FhePlayground

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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

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Math formulas

${+}\colon {\mathfrak {R}}\times R\to R.$

Math formulas2

${+}\colon \mathbb {R} \times R\to R.$

${\sqrt[{3}]{x^{3}+y^{3} \over 2}}$

$\int \limits _{1}^{3}{x}\qquad \int *{\sqrt {x^{2}+{\frac {1}{3}}}}\qquad \int [{\Bigg ]}{\sqrt {x^{2}}}=|x|$

$\prime ,\backprime ,f^{\prime },f',f'',f^{(3)},{\dot {y}},{\ddot {y}}$

$x_{1}^{n}$

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

ElGamal

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

${\begin{aligned}{\mathcal {E}}(m_{1})\cdot {\mathcal {E}}(m_{2})&=(g^{r_{1}},m_{1}\cdot h^{r_{1}})(g^{r_{2}},m_{2}\cdot h^{r_{2}})\\[6pt]&=(g^{r_{1}+r_{2}},(m_{1}\cdot m_{2})h^{r_{1}+r_{2}})\\[6pt]&={\mathcal {E}}(m_{1}\cdot m_{2}).\end{aligned}}$

Goldwasser–Micali

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

${\begin{aligned}{\mathcal {E}}(b_{1})\cdot {\mathcal {E}}(b_{2})&=x^{b_{1}}r_{1}^{2}x^{b_{2}}r_{2}^{2}\;{\bmod {\;}}n\\[6pt]&=x^{b_{1}+b_{2}}(r_{1}r_{2})^{2}\;{\bmod {\;}}n\\[6pt]&={\mathcal {E}}(b_{1}\oplus b_{2}).\end{aligned}}$

where $\oplus$ denotes addition modulo 2, (i.e. exclusive-or).