Georgeg at 08:02, 27 February 2020

2020-02-27T08:02:42Z

Georgeg: Created page with "This page paragraph should appear in the preview popup, alongside the picture that that it contains. My guess is that the preview of a page contains the first paragraph and th..."

2020-02-21T17:57:25Z

Created page with "This page paragraph should appear in the preview popup, alongside the picture that that it contains. My guess is that the preview of a page contains the first paragraph and th..."

New page

This page paragraph should appear in the preview popup, alongside the picture that that it contains. My guess is that the preview of a page contains the first paragraph and the media in that page, up to the first paragraph marked between double '=' equal sign

A fully homomorphic encryption (FHE) scheme is an encryption scheme which supports computation on encrypted data: given a ciphertext that encrypts some data μ, one can compute a ciphertext that encrypts f(μ) for any efficiently computable function f, without ever needing to decrypt the data or know the decryption key. FHE has numerous theoretical and practical applications, the canonical one being to the problem of outsourcing computation to a remote server without compromising one’s privacy. In 2009, Gentry put forth the first candidate construction of FHE based on ideal lattices [Gen09]. Since then, substantial progress has been made [vDGHV10, SS10, SV10, BV11a, BV11b, BGV12, GHS12, GSW13, BV14, AP14], offering various improvements in conceptual and technical simplicity, efficiency, security guarantees, assumptions, etc; in particular, Gentry, Sahai and Waters presented a very simple FHE (hereafter called the GSW cryptosystem) based on the standard learning with errors (LWE) assumption.
[[File:Fhe0.jpg|thumb]]

== Paragraph ==
This new paragraph should not appear in the page preview popup

== Math formulas ==
<math>
{+}\colon \mathfrak{R}\times R\to R.
</math>

== Math formulas2 ==
<math>
{+}\colon \mathbb{R}\times R\to R.
</math>

<math>
\sqrt[3]{x^3+y^3 \over 2}
</math>

<math>
\int\limits_{1}^{3}{x}\qquad \int*{\sqrt{x^2 + \frac{1}{3}}}\qquad\int[\Bigg]{\sqrt{x^2}} = |x|
</math>

<math>\prime, \backprime, f^\prime, f', f'', f^{(3)}, \dot y, \ddot y</math>

<math>x_1^n</math>

Given a plaintext vector <math> \vec z = (z_1,z_2,...,z_{n/2}) \in \mathbb{C}^{n/2} </math> and a scaling factor <math> \Delta > 1 </math>, the plaintext vector is encoded as a polynomial <math> m(X) \in R:= \mathbb{Z}[X]/(X^n+1) </math>
by computing <math> m(X) = \lfloor \Delta \cdot \phi^{-1}(\vec z) \rceil \in R</math> where <math> \lfloor \cdot \rceil </math> denotes the coefficient-wise rounding function.

'''ElGamal'''

In the [[ElGamal encryption|ElGamal cryptosystem]], in a cyclic group <math>G</math> of order <math>q</math> with generator <math>g</math>, if the public key is <math>(G, q, g, h)</math>, where <math>h = g^x</math>, and <math>x</math> is the secret key, then the encryption of a message <math>m</math> is <math>\mathcal{E}(m) = (g^r,m\cdot h^r)</math>, for some random <math>r \in \{0, \ldots, q-1\}</math>. The homomorphic property is then

:<math>
\begin{align}
\mathcal{E}(m_1) \cdot \mathcal{E}(m_2) &= (g^{r_1},m_1\cdot h^{r_1})(g^{r_2},m_2 \cdot h^{r_2}) \\[6pt]
&= (g^{r_1+r_2},(m_1\cdot m_2) h^{r_1+r_2}) \\[6pt]
&= \mathcal{E}(m_1 \cdot m_2).
\end{align}
</math>

'''Goldwasser–Micali'''

In the [[Goldwasser–Micali cryptosystem]], if the public key is the modulus <math>n</math> and quadratic non-residue <math>x</math>, then the encryption of a bit <math>b</math> is <math>\mathcal{E}(b) = x^b r^2 \;\bmod\; n</math>, for some random <math>r \in \{0, \ldots, n-1\}</math>. The homomorphic property is then

:<math>
\begin{align}
\mathcal{E}(b_1)\cdot \mathcal{E}(b_2) &= x^{b_1} r_1^2 x^{b_2} r_2^2 \;\bmod\; n \\[6pt]
&= x^{b_1+b_2} (r_1r_2)^2 \;\bmod\; n \\[6pt]
&= \mathcal{E}(b_1 \oplus b_2).
\end{align}
</math>

where <math>\oplus</math> denotes addition modulo 2, (i.e. [[Exclusive disjunction|exclusive-or]]).

@@ Line 6: / Line 6: @@
 == Paragraph ==
 This new paragraph should not appear in the page preview popup
 == Math formulas ==
@@ Line 58: / Line 60: @@
 where <math>\oplus</math> denotes addition modulo 2, (i.e. [[Exclusive disjunction|exclusive-or]]).

FhePlayground - Revision history

Georgeg at 08:02, 27 February 2020

Georgeg: Created page with "This page paragraph should appear in the preview popup, alongside the picture that that it contains. My guess is that the preview of a page contains the first paragraph and th..."