) and O(n2), respectively). In fact, under the assumption that all parties share a random bit string of length O(n2), the size of the public key can be reduced to O(n).

The “learning with errors” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. A main open question was whether LWE and its applications could be made truly efficient by exploiting extra algebraic structure, as was done for lattice-based hash functions (and related primitives).We resolve this question in the affirmative by introducing an algebraic variant of LWE called ring-LWE, and proving that it too enjoys very strong hardness guarantees. Specifically, we show that the ring-LWE distribution is pseudorandom, assuming that worst-case problems on ideal lattices are hard for polynomial-time quantum algorithms. Applications include the first truly practical lattice-based public-key cryptosystem with an efficient security reduction; moreover, many of the other applications of LWE can be made much more efficient through the use of ring-LWE.

/pdf/on-ideal-lattices-and-learning-with-errors-over-rings-5dbzdtcjwk.pdf

On Ideal Lattices and Learning with Errors over Rings

The “learning with errors” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. A main open question was whether LWE and its applications could be made truly efficient by exploiting extra algebraic structure, as was done for lattice-based hash functions (and related primitives).

We resolve this question in the affirmative by introducing an algebraic variant of LWE called ring-LWE, and proving that it too enjoys very strong hardness guarantees. Specifically, we show that the ring-LWE distribution is pseudorandom, assuming that worst-case problems on ideal lattices are hard for polynomial-time quantum algorithms. Applications include the first truly practical lattice-based public-key cryptosystem with an efficient security reduction; moreover, many of the other applications of LWE can be made much more efficient through the use of ring-LWE. Finally, the algebraic structure of ring-LWE might lead to new cryptographic applications previously not known to be based on LWE.

/pdf/on-ideal-lattices-and-learning-with-errors-over-rings-nu7g19xlwr.pdf

On ideal lattices and learning with errors over rings

http://www.cs.toronto.edu/~toni/Courses/PCP/handouts/ugc-vc.pdf

Oded Regev

Papers

On lattices, learning with errors, random linear codes, and cryptography

On lattices, learning with errors, random linear codes, and cryptography

On Ideal Lattices and Learning with Errors over Rings

On ideal lattices and learning with errors over rings

Vertex cover might be hard to approximate to within 2-ε