Difference between revisions of "Fully Homomorphic Encryption without Modulus Switching"
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== Preliminaries == | == Preliminaries == | ||
| + | For an integer <math>q </math>, write <math> \mathbb Z_q := \left(-q/2, q/2 \right] \cap \mathbb Z</math>. This is not the same with the ring <math>\mathbb Z/q \mathbb Z </math>. For any <math> x \in \mathbb Q </math>, write <math> [x]_p </math> for the unique value in <math>\left(-q/2, q/2 \right] \cap \mathbb Z </math> that is congruent to <math>x </math> modulo <math>q</math>. | ||
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| + | If <math>v,w </math> are two <math>n</math>-dimensional vectors, then the tensor product <math> v \otimes w </math> is the <math> n^2</math> dimensional vector containing all elements of the form <math>v[i]w[j]</math>. Note that | ||
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| + | <center><math> <v \otimes w, x \otimes y> = <v,x> \cdot <w,y> .</math> </center> | ||
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| + | == Building Blocks of a homomorphic encryption scheme == | ||
| + | |||
| + | We start by presenting Regev's <ref name='Regev'> O. Regev. On lattices, learning with errors, random linear codes, and cryptography. | ||
| + | In Harold N. Gabow and Ronald Fagin, editors, STOC, pages 84–93. ACM, 2005 </ref> basic public-key encryption scheme. | ||
==References== | ==References== | ||
Revision as of 12:57, 18 January 2021
This scheme proposed by Brakerski [1] has a number of advantages over previous candidates such as BGV. In particular, it uses the same modulus throughout the evaluation process, so there's no need for modulus switching. Security of these scheme is baed on the hardness of the GapSVP problem.
Preliminaries
For an integer , write . This is not the same with the ring . For any , write for the unique value in that is congruent to modulo .
![{\displaystyle \mathbb Z_q := \left(-q/2, q/2 \right] \cap \mathbb Z}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1c20f7fa209f95e0ff9082a79344b2bbf5cedad3)
![{\displaystyle [x]_p }](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c99b8e30f34e72b8a432c11d4ca30d85584abe75)
![{\displaystyle \left(-q/2, q/2 \right] \cap \mathbb Z }](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ce54aee0e87be779c46b5ee5342fa792a6d7e956)
If are two -dimensional vectors, then the tensor product is the dimensional vector containing all elements of the form . Note that
![{\displaystyle v[i]w[j]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/77bfd669704fc35727c7d15598f7196c131d7ef1)
Building Blocks of a homomorphic encryption scheme
We start by presenting Regev's [2] basic public-key encryption scheme.
References
- ↑ Z. Brakerski, Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP. In: Safavi-Naini R., Canetti R. (eds) Advances in Cryptology – CRYPTO 2012. CRYPTO 2012. Lecture Notes in Computer Science, vol 7417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32009-5_50
- ↑ O. Regev. On lattices, learning with errors, random linear codes, and cryptography. In Harold N. Gabow and Ronald Fagin, editors, STOC, pages 84–93. ACM, 2005
