Difference between revisions of "GSW"
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<center> <math>BitDecomp(\vec a) = (a_{1,0}, \dots, a_{1,l-1}, \dots, a_{k,0},\dots, a_{k,l-1}),</math> where <math>a_{i,j}</math> is the <math>j</math>-th bit in the binary decomposition of <math>a_i </math>, so a <math>N</math>-dimensional vector. </center> | <center> <math>BitDecomp(\vec a) = (a_{1,0}, \dots, a_{1,l-1}, \dots, a_{k,0},\dots, a_{k,l-1}),</math> where <math>a_{i,j}</math> is the <math>j</math>-th bit in the binary decomposition of <math>a_i </math>, so a <math>N</math>-dimensional vector. </center> | ||
| + | |||
| + | For any <math>N</math>-dimensional vector <math>\vec{a'}= (a_{1,0}, \dots, a_{1,l-1}, \dots, a_{k,0},\dots, a_{k,l-1}) </math>, define | ||
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| + | <center><math>BitDecomp^{-1}(\vec{a'}) = \left( \sum 2^j \cdot a_{1,j}, \dots, \sum 2^j \cdot a_{k,j} \right) </math>, now a <math>k</math> dimensional vector, </center> | ||
| + | and | ||
| + | |||
| + | <center><math>Flatten(\vec{a'}) = BitDecomp(BitDecomp^{-1}(\vec{a})) \in \mathbb Z_q^{N} </math>. </center> | ||
| + | |||
| + | The authors also introduce | ||
| + | |||
| + | <center><math>Powersof2(\vec{b}) = (b_1, 2b_1, \dots, 2^{l-1}b_1, \dots, b_k, 2b_k, \dots, 2^{l-1}b_k) </math>, an <math>N</math>-dimensional vector. </center> | ||
| + | |||
| + | Some obvious properties of these definitions are | ||
| + | |||
| + | *<math> \langle BitDecomp(\vec{a}), Powersof2(\vec{b}) \rangle = \langle a,b \rangle \rangle </math>; | ||
| + | |||
| + | * For any <math>N</math>-dimensional <math>\vec{a'} </math>, we have that | ||
| + | <math> \langle \vec{a'}, Powersof2(\vec{b}) \rangle = \langle BitDecomp^{-1}(\vec{a'}),\vec{b} \rangle = \langle Flatten(\vec{a'}), Powersof2(\vec{b}) \rangle.</math> | ||
| + | |||
| + | The latter is showing us an interesting feature of <math> Flatten </math>, i.e. it makes the coefficients of a vector "small", without changing its inner product with <math>Powersof2(\vec{b})</math> | ||
| + | |||
| + | We will have to "flatten" ciphertexts, which are matrices. When <math>Flatten </math> is applied to a <math>N \times N </math> matrix <math>C</math> , this will be done by applying <math> Flatten</math> to each row of <math>C</math>. Recall that a step in the decryption process is multiplying the ciphertext matrix <math>C</math> with a secret key vector <math>\vec v</math>. We will always use <math> \vec{v} = Powersof2(\vec{s}) </math>, a vector of this special form in order for flattening to preserve the product between <math> C</math> and <math>\vec{v} </math> required for decryption. | ||
| + | |||
| + | Now, for any <math>N \times N</math> ciphertext matrix <math>C</math>, we have | ||
| + | |||
| + | <center><math>Flatten(C) \cdot \vec{v} = C \cdot \vec{v}</math> </center> | ||
| + | |||
| + | After computing a ciphertext <math>C_3 \leftarrow I_N - C_1 \cdot C_2 </math> for the NAND gate, we will immediately flatten it, namely we will set | ||
| + | |||
| + | <center><math> C^{NAND} \leftarrow Flatten(C_3) </math>.</center> | ||
== References == | == References == | ||
Revision as of 10:47, 5 June 2020
Around 2013, Gentry, Sahai and Waters [1] proposed a new way of building FHE schemes whose homomorphic multiplication algorithms are more natural than those presented in BFV or BGV. A distinguished feature of the scheme we are about to present is an asymmetric formula for the growth of the noise when multiplying two ciphertexts. Due to this feature, certain types of circuits have a very slow noise growth rate. Based on this asymmetry, Alperin-Sheriff and Peikert [2] found a very efficient bootstrapping technique for the GSW scheme.
More efficient FHE schemes based on ring variants of GSW have been proposed since then. These achieve very fast bootstrapping via various optimisation techniques, such as "refreshing the ciphertexts" after every single homomorphic operation. To our knowledge, among the schemes based on GSW (and not only), TFHE [3] holds the record for fastest bootstrapping. In the literature, the schemes based on the aformentioned work of Gentry, Sahai and Waters are commonly referred to as "third generation FHE". Their security is based on the so-called approximate eigenvector problem.
Overview of the GSW scheme
We follow the exposition in the paper "Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based" [1] by Gentry, Sahai and Waters.
The description of this scheme is strikingly simple. In GSW, homomorphic encryption based on LWE is achieved while the homomorphic addition and multiplications correspond to matrix addition and multiplication, respectively.
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 } be a natural number, representing some modulus 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 N } a dimension parameter. A ciphertext is a matrix 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 } of dimension 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 N \times N } with "small" entries from 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 \mathbb Z_q } . A 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 \vec{v} } is a 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 N} -dimensional vector over 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 \mathbb Z_q } with one big coefficient 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 v_i} . We can intuitively think of "small" as meaning much smaller (in order of magnitude) than 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} and "big" meaning the same order of magnitude 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 q } . In fact, we will make use of the case when the entries 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 C } belong 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 \{0,1 \} } . We also restrict the 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 \mu } to be a "small" 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 C } encrypts 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 \mu } under 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 \vec{v}} if 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 \cdot \vec{v} = \mu \cdot \vec{v} + \vec{e} } , 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 \vec{e} } is a small error vector.
To decrypt, one first exacts the 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 i} -th row 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_i} 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 C} . Compute 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 x \leftarrow \langle C_i, \vec{v} \rangle = \mu \cdot v_i + e_i } . Now, 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 v_i} is large 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 e_i } small, we have
Basically, 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 \vec{v} } is almost an eigenvector for 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 } corresponding to the eigenvalue 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 \mu } . This is commonly referred to being an approximate eigenvector.
Observe that if 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 }
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 C_2 }
encrypt 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 \mu_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 \mu_2 }
respectively under the same key vector 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 v }
, then
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 \vec{e_i} } represents the error vector 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 C_i} , for 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 i \in \{ 1,2\} } . As long as the sum 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 \vec{e_1} + \vec{e_2} } is small, the sum matrix 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 } is an encryption 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 \mu_1 +\mu_2} .
Looking at the equality
,
we observe 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_1 \cdot C_2 } is an encryption of the product 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 \mu_1 \mu_2 } , which can be decrypted correctly if the 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 i} -th component of the vector 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 \mu_2 \cdot \vec{e_1} + C_1 \cdot \vec{e_2} } is smaller than 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 v_i} .
Remark 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_2 \cdot C_1 } is also an encryption 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 \mu_1 \mu_2} , even though matrix multiplication is not commutative. The essence of the previously anticipated asymmetry is captured by the fact that the noises 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 C_1 \cdot C_2 } 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 C_2 \cdot C_1} are very different.
Flattening ciphertexts and keeping the noise small
We can decrypt homomorphically evaluated 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} as long as the 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 i} -th component of the error vector 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 \vec{e}} is small enough. As we saw in the overview section, the noise might grow quickly with each multiplication. To mitigate this and to perform homomorphically evaluations of deep circuits (in the sense of multiplication depth), observing that the noise growth depends of the size of the coefficients of the ciphertexts, the size of the messages and the fresh error size (in an asymmetric way), we can try to restrict the message space and the entries of the ciphertexts 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 \{0,1 \} } . We also impose a bound 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 B} for the entries of the fresh fresh error vectors 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 \vec{e}} .
Such ciphertexts will be further called 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 B} -strongly bounded, terminology that is used throughout the paper GSW [1].
The homomorphic operations performed on ciphertexts should correspond to operations "in the clear" that keep the message within the 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 \{0,1 \}} . To achieve this, we will use only NAND gates to evaluate any circuit.
Notice that if 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} are two 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 B} -strongly bounded ciphertexts, 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_3 \leftarrow I_N - C_1 C_2 } obtained by evaluating a NAND gate has underlying message 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 \{0,1 \} } , but the coefficients 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 C_3} 's error vector have magnitude 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 (N+1)B} . If one could ensure that the coefficients 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 C_3} have magnitude 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 1} , then the noise after evaluating a NAND gate will remain quite small, allowing us to evaluate deeper circuits.
The authors introduce an operation on ciphertexts called flattening, inspired by ideas in second generation FHE schemes such as BFV and BGV. Flattening is basically an operation that modify vectors without affecting their dot product.
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 N = k \cdot l} , 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 = \lfloor \log_2{q}+1 \rfloor } 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 k} is a positive integer. We will consider 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 \vec{a}, \vec{b} \in \mathbb Z_q^k } , two -dimensional vector spaces.
For any 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 N} -dimensional vector 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 \vec{a'}= (a_{1,0}, \dots, a_{1,l-1}, \dots, a_{k,0},\dots, a_{k,l-1}) } , define
and
The authors also introduce
Some obvious properties of these definitions are
- 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 \langle BitDecomp(\vec{a}), Powersof2(\vec{b}) \rangle = \langle a,b \rangle \rangle } ;
- For any 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 N} -dimensional 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 \vec{a'} } , 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 \langle \vec{a'}, Powersof2(\vec{b}) \rangle = \langle BitDecomp^{-1}(\vec{a'}),\vec{b} \rangle = \langle Flatten(\vec{a'}), Powersof2(\vec{b}) \rangle.}
The latter is showing us an interesting feature 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 Flatten } , i.e. it makes the coefficients of a vector "small", without changing its inner product with 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 Powersof2(\vec{b})}
We will have to "flatten" ciphertexts, which are matrices. When 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 Flatten } is applied to a 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 N \times N } matrix 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} , this will be done by applying 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 Flatten} to each row 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 C} . Recall that a step in the decryption process is multiplying the ciphertext matrix 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} with a secret key vector 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 \vec v} . We will always 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 \vec{v} = Powersof2(\vec{s}) } , a vector of this special form in order for flattening to preserve the product between 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} 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 \vec{v} } required for decryption.
Now, for any 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 N \times N} ciphertext matrix , we have
After computing 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_3 \leftarrow I_N - C_1 \cdot C_2 } for the NAND gate, we will immediately flatten it, namely we will set
References
- ↑ 1.0 1.1 1.2 C. Gentry, A. Sahai, and B. Waters. Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based. In CRYPTO 2013 (Springer). https://eprint.iacr.org/2013/340
- ↑ J. Alperin-Sheriff and C. Peikert. Faster Bootstrapping with Polynomial Error. In CRYPTO 2014 (Springer). https://eprint.iacr.org/2014/094
- ↑ I. Chillotti, N. Gama, M. Georgieva, and M. Izabachène. TFHE: Fast Fully Homomorphic Encryptionover the Torus. In Journal of Cryptology, volume 33, pages 34–91 (2020). https://eprint.iacr.org/2018/421
