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 work that builds upon ideal lattices. Along with this introductory work, we will discuss a more intuitive and easier to understand approach based on integer encryption. Taken together, it is possible to understand FHEs without understanding ideal lattices, which we will discuss separately.

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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Chapters (private)

• Principles of Fully Homomorphic Encryption

• Components of FHEs

  • Somewhat Homomorphic Encryption Schemes (SWHEs)
  • Integer SWHE of [DGHV09]
  • Bootstrapping and the
    $Recrypt$ Algorithm
• Discussion on Lattices and Ideal Lattices
  • Hard Problems on Lattices
  • SWHE of [Gen09]
• FHE over a torus
• Faster Bootstrapping

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References

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.