FHE History

Pre-FHE

• Ronald Rivest, Leonard Adleman and Mike Dertouzos
  

  On Data Banks and Privacy Homomorphisms
  

  http://people.csail.mit.edu/rivest/RivestAdlemanDertouzos-OnDataBanksAndPrivacyHomomorphisms.pdf
  

• Shafi Goldwasser and Silvio Micali
  

  Probabilistic Encryption
  

  http://groups.csail.mit.edu/cis/pubs/shafi/1984-jcss.pdf
  

• Taher El Gamal
  

  A Public-key Cryptosystem and a Signature Scheme based on Discrete Logarithms
  

  https://link.springer.com/chapter/10.1007/3-540-39568-7_2
  

• Pascal Paillier
  

  Public-key Cryptosystems based on Composite Degree Residuosity Classes
  

  Springer Link
  

• Ivan Damgard and Mads Jurik
  

  A Generalisation, a Simplification and Some Applications of Paillier's Probabilistic Public-Key System
  

  http://www.brics.dk/RS/00/45/
  

• Dan Boneh, Eu Jin Goh and Kobbi Nissim
  

  Evaluating 2-DNF Formulas on Ciphertexts
  

  http://crypto.stanford.edu/~dabo/abstracts/2dnf.html
  

• Craig Gentry, Shai Halevi and Vinod Vaikuntanathan
  

  A Simple BGN-Type Cryptosystem from LWE
  

  https://eprint.iacr.org/2010/182
  

  Note: This cryptosystem was discovered after Gentry's work on FHE, as a first attempt to base FHE on standard assumptions such as learning with errors (LWE).
  

Gen I

• Craig Gentry
  

  A fully homomorphic encryption scheme
  

  https://crypto.stanford.edu/craig/craig-thesis.pdf
  

• Marten van Dijk, Craig Gentry, Shai Halevi and Vinod Vaikuntanathan
  

  Fully Homomorphic Encryption over the Integers
  

  https://eprint.iacr.org/2009/616
  

• Nigel Smart and Frederik Vercauteren
  

  Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes
  

  https://eprint.iacr.org/2009/571
  

• Craig Gentry
  

  Toward basing fully homomorphic encryption on worst-case hardness
  

  http://www.iacr.org/archive/crypto2010/62230116/62230116.pdf
  

• Shai Halevi and Craig Gentry
  

  Fully Homomorphic Encryption without Squashing Using Depth-3 Arithmetic Circuits
  

  https://eprint.iacr.org/2011/279
  

  Note: I would classify this scheme as "somewhere between" the first and second generations, in the sense that it relies on fewer assumptions than the older schemes, yet it still uses ideal lattices.
  

Gen II

Key Papers:

• Zvika Brakerski and Vinod Vaikuntanathan
  

  Efficient Fully Homomorphic Encryption from (Standard) LWE
  

  https://eprint.iacr.org/2011/344
  

• Zvika Brakerski, Craig Gentry and Vinod Vaikuntanathan
  

  Fully Homomorphic Encryption without Bootstrapping
  

  https://eprint.iacr.org/2011/277
  

• Zvika Brakerski
  

  Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP
  

  https://eprint.iacr.org/2012/078
  

• Craig Gentry, Shai Halevi and Nigel Smart
  

  Fully Homomorphic Encryption with Polylog Overhead
  

  https://eprint.iacr.org/2011/566
  

• Craig Gentry, Shai Halevi and Nigel Smart
  

  Homomorphic Evaluation of the AES Circuit
  

  https://eprint.iacr.org/2012/099
  

Other Works:

• Craig Gentry, Shai Halevi, Chris Peikert and Nigel P. Smart
  

  Field Switching in BGV-Style Homomorphic Encryption
  

  http://eprint.iacr.org/2012/240
  

• Zvika Brakerski, Craig Gentry, and Shai Halevi
  

  Packed Ciphertexts in LWE-Based Homomorphic Encryption
  

  https://eprint.iacr.org/2012/565
  

• Adriana Lopez-Alt, Eran Tromer and Vinod Vaikuntanathan
  

  Multikey Fully Homomorphic Encryption and On-the-Fly Multiparty Computation
  

  https://eprint.iacr.org/2013/094
  

• Junfeng Fan and Frederik Vercauteren
  

  Somewhat Practical Fully Homomorphic Encryption
  

  https://eprint.iacr.org/2012/144
  

• Tancrede Lepoint and Michael Naehrig
  

  A Comparison of the Homomorphic Encryption Schemes FV and YASHE
  

  https://eprint.iacr.org/2014/062
  

Gen III

• Craig Gentry, Amit Sahai and Brent Waters
  

  Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based
  

  https://eprint.iacr.org/2013/340
  

• Zvika Brakerski and Vinod Vaikuntanathan
  

  Lattice-Based FHE as Secure as PKE
  

  https://eprint.iacr.org/2013/541
  

• Jacob Alperin-Sheriff and Chris Peikert
  

  Faster Bootstrapping with Polynomial Error
  

  https://eprint.iacr.org/2014/094
  

• Leo Ducas and Daniele Micciancio
  

  FHEW: Bootstrapping Homomorphic Encryption in less than a second
  

  https://eprint.iacr.org/2014/816
  

• Ryo Hiromasa, Masayuki Abe and Tatsuaki Okamoto
  

  Packing Messages and Optimizing Bootstrapping in GSW-FHE
  

  Talk Slides and Springer Link
  

• Ilaria Chillotti and Nicolas Gama and Mariya Georgieva and Malika Izabachène
  

  Faster Fully Homomorphic Encryption: Bootstrapping in less than 0.1 Seconds
  

  https://eprint.iacr.org/2016/870