FHE

Let us fix a security parameter $\lambda$, a variable that measures the input size of the computational problem. In the case of the encryption scheme, both the requirements of the cryptographic algorithms as well as the security of the scheme are expressed in terms of $\lambda$.

Intuition

Let ${\mathcal {E}}=({\mathcal {C}},{\mathcal {P}},KeyGen,Encrypt,Decrypt,Evaluate)$ be a homomorphic encryption scheme. A desirable property of ${\mathcal {E}}$ is the possibility of evaluation arbitrary functions on ciphertexts from ${\mathcal {C}}$ in a meaningful way. Clearly the complexity of $Evaluate(f,c)$ must depend on the complexity of the function $f$. To measure the latter, we use 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_f } , the size of a boolean circuit that computes 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 f } . The algorithm 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 Evaluate } is efficient if there exists a polynomial 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 g } such that for any function 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 f } that is represented by a circuit of size 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_f } , the 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 Evaluate(f,c_1, \dots, c_t, pk) } is 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 S_f \cdot g(\lambda) } .

The circuit representation 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 f } breaks down the computation 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 f } into AND, OR, and NOT gates. To evaluate these it is enough to be able to add, subtract and multiply. Therefore, to obtain a fully homomorphic encryption scheme, all we need is a scheme that supports arbitrary additions, subtractions and multiplications on the ciphertexts, so as it does on their underlying encrypted messages.

Definitions

L Assume that 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 \mathcal C } and the plaintext 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 P } have the algebraic structure of a ring and that 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 Decrypt : \mathcal C \to \mathcal P} is a ring homomorphism.

Definition. A homomorphic encryption scheme 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 = (\mathcal C, \mathcal P, KeyGen, Encrypt, Decrypt, Evaluate) } is called leveled if the decryption algorithm is correct for a certain number of ring operations made 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 \mathcal C } .

TODO: Write about the practical reasons for using a leveled scheme.

Definition. A homomorphic encryption scheme 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} is called compact if there exists a polynomial function 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=s(\lambda) } such that the output length 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 Eval(f,c,pk)} is 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 s(\lambda) } bits long, regardless of the ring homomorphism 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 f : \mathcal C^{t} \to \mathcal C } or the number of ciphertext inputs 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 t} , for any tuple $c=(c_{1},\dots ,c_{t})\in {\mathcal {C}}^{t}$ of ciphertexts.

A homomorphic encryption scheme as above, but for which we do not require that $s(\lambda )$ is polynomial in $\lambda$ is called bounded .

Definition. A scheme that is compact and for which the decryption algorithm is correct for any number of ring operations made on ${\mathcal {C}}$ is called fully homomorphic.

In one of his seminal papers, Gentry showed that under a "circular security" assumption, a leveled homomorphic encryption scheme ${\mathcal {E}}$ which is capable of homomorphically evaluating its $Decryption$ can be transformed into a fully homomorphic encryption scheme ${\mathcal {E}}'$. Gentry called the process of transforming ${\mathcal {E}}\to {\mathcal {E}}'$ bootstrapping .

Examples

We start by presenting an encryption scheme ${\mathcal {E}}$ proposed by van Dijk, Gentry and Halevi (TODO:cite). The encryption scheme ${\mathcal {E}}$ goes as follows.

We fix the security parameter $\lambda$ and set $N=\lambda ,P=\lambda ^{2}$ and $Q=\lambda ^{5}$.

$Keygen(\lambda ):$ Output a random odd $P$-bit integer $p$.

$Encrypt(p,m):$ To encrypt a bit $m\in \{0,1\}$, let $m'$ be a random $N$-bit integer such that $m'=m{\pmod {2}}$. Output the ciphertext $c\leftarrow m'+pq$, 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 q } is a random 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} -bit integer number.

$Decypt(p,c):$ Ouput (c mod p) mod 2, where (c mod p) is the unique integer $c'\in (-p/2,p/2)$ such that $p$ divides $c-c'$.

Remark that ciphertexts from outputed by 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 Encrypt } are near-multiples 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 p } . The distance from 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 } to the nearest multiple 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 p} is commonly called noise . The decryption algorithm is correct because the noise 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' } has the same parity as the message (in plaintext) 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 } .

This scheme is homomorphic since by simply adding, subtracting or multiplying the ciphertexts as integers we can add, subtract or multiply modulo 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} the underlying plaintext messages. However, such operations increase the noise on the ciphertexts. Complications arise, i.e. 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 Decrypt } might not work correctly, when the noise 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'} is too large when compared to 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 p} . This is a recurrent issue in all homomorphic encription schemes proposed so far in the literature.

Let us emphasize this on the multiplication operation 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 Mult } . 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 c_1,c_2 } be two ciphertexts with associated noise 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_1' } and 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_2' } respectively. We have that

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 Mult(c_1,c_2) = m_1' \cdot m_2' + p \cdot q' } for some 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' } . As long 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_1' \cdot m_2'| < p/2} , we have that

(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 }
 mod 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  p }
) = 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_1' \cdot m_2' }
 as it should be for correct decryption.

If we want to homomorphically evaluate a function 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 f } whose circuit representation has many additions, subtractions and multiplication operation, we have to be careful that the magnitude of the noise does not surpass 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 } . This is why the scheme we just presented is not fully homomorphic, but only leveled.

The semantic security of the scheme can be reduced to an approximate gcd problem. That is, it was proved that an attacker cannot efficiently break the semantic security 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 } unless the approximate gcd problem is easy.

TODO: Bootstrappable encryption