This work describes a fast fully homomorphic encryption scheme over the torus (TFHE) that revisits, generalizes and improves the fully homomorphic encryption (FHE) based on GSW and its ring variants. The simplest FHE schemes consist in bootstrapped binary gates. In this gate bootstrapping mode, we show that the scheme FHEW of Ducas and Micciancio (Eurocrypt, 2015) can be expressed only in terms of external product between a GSW and an LWE ciphertext. As a consequence of this result and of other optimizations, we decrease the running time of their bootstrapping from 690 to 13 ms single core, using 16 MB bootstrapping key instead of 1 GB, and preserving the security parameter. In leveled homomorphic mode, we propose two methods to manipulate packed data, in order to decrease the ciphertext expansion and to optimize the evaluation of lookup tables and arbitrary functions in $${\mathrm {RingGSW}}$$-based homomorphic schemes. We also extend the automata logic, introduced in Gama et al. (Eurocrypt, 2016), to the efficient leveled evaluation of weighted automata, and present a new homomorphic counter called $$\mathrm {TBSR}$$, that supports all the elementary operations that occur in a multiplication. These improvements speed up the evaluation of most arithmetic functions in a packed leveled mode, with a noise overhead that remains additive. We finally present a new circuit bootstrapping that converts $$\mathsf {LWE}$$ ciphertexts into low-noise $${\mathrm {RingGSW}}$$ ciphertexts in just 137 ms, which makes the leveled mode of TFHE composable and which is fast enough to speed up arithmetic functions, compared to the gate bootstrapping approach. Finally, we provide an alternative practical analysis of LWE based schemes, which directly relates the security parameter to the error rate of LWE and the entropy of the LWE secret key, and we propose concrete parameter sets and timing comparison for all our constructions.

/pdf/tfhe-fast-fully-homomorphic-encryption-over-the-torus-3kwct59cil.pdf

TFHE: Fast Fully Homomorphic Encryption Over the Torus

In this paper, we revisit fully homomorphic encryption (FHE) based on GSW and its ring variants. We notice that the internal product of GSW can be replaced by a simpler external product between a GSW and an LWE ciphertext.

https://link.springer.com/content/pdf/10.1007%2F978-3-662-53887-6_1.pdf

Faster Fully Homomorphic Encryption: Bootstrapping in Less Than 0.1 Seconds

Secure multi-party computation (MPC) has evolved from a theoretical curiosity in the 1980s to a tool for building real systems today. Over the past decade, MPC has been one of the most active resea...

/pdf/a-pragmatic-introduction-to-secure-multi-party-computation-l30nxb601s.pdf

A Pragmatic Introduction to Secure Multi-Party Computation

The rise of machine learning as a service multiplies scenarios where one faces a privacy dilemma: either sensitive user data must be revealed to the entity that evaluates the cognitive model (e.g., in the Cloud), or the model itself must be revealed to the user so that the evaluation can take place locally. Fully Homomorphic Encryption (FHE) offers an elegant way to reconcile these conflicting interests in the Cloud-based scenario and also preserve non-interactivity. However, due to the inefficiency of existing FHE schemes, most applications prefer to use Somewhat Homomorphic Encryption (SHE), where the complexity of the computation to be performed has to be known in advance, and the efficiency of the scheme depends on this global complexity.

/pdf/fast-homomorphic-evaluation-of-deep-discretized-neural-4sysnjpjyw.pdf

Fast Homomorphic Evaluation of Deep Discretized Neural Networks

This paper extends the leveled homomorphic encryption scheme for an approximate arithmetic of Cheon et al. (ASIACRYPT 2017) to a fully homomorphic encryption, i.e., we propose a new technique to refresh low-level ciphertexts based on Gentry’s bootstrapping procedure. The modular reduction operation is the main bottleneck in the homomorphic evaluation of the decryption circuit. We exploit a scaled sine function as an approximation of the modular reduction operation and present an efficient evaluation strategy. Our method requires only one homomorphic multiplication for each of iterations and so the total computation cost grows linearly with the depth of the decryption circuit. We also show how to recrypt packed ciphertexts on the RLWE construction with an open-source implementation. For example, it takes 139.8 s to refresh a ciphertext that encrypts 128 numbers with 12 bits of precision, yielding an amortized rate of 1.1 seconds per slot.

/pdf/bootstrapping-for-approximate-homomorphic-encryption-hnqwkjg8xq.pdf

Bootstrapping for approximate homomorphic encryption

In this paper, we present several methods to improve the evaluation of homomorphic functions in TFHE, both for fully and for leveled homomorphic encryption. We propose two methods to manipulate packed data, in order to decrease the ciphertext expansion and optimize the evaluation of look-up tables and arbitrary functions in ${\mathrm {RingGSW}}$ based homomorphic schemes. We also extend the automata logic, introduced in [12, 19], to the efficient leveled evaluation of weighted automata, and present a new homomorphic counter called $\mathrm {TBSR}$, that supports all the elementary operations that occur in a multiplication. These improvements speed-up the evaluation of most arithmetic functions in a packed leveled mode, with a noise overhead that remains additive. We finally present a new circuit bootstrapping that converts $\mathsf {LWE}$ into low-noise ${\mathrm {RingGSW}}$ ciphertexts in just 137 ms, which makes the leveled mode of TFHE composable, and which is fast enough to speed-up arithmetic functions, compared to the gate-by-gate bootstrapping given in [12]. Finally, we propose concrete parameter sets and timing comparison for all our constructions.

/pdf/faster-packed-homomorphic-operations-and-efficient-circuit-daysnmkqz0.pdf

Faster Packed Homomorphic Operations and Efficient Circuit Bootstrapping for TFHE

In this paper we present a new post-quantum electronic-voting protocol. Our construction is based on LWE fully homomorphic encryption and the protocol is inspired by existing e-voting schemes, in particular Helios. The strengths of our scheme are its simplicity and transparency, since it relies on public homomorphic operations. Furthermore, the use of lattice-based primitives greatly simplifies the proofs of correctness, privacy and verifiability, as no zero-knowledge proof are needed to prove the validity of individual ballots or the correctness of the final election result. The security of our scheme is based on classical SIS/LWE assumptions, which are asymptotically as hard as worst-case lattice problems and relies on the random oracle heuristic. We also propose a new procedure to distribute the decryption task, where each trustee provides an independent proof of correct decryption in the form of a publicly verifiable ciphertext trapdoor. In particular, our protocol requires only two trustees, unlike classical proposals using threshold decryption via Shamir's secret sharing.

Ilaria Chillotti

Papers

Faster Packed Homomorphic Operations and Efficient Circuit Bootstrapping for TFHE

A Homomorphic LWE Based E-voting Scheme