<!-- <span style="color: #0; font-family: Times New Roman; font-size: 1.3em;">Fully Homomorphic Encryption Schemes
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In this series of notes, we will discuss *fully homomorphic encryption* (FHE) schemes, beginning with the first such scheme introduced by Craig Gentry in his [thesis](https://crypto.stanford.edu/craig/craig-thesis.pdf) work that builds upon ideal lattices. Along with this introductory work, we will discuss a more intuitive and easier to understand [approach](https://eprint.iacr.org/2009/616.pdf) based on integer encryption. Taken together, it is possible to understand FHEs without understanding ideal lattices, which we will discuss separately.

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In the chapters that follow, we will discuss some of the more recent advances in FHEs, such as FHEs over a torus, as well as some of the methods that speed up the bootstrapping step, typically the most expensive step computationally.

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In the first chapter, we will attempt to formalize FHEs, particularly . We will discuss lattices in the second chapter. In the last chapter, we will discuss the fully homomorphic encryption scheme introduced by Gentry.
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<span style="color: #0; font-family: Times New Roman; font-size: 1.1em;">Chapters (private)
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- [Principles of Fully Homomorphic Encryption](/@kullervo/principlesFHE) [target=_blank]
- ### <span style="color: #0; font-family: Times New Roman; font-size: 1.1em;">Components of FHEs </span>
- [Somewhat Homomorphic Encryption Schemes (SWHEs)](/@kullervo/SWHEs) [target=_blank]
- [Integer SWHE of [DGHV09]](/@kullervo/integerSWHE) [target=_blank]
- [Bootstrapping and the $Recrypt$ Algorithm](/@kullervo/bootstrappingFHE) [target=_blank]
- [Discussion on Lattices and Ideal Lattices](/@kullervo/lattices) [target=_blank]
- [Hard Problems on Lattices](/@kullervo/hardLatticeProblems) [target=_blank]
- [SWHE of [Gen09]](/@kullervo/latticeSWHE) [target=_blank]
- [FHE over a torus](/@kullervo/torusFHE) [target=_blank]
- [Faster Bootstrapping](/@kullervo/fasterBootsFHE) [target=_blank]

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<span style="color: #0; font-family: Times New Roman; font-size: 1.1em;">Remarks
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This series reflects our introductory views on FHEs. Some of these views are possibly over-simplifications. Further, despite our efforts to minimize them, they may contain certain inaccuracies and we would be happy to remedy any such mistakes. Please feel free to contact --- at gmail dot com if you would like to contribute to this series in any way, with due acknowledgement on our part. Thank you in advance.
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<span style="color: #0; font-family: Times New Roman; font-size: 1.1em;">References
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1. [Gen09] Craig Gentry. A Fully Homomorphic Encryption Scheme. https://crypto.stanford.edu/craig/craig-thesis.pdf, 2009.
2. [DGHV09] Marten van Dijk, Craig Gentry, Shai Halevi and Vinod Vaikuntanathan. Fully Homomorphic Encryption over the Integers. https://eprint.iacr.org/2009/616.pdf, in IACR, 2009.
3. [GSW13] Craig Gentry, Amit Sahai and Brent Waters. Homomorphic Encryption from Learning with Errors. https://eprint.iacr.org/2013/340.pdf, in IACR, 2013.
4. [DM14] Leo Ducas and Daniele Micciancio. FHEW: Bootstrapping Homomorphic Encryption in less than a second. https://eprint.iacr.org/2014/816.pdf, in IACR, 2014.
5. [CGGI16] Ilaria Chillotti, Nicolas Gama, Mariya Georgieva, and Malika Izabachene. Faster Fully Homomorphic Encryption: Bootstrapping in less than 0.1 Seconds. https://eprint.iacr.org/2016/870.pdf, in IACR, 2016.
6. [CGGI18] Ilaria Chillotti, Nicolas Gama, Mariya Georgieva and Malika Izabachene. TFHE: Fast Fully Homomorphic Encryption over the Torus. https://eprint.iacr.org/2018/421.pdf, in IACR, 2018.