Learning with Errors is easy with quantum samples

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE

Encrypted statistical machine learning: new privacy preserving methods

We present two new statistical machine learning methods designed to learn on fully homomorphic encrypted (FHE) data. The introduction of FHE schemes following Gentry (2009) opens up the prospect of privacy preserving statistical machine learning analysis and modelling of encrypted data without compromising security constraints. We propose tailored algorithms for applying extremely random forests, involving a new cryptographic stochastic fraction estimator, and na\"{i}ve Bayes, involving a semi-parametric model for the class decision boundary, and show how they can be used to learn and predict from encrypted data. We demonstrate that these techniques perform competitively on a variety of classification data sets and provide detailed information about the computational practicalities of these and other FHE methods.Comment: 39 page

High-Precision Arithmetic in Homomorphic Encryption

In most RLWE-based homomorphic encryption schemes the native plaintext elements are polynomials in a ring $\mathbb{Z}_t[x]/(x^n+1)$, where $n$ is a power of $2$, and $t$ an integer modulus. For performing integer or rational number arithmetic one typically uses an encoding scheme, which converts the inputs to polynomials, and allows the result of the homomorphic computation to be decoded to recover the result as an integer or rational number respectively. The problem is that the modulus $t$ often needs to be extremely large to prevent the plaintext polynomial coefficients from being reduced modulo~$t$ during the computation, which is a requirement for the decoding operation to work correctly. This results in larger noise growth, and prevents the evaluation of deep circuits, unless the encryption parameters are significantly increased. We combine a trick of Hoffstein and Silverman, where the modulus $t$ is replaced by a polynomial $x-b$, with the Fan-Vercauteren homomorphic encryption scheme. This yields a new scheme with a very convenient plaintext space $\mathbb{Z}/(b^n+1)\mathbb{Z}$. We then show how rational numbers can be encoded as elements of this plaintext space, enabling homomorphic evaluation of deep circuits with high-precision rational number inputs. We perform a fair and detailed comparison to the Fan-Vercauteren scheme with the Non-Adjacent Form encoder, and find that the new scheme significantly outperforms this approach. For example, when the new scheme allows us to evaluate circuits of depth $9$ with $32$-bit integer inputs, in the same parameter setting the Fan-Vercauteren scheme only allows us to go up to depth $2$. We conclude by discussing how known applications can benefit from the new scheme