BGV

In 2011, Brakerski, Gentry and Vaikuntanathan (BGV) published the paper in which they introduce a new (leveled) fully homomorphic encryption (FHE) that improves performance and bases security on weaker assumptions than schemes from the previous generation.

A central conceptual contribution of this work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), without Gentry’s bootstrapping procedure.

Until recently, the BGV scheme was considered to be the most efficient homomorphic encryption scheme when performing the same operations on multiple ciphertexts at once.

Modulus switching

@TODO

New noise management technique

@TODO

Leveled Fully Homomorphic Encryption

Most of the work done by the will focus on the construction of a leveled fully homomorphic scheme, in the sense that the parameters of the scheme depend (polynomially) on the depth of the circuits that the scheme is capable of evaluating.

Definition. We say that a family of homomorphic encryption schemes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{\mathcal E^{(L)}: L \in \mathbb Z^{+} \} } is leveled fully homomorphic if, for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L \in \mathbb Z^{+} } , they all use the same decryption circuit, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathcal E^{(L)} } compactly evaluates all circuits of depth at most Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L } (that use some specified complete set of gates), and the computational complexity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathcal E^{(L)} } 's algorithms is polynomial (a fixed polynomial for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L } ) in the security parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L} , and the size of the circuit (in the case of the evaluation algorithm).

The construction: FHE without bootstrapping

The authors base the security of their scheme on the hardness of Ring-Learning with errors problems, a generalisation of the classical LWE problem.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda } be a security parameter, representing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^{\lambda}} security against known attacks.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R } be a ring. For any integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q } , we write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_q } for the quotient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R/qR } .

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q=q(\lambda) } be an odd modulus and $\chi =\chi (\lambda )$ a ``noise" distribution over $R$. Let $N=N(\lambda )$ be an additional parameter of the system which is larger than $3\cdot \log {q}$.

Let us assume that the plaintext is $R_{2}=R/2R$.

• E.Setup($1^{\lambda },1^{\mu }$): Choose a $\mu$-bit modulus $q$ and choose the other parameters $d=d(\lambda ,\mu )$, and $N=\lceil 3\log q\rceil$, $\chi (\lambda ,\mu )$ appropriately to ensure that the scheme is based on a Ring-LWE instance that achieves $2^{\lambda }$ security against known attacks. Let $R=\mathbb {Z} [X]/(x^{d}+1)$ and let params = $(q,d,N,\chi )$.

• E.SecretKetGen(params): Draw $s'\leftarrow \chi$. Set $sk=s\leftarrow (1,s')\in R_{q}^{2}$.

• E.PublicKeyGen(params, sk): Recall that the secret key is $sk=s=(1,s')$. This algorithm generates a (column) vector $A'\leftarrow R_{q}^{N}$, uniformly and a vector $e\leftarrow \chi ^{N}$. The algorithm computes

$b=A's'+2e$. Then, set $A$ to be the $N\times 2$ matrix obtained by setting $b$ on the first column followed by the entries of $-A'$. We remark that $A\cdot s=2e$. The algorithm outputs the public key Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle pk=A} .

• E.Enc(params, pk,m): To encrypt a message Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m \in R_2 = R/2R } , set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m \leftarrow (m,0) \in R_q^2 } , sample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r \leftarrow R_2^N } uniformly and output the ciphertext Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c \leftarrow m+A^T \cdot r \in R_q^2 } .

• E.Dec(params, sk, c): Output Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m \leftarrow [[<c,s>]_q]_2 } . Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [\cdot]_q} denotes reduction into the range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (- q/2,q/2] } .

Correctness is easy to verify, whereas we refer to the paper for details upon security.

Key switching (Dimension reduction)

Recall that in the above scheme, the decryption equation for a ciphertext Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c } that encrypts a message Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m } under the secret key Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s} can be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m = E.Dec(params, s,c) = [[L_{c}(s)]_q]_2,} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_c } is a linear operator which depends on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c } . To be precise, the latter is just the inner product $L_{c}(s)=<c,s>$ between two vectors in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_q^2} .