Chris Peikert

We construct public-key cryptosystems that are secure assuming the

\emph{worst-case} hardness of approximating the length of a shortest

nonzero vector in an $n$-dimensional lattice to within a small

$\poly(n)$ factor. Prior cryptosystems with worst-case connections

were based either on the shortest vector problem for a \emph{special

class} of lattices ...
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Sanjeev Arora, Rong Ge

In the {\em learning parities with noise} problem ---well-studied in learning theory and cryptography--- we

have access to an oracle that, each time we press a button,

returns a random vector $ a \in \GF(2)^n$ together with a bit $b \in \GF(2)$ that was computed as

$a\cdot u +\eta$, where ...
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Zvika Brakerski, Vinod Vaikuntanathan

We present a fully homomorphic encryption scheme that is based solely on the (standard) learning with errors (LWE) assumption. Applying known results on LWE, the security of our scheme is based on the worst-case hardness of ``short vector problems'' on arbitrary lattices.

Our construction improves on previous works in two ... more >>>

Craig Gentry

This survey, aimed mainly at mathematicians rather than practitioners, covers recent developments in homomorphic encryption (computing on encrypted data) and program obfuscation (generating encrypted but functional programs). Current schemes for encrypted computation all use essentially the same "noisy" approach: they encrypt via a noisy encoding of the message, they decrypt ... more >>>

Srinivasan Arunachalam, Alex Bredariol Grilo, Aarthi Sundaram

In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results.

1) Hardness of learning ... more >>>

Joan Bruna, Oded Regev, Min Jae Song, Yi Tang

We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to those of LWE. Alternatively, our result can also be seen as opening new avenues ... more >>>

Yilei Chen, Qipeng Liu, Mark Zhandry

We show polynomial-time quantum algorithms for the following problems:

(*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a ...
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Noel Arteche, Gaia Carenini, Matthew Gray

We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate $\mathbf{TC}^0$-Frege. This extends the line of results of Kraí?ek and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and ... more >>>