BGV - Revision history

Gturcas: /* Implementation */

2021-02-07T16:04:11Z

Implementation

Gturcas: /* Implementation */

2021-01-11T14:52:46Z

Implementation

Gturcas: /* Implementation */

2021-01-11T14:52:03Z

Implementation

Gturcas: /* Implementation */

2021-01-11T14:51:29Z

Implementation

Gturcas: /* Implementation */

2021-01-11T14:47:20Z

Implementation

Gturcas: /* Leveled FHE */

2021-01-11T14:44:24Z

Leveled FHE

Gturcas: /* New noise management technique */

2021-01-11T14:25:43Z

New noise management technique

Gturcas: /* Leveled FHE */

2021-01-11T14:25:14Z

Leveled FHE

Gturcas: /* Leveled FHE */

2021-01-11T14:23:01Z

Leveled FHE

Gturcas: /* Leveled FHE */

2021-01-11T14:22:01Z

Leveled FHE

← Older revision		Revision as of 16:04, 7 February 2021
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	The implementation incorporates many optimizations that make homomorphic evaluation run faster, focusing mostly on effective use of the [https://eprint.iacr.org/2011/133 Smart-Vercauteren ciphertext packing] techniques and the [https://eprint.iacr.org/2012/099 Gentry-Halevi-Smart optimizations]. The latter were introduce in the seminal paper "Homomorphic Evaluation of the AES circuit"<ref>C. Gentry, S. Halevi, N. Smart. Homomorphic Evaluation of the AES Circuit, Advances in Cryptology – CRYPTO 2012, pp. 850-867.</ref>.		The implementation incorporates many optimizations that make homomorphic evaluation run faster, focusing mostly on effective use of the [https://eprint.iacr.org/2011/133 Smart-Vercauteren ciphertext packing] techniques and the [https://eprint.iacr.org/2012/099 Gentry-Halevi-Smart optimizations]. The latter were introduce in the seminal paper "Homomorphic Evaluation of the AES circuit"<ref>C. Gentry, S. Halevi, N. Smart. Homomorphic Evaluation of the AES Circuit, Advances in Cryptology – CRYPTO 2012, pp. 850-867.</ref>.
		+
		+	==Example use of HElib==
		+
		+	We refer to
		+
		+	https://tommd.github.io/posts/HELib-Intro.html
		+
		+	for short tutorial on how to use HElib for simple computations. More precisely, at the above link the reader can find an example of program which fills two vectors with integers, encrypts them using the [BGV] implementation and adds them (component by component) homomorphically.

	== References ==		== References ==

@@ Line 166: / Line 166: @@
 [https://github.com/homenc/HElib HElib] is an open-source software library that implements homomorphic encryption (HE) which contains the implementation of the Brakerski-Gentry-Vaikuntanathan (BGV) scheme with bootstrapping.
-The implementation incorporates many optimizations that make homomorphic evaluation run faster, focusing mostly on effective use of the [https://eprint.iacr.org/2011/133 Smart-Vercauteren ciphertext packing] techniques and the [https://eprint.iacr.org/2012/099 Gentry-Halevi-Smart optimizations]. The latter were introduce in the seminal paper
+The implementation incorporates many optimizations that make homomorphic evaluation run faster, focusing mostly on effective use of the [https://eprint.iacr.org/2011/133 Smart-Vercauteren ciphertext packing] techniques and the [https://eprint.iacr.org/2012/099 Gentry-Halevi-Smart optimizations]. The latter were introduce in the seminal paper "Homomorphic Evaluation of the AES circuit"<ref>C. Gentry, S. Halevi, N. Smart. Homomorphic Evaluation of the AES Circuit, Advances in Cryptology – CRYPTO 2012, pp. 850-867.</ref>.
 == References ==

← Older revision		Revision as of 14:52, 11 January 2021
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	<ref>C. Gentry, S. Halevi, N. Smart. Homomorphic Evaluation of the AES Circuit, Advances in Cryptology – CRYPTO 2012, pp. 850-867.</ref>		<ref>C. Gentry, S. Halevi, N. Smart. Homomorphic Evaluation of the AES Circuit, Advances in Cryptology – CRYPTO 2012, pp. 850-867.</ref>
−
−	~~We refer to their github page for more details: https://github.com/homenc/HElib~~

	== References ==		== References ==

@@ Line 164: / Line 164: @@
 == Implementation ==
-[https://github.com/homenc/HElib HElib] is an open-source software library that implements homomorphic encryption (HE) which contains the implementation of the Brakerski-Gentry-Vaikuntanathan (BGV) scheme with bootstrapping. We refer to their github page for more details: https://github.com/homenc/HElib
+[https://github.com/homenc/HElib HElib] is an open-source software library that implements homomorphic encryption (HE) which contains the implementation of the Brakerski-Gentry-Vaikuntanathan (BGV) scheme with bootstrapping.
 == References ==

← Older revision		Revision as of 14:47, 11 January 2021
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	== Implementation ==		== Implementation ==
		+
		+	[https://github.com/homenc/HElib HElib] is an open-source software library that implements homomorphic encryption (HE) which contains the implementation of the Brakerski-Gentry-Vaikuntanathan (BGV) scheme with bootstrapping. We refer to their github page for more details: https://github.com/homenc/HElib

	== References ==		== References ==

@@ Line 139: / Line 139: @@
 Remember one can see <math>L_{c_1}(x) \cdot L_{c_2}(x) = <c_1,x> \cdot <c_2,x> </math> as a linear equation over the coefficients of <math>x \otimes x</math>, the tensoring of <math>x </math> with itself. This is the meaning of the linear functional <math>L_{c_1,c_2}</math>. Then, output
-<center><math>c_4 \leftarrow</math> FHE.Refresh(<math>c_3 \tau_{s_{j}'' \to s_{j-1}}, q_j, q_{j-1}</math> ) </center>
+<center><math>c_4 \leftarrow</math> FHE.Refresh(<math>c_3 \tau_{s_{j}'' \to s_{j-1}}, q_j, q_{j-1}</math> ).</center>
 == References ==

← Older revision		Revision as of 14:25, 11 January 2021
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	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.		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.
−
−	~~== New noise management technique ==~~
−
−	~~@TODO~~

	== Leveled Fully Homomorphic Encryption ==		== Leveled Fully Homomorphic Encryption ==

@@ Line 133: / Line 133: @@
 Output
-<center><math>  c_4 := FHE.Refresh(c_3, \tau_{s_{j}'' \to s_{j-1}}, q_j, q_{j-1}).</math> </center>
+<center><math>  c_4 \leftarrow FHE.Refresh(c_3, \tau_{s_{j}'' \to s_{j-1}}, q_j, q_{j-1}).</math> </center>
 * FHE. Mult(<math>pk, c_1, c_2 </math>): As before, the ciphertexts must be encrypted under the same <math> s_j</math>. If they are not, one uses FHE.Refresh to make it so.
@@ Line 141: / Line 141: @@
 <center><math>L^{long}_{c_1,c_2}(x \otimes x).</math></center>
-Remember one can see <math>L_{c_1}(x) \cdot L_{c_2}(x) = <c_1,x> \cdot <c_2,x> </math> as a linear equation over the coefficients of <math>x \otimes x</math>, the tensoring of <math>x </math> with itself. This is the meaning of the linear functional <math>L_{c_1,c_2}</math>.
+Remember one can see <math>L_{c_1}(x) \cdot L_{c_2}(x) = <c_1,x> \cdot <c_2,x> </math> as a linear equation over the coefficients of <math>x \otimes x</math>, the tensoring of <math>x </math> with itself. This is the meaning of the linear functional <math>L_{c_1,c_2}</math>. Then, output
 == References ==

← Older revision		Revision as of 14:22, 11 January 2021
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	First, multiply: the new ciphertext, under the secret key <math>s_j' = s_j \otimes s_j </math>, is the coefficient vector <math>c_3 </math> of the linear equation		First, multiply: the new ciphertext, under the secret key <math>s_j' = s_j \otimes s_j </math>, is the coefficient vector <math>c_3 </math> of the linear equation
		+
		+	<center><math>L^{long}_{c_1,c_2}(x \otimes x).</math></center>
		+
		+	Remember one can see <math>L_{c_1}(x) \cdot L_{c_2}(x) = <c_1,x> \cdot <c_2,x> </math> as a linear equation over the coefficients of <math>x \otimes x</math>, the tensoring of <math>x </math> with itself. This is the meaning of the linear functional <math>L_{c_1,c_2}</math>.

	== References ==		== References ==