# Fully Homomorphic Encryption over the Integers

@inproceedings{Dijk2010FullyHE, title={Fully Homomorphic Encryption over the Integers}, author={Marten van Dijk and Craig Gentry and Shai Halevi and Vinod Vaikuntanathan}, booktitle={International Conference on the Theory and Application of Cryptographic Techniques}, year={2010} }

We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct a fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of our scheme is the conceptual simplicity.
We reduce the security of our scheme to finding an…

## 1,717 Citations

### Practical Homomorphic Encryption Over the Integers

- Computer Science, MathematicsArXiv
- 2017

A sequence of generalisations is presented of basic homomorphic encryption schemes for integer arithmetic, intended for use in secure single-party computation in the cloud, with increasing levels of security, but decreasing practicality.

### Parameter Constraints on Homomorphic Encryption Over the Integers

- Mathematics, Computer Science
- 2017

The research paper Fully Homomorphic Encryption over the Integers by van Dijk, Gentry, Halevi, and Vaikuntanathan [31] explores the construction of an encryption scheme over the integers that is…

### Fully Homomorphic Encryption over the Integers with Shorter Public Keys

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2011

It is shown that fully homomorphic encryption can be implemented using simple arithmetic operations, and some optimizations from the recent Gentry-Halevi implementation of Gentry's scheme are obtained, roughly the same level of efficiency.

### Integer-based fully homomorphic encryption

- Computer Science, Mathematics
- 2011

This paper explores one such bootstrappable system based on simple integer arithmetic in a manner that someone without a high level of experience in homomorphic encryption can readily understand and presents an implementation of the system as well as a latticebased attack.

### Fully homomorphic encryption modulo Fermat numbers

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2019

This paper recast state-of-the-art constructions for fully homomorphic encryption in the simple language of arithmetic modulo large Fermat numbers in a simple ring to make its implementation much easier.

### Fully Homomorphic Encryption from Approximate Ideal Lattices

- Mathematics, Computer Science
- 2015

The security of the somewhat homomorphic encryption scheme is reduced to solving a partial approximate ideal lattice problem and the performance of the proposed scheme is demonstrated to be better than that of previous schemes by computational experimental.

### Faster Fully Homomorphic Encryption

- Computer Science, MathematicsASIACRYPT
- 2010

Two improvements to Gentry’s fully homomorphic scheme based on ideal lattices are described: a more aggressive analysis of one of the hardness assumptions and a probabilistic decryption algorithm that can be implemented with an algebraic circuit of low multiplicative degree.

### Symmetric Somewhat Homomorphic Encryption over the Integers

- Computer Science, Mathematics
- 2015

A symmetric variant of homomomorphic encryption scheme by van Dijk et al.

### New Fully Homomorphic Encryption over the Integers

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2011

A new fully homomorphic encryption scheme is constructed, whose security depends on the hardness of the decisional approximate principle ideal lattice polynomial (APIP), given a list of approximate multiples of a principal ideal lattICE.

### A New Somewhat Homomorphic Encryption Scheme over Integers

- Computer Science, Mathematics2012 International Conference on Computer Distributed Control and Intelligent Environmental Monitoring
- 2012

This paper reduces the public key size to Õ(λ<sup>3</sup>) by encrypting with a new form using Gentry's techniques, and can easily convert the somewhat scheme into a practical fully homomorphic encryption scheme available in cloud computing.

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