Homomorphic Encryption Mike Merry Motivations for homomorphic encryption
![Homomorphic Encryption Mike Merry Homomorphic Encryption Mike Merry](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-1.jpg)
![• Motivations for homomorphic encryption • What do we need? • What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-2.jpg)
![• Motivations for homomorphic encryption • What is necessary? • What is homomorphic • Motivations for homomorphic encryption • What is necessary? • What is homomorphic](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-3.jpg)
![Motivations • Medical data is sensitive • Would like to use machinelearning to process Motivations • Medical data is sensitive • Would like to use machinelearning to process](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-4.jpg)
![e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’ e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-5.jpg)
![g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’ g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-6.jpg)
![g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’ g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-7.jpg)
![• Motivations for homomorphic encryption • What do we need? • What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-8.jpg)
![What do we need? • Process data: • • In any way we want What do we need? • Process data: • • In any way we want](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-9.jpg)
![• Motivations for homomorphic encryption • What do we need? • What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-10.jpg)
![A (reasonably loose) definition • Let U be unencrypted space, and be E encrypted A (reasonably loose) definition • Let U be unencrypted space, and be E encrypted](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-11.jpg)
![A formal definition [1] • [1] Armknecht, F. , Boyd, C. , Carr, C. A formal definition [1] • [1] Armknecht, F. , Boyd, C. , Carr, C.](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-12.jpg)
![Properties: correctness • Properties: correctness •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-13.jpg)
![Properties: Compactness • Properties: Compactness •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-14.jpg)
![Properties: Privacy • Circuit privacy: The distribution of the evaluation of a circuit in Properties: Privacy • Circuit privacy: The distribution of the evaluation of a circuit in](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-15.jpg)
![• Motivations for homomorphic encryption • What do we need? • What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-16.jpg)
![Bootstrapping [2] • [2] Gentry, C. (2009). Fully homomorphic encryption using ideal lattices. In Bootstrapping [2] • [2] Gentry, C. (2009). Fully homomorphic encryption using ideal lattices. In](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-17.jpg)
![m m](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-18.jpg)
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![• Motivations for homomorphic encryption • What do we need? • What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-24.jpg)
![Dijk and Gentry’s FHE over the integers [3] • [3] Van Dijk, M. , Dijk and Gentry’s FHE over the integers [3] • [3] Van Dijk, M. ,](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-25.jpg)
![Dijk and Gentry’s FHE over the integers • Dijk and Gentry’s FHE over the integers •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-26.jpg)
![Dijk and Gentry’s FHE over the integers • Dijk and Gentry’s FHE over the integers •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-27.jpg)
![References 1. Armknecht, F. , Boyd, C. , Carr, C. , Gjosteen, K. , References 1. Armknecht, F. , Boyd, C. , Carr, C. , Gjosteen, K. ,](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-28.jpg)
- Slides: 28
![Homomorphic Encryption Mike Merry Homomorphic Encryption Mike Merry](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-1.jpg)
Homomorphic Encryption Mike Merry
![Motivations for homomorphic encryption What do we need What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-2.jpg)
• Motivations for homomorphic encryption • What do we need? • What is homomorphic encryption? • How do we make it work? • An example of how to do it
![Motivations for homomorphic encryption What is necessary What is homomorphic • Motivations for homomorphic encryption • What is necessary? • What is homomorphic](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-3.jpg)
• Motivations for homomorphic encryption • What is necessary? • What is homomorphic encryption? • How do we make it work? • An example of how to do it
![Motivations Medical data is sensitive Would like to use machinelearning to process Motivations • Medical data is sensitive • Would like to use machinelearning to process](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-4.jpg)
Motivations • Medical data is sensitive • Would like to use machinelearning to process it • Do you trust putting it in the cloud? • Do you trust the integrity of these servers all the time, in all jurisdictions? • Would still like to use cloud services on the data Medical data
![e e Enck m e Deck e m m fm m e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-5.jpg)
e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’
![ge e e Enck m e Deck e m m fm m g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-6.jpg)
g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’
![ge e e Enck m e Deck e m m fm m g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-7.jpg)
g(e) e’ e Enc(k, m = e) Dec(k, e) = m m f(m) m’
![Motivations for homomorphic encryption What do we need What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-8.jpg)
• Motivations for homomorphic encryption • What do we need? • What is homomorphic encryption? • How do we make it work? • An example of how to do it
![What do we need Process data In any way we want What do we need? • Process data: • • In any way we want](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-9.jpg)
What do we need? • Process data: • • In any way we want On an insecure environment Without leaking information about the data In a time- and space-efficient manner • We are not (necessarily) concerned with hiding what we are doing • This is called Functional Encryption (FE) and is related, but distinct
![Motivations for homomorphic encryption What do we need What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-10.jpg)
• Motivations for homomorphic encryption • What do we need? • What is homomorphic encryption? • How do we make it work? • An example of how to do it
![A reasonably loose definition Let U be unencrypted space and be E encrypted A (reasonably loose) definition • Let U be unencrypted space, and be E encrypted](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-11.jpg)
A (reasonably loose) definition • Let U be unencrypted space, and be E encrypted space. Let f: U → U be a function that we want to compute on our data • We would like to find g: E → E such that g(e) = Enc(k, f(m)), that is efficient and secure
![A formal definition 1 1 Armknecht F Boyd C Carr C A formal definition [1] • [1] Armknecht, F. , Boyd, C. , Carr, C.](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-12.jpg)
A formal definition [1] • [1] Armknecht, F. , Boyd, C. , Carr, C. , Gjosteen, K. , Jaschke, A. , Reuter, C. A. , & Strand, M. (2015). A Guide to Fully Homomorphic Encryption. Cryptology e. Print Archive, 1– 35. Retrieved from https: //eprint. iacr. org/2015/1192
![Properties correctness Properties: correctness •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-13.jpg)
Properties: correctness •
![Properties Compactness Properties: Compactness •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-14.jpg)
Properties: Compactness •
![Properties Privacy Circuit privacy The distribution of the evaluation of a circuit in Properties: Privacy • Circuit privacy: The distribution of the evaluation of a circuit in](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-15.jpg)
Properties: Privacy • Circuit privacy: The distribution of the evaluation of a circuit in homomorphic space is indistinguishable from the encryption of the output of the circuit • A different way of viewing Circuit Privacy is to consider whether you would be happy to release the encrypted results of a message-space function. If you would, then as the distribution of the evaluation space is the same, you should be happy to release those results as well. • This doesn’t mean the distributions are equivalent to fresh cipher texts as the distribution of f(x) is not random.
![Motivations for homomorphic encryption What do we need What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-16.jpg)
• Motivations for homomorphic encryption • What do we need? • What is homomorphic encryption? • How do we make it work? • An example of how to do it
![Bootstrapping 2 2 Gentry C 2009 Fully homomorphic encryption using ideal lattices In Bootstrapping [2] • [2] Gentry, C. (2009). Fully homomorphic encryption using ideal lattices. In](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-17.jpg)
Bootstrapping [2] • [2] Gentry, C. (2009). Fully homomorphic encryption using ideal lattices. In Proceedings of the 41 st annual ACM symposium on Symposium on theory of computing - STOC ’ 09 (Vol. 19, p. 169). New York, USA: ACM Press.
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![Motivations for homomorphic encryption What do we need What is • Motivations for homomorphic encryption • What do we need? • What is](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-24.jpg)
• Motivations for homomorphic encryption • What do we need? • What is homomorphic encryption? • How do we make it work? • An example of how to do it
![Dijk and Gentrys FHE over the integers 3 3 Van Dijk M Dijk and Gentry’s FHE over the integers [3] • [3] Van Dijk, M. ,](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-25.jpg)
Dijk and Gentry’s FHE over the integers [3] • [3] Van Dijk, M. , Gentry, C. , Halevi, S. , & Vaikuntanathan, V. (2010). Fully Homomorphic Encryption over the Integers. In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 24– 43). Springer Berlin Heidelberg. http: //doi. org/10. 1007/978 -3 -642 -13190 -5_2
![Dijk and Gentrys FHE over the integers Dijk and Gentry’s FHE over the integers •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-26.jpg)
Dijk and Gentry’s FHE over the integers •
![Dijk and Gentrys FHE over the integers Dijk and Gentry’s FHE over the integers •](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-27.jpg)
Dijk and Gentry’s FHE over the integers •
![References 1 Armknecht F Boyd C Carr C Gjosteen K References 1. Armknecht, F. , Boyd, C. , Carr, C. , Gjosteen, K. ,](https://slidetodoc.com/presentation_image/a951b25bd8324882eeff0acfcb7c2822/image-28.jpg)
References 1. Armknecht, F. , Boyd, C. , Carr, C. , Gjosteen, K. , Jaschke, A. , Reuter, C. A. , & Strand, M. (2015). A Guide to Fully Homomorphic Encryption. Cryptology e. Print Archive, 1– 35. Retrieved from https: //eprint. iacr. org/2015/1192 2. Gentry, C. (2009). Fully homomorphic encryption using ideal lattices. In Proceedings of the 41 st annual ACM symposium on Symposium on theory of computing - STOC ’ 09 (Vol. 19, p. 169). New York, USA: ACM Press. http: //doi. org/10. 1145/1536414. 1536440 3. Van Dijk, M. , Gentry, C. , Halevi, S. , & Vaikuntanathan, V. (2010). Fully Homomorphic Encryption over the Integers. In Annual International Conference on the Theory and Applications of Cryptographic Techniques (pp. 24– 43). Springer Berlin Heidelberg. http: //doi. org/10. 1007/978 -3642 -13190 -5_2
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