Homomorphic encryption scheme

Presentation
on
A Secure Database
System using
Homomorphic
Encryption
Schemes
1

23 January 2020
GROUP INFORMATION 2
Md. Shahin Kadir
Student ID: 183235
Md. Ibrahim Ali
Student ID: 183202
Md. Shams Sayied Haque
Student ID: 183227
Saleh Ahmmed Miajee
Student ID: 183228
IIT, Jahangirnagar University

23 January 2020
INTRODUCTION 3
Cloud computing is an attractive
solution that can provide low cost
storage and processing capabilities
for government agencies, hospitals,
and small and medium enterprises.

23 January 2020
INTRODUCTION 4
The confidentiality of information
as well as the liability for incidents
affecting the infrastructure arise
major problems.
So, the ultimate objective is to present a relational database system based on
homomorphism encryption schemes to preserve the integrity and confidentiality of
the data.

23 January 2020
INTRODUCTION 5
The data can be encrypted by the client, and then
sent to the cloud’s provider for storage or processing.
Only the client holds the decryption keys necessary
to read the data. Despite the fact that this type of
processing may increase the amount of computing
time, the benefits associated with it are worth the
processing overhead. Indeed, this model of
computing can preserve the confidentiality and
integrity of the data while delegating the storage and
processing to an un-trusted third party.

23 January 2020
INTRODUCTION 6
In this paper, it is presented a novel
technique to execute SQL statements over
encrypted data and developed a secure
database system that processes these
queries. The parameters of SQL queries are
encrypted by the client and sent to the
server for processing. The latter performs
the requested operation over an encrypted
database and returns an encrypted result to
the client.

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PRIVATE INFORMATION RETRIEVAL 7
Chow et al. [8]
discussed the
importance of cloud
computing, and how
this technology can be
enticing due to its
flexibility and cost-
efficiency. The
authors highlight that
people require explicit
guarantees that their
data will be protected
under well-defined
policies and
mechanisms.
The private
information retrieval
(PIR) approach,
introduced by Chor et
al. in [9], achieves the
retrieval of an ith bit
in a block without
revealing information
about the bit
retrieved or about the
request for the bit
itself.
Raykova et al. [10]
extended the PIR
approach by
proposing a secure
anonymous search
system. The system
employs keyword
search such that only
authorized clients
have access to their
blocks.

23 January 2020
Shang et al. [11]
tackled the problem
of protecting the
database itself. The
information attained
from the monitoring
process is used to
understand how a
malicious querier can
conduct attacks to
retrieve excessive
amount of data from
the server.
Nakamura et al. [12]
constructed a system
with three
components, a
querier that initiates
requests, an
authentication-server
that processes these
requests, and a
database that returns
the appropriate data
in response to the
request.
Yinan and Cao [13]
used the PIR approach
to propose a system
that controls the
access to the
database.

23 January 2020
Among the most important criteria in PIR protocol are the communication cost and the
amount of data sent back to the querier.
Gentry et al. [14] proposed a
scheme to retrieve a bit or a
block from a database with a
constant communication rate.
Melchor et al. [15] proposed a
scheme that reaches the
available data with a
reasonable communication
cost while achieving lower
computational cost compared
to other PIR protocols.

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SECURED SQL OPERATIONS 10
Cloud
Provider
Client
Encrypted Data
Encrypted Request
Encrypted Result
Figure: Secure Data Retrieval

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Here describe below the circuit of a simple SQL SELECT query:
SELECT * from T where c = v
Here the value v is in encrypted
form.
The processing of the SELECT query is divided into three sub-circuits. Firstly, we calculate the
following index for each record R in the table T:
Where size is the number of bits in column c; 𝑐𝑖 and 𝑣𝑖 are the ith bits of column c and search criteria v, respectively. 𝐼𝑅
is a one bit value that is equal to 1 if v matches the value of column c, 0 otherwise.

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Next, identify the nth record that matches the selection criteria. For that purpose, we consider 𝜂
= 𝜀𝑝𝑘(𝑛) to be the encryption of n under public key pk. For each record R we calculate the
following sum:
Calculate a second index 𝐼′𝑅:
𝐼′𝑅 is equal to 1 if the record R is the nth record that matches the selection criteria, 0 otherwise.

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Then, we multiply every bit of each record R in table T by the corresponding value 𝐼′𝑅.
This latter operation forms a table 𝑇′ that is related to the original table T as follows:

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Finally, by adding all records of table 𝑇′, we retrieve the nth record Rs that matches the
selection criteria:
If no record matches the selection criteria, a record containing zeros will be returned to
the requester.

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HOMOMORPHIC ENCRYPTION SCHEME 15
Key Generation
The private key Sk is a random P-bit odd number.Private Key
Public Key
The public key consists of a list of integers that are the using the
encryption scheme with the secret key sk as a public key.
Generate a set 𝑦⃗={𝑦1,…,yβ} of rational numbers in [0,2[ such that there is a sparse subset
𝑆 ∈ {1,…,𝛽} of size ∝ with
Set sk* to be the sparse subset S, encoded as a vector 𝑠 ∈ {0,1} 𝛽 with hamming weight α.
Set 𝑝𝑘∗← to be the public key.

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Encryption (pk*,m)
Set 𝑚′ to be a random N-bit number such that m and m’ have the same parity,
Here q is a random Q-bit number. Then the cipher text c is post-processed to produce a
vector 𝑧⃗={𝑧1,…,𝑧𝛽}, defined by,
The output cipher text c* consists of c and 𝑧⃗={ 𝑧1,…, 𝑧𝛽}
Now compute c as,

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Decryption (sk*, c*)
Addition
Multiplication
The output cipher text c* consists of c together with the result of post-processing the
resulting cipher text with 𝑦⃗.
Arithmetic Operations

23 January 2020
Bootstrapping the Encryption Scheme
C1
Decryption circuit
m
(Noise)
C2
m
Figure: Removing noise from original cipher text (bootstrapping)

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IMPLEMENTATION 19
The application supports the following SQL
operations:
SELECT with wildcard characters (*, ?)
and relational operators (< >).
UPDATE with wildcard characters (*, ?)
and relational operators (< >).
DELETE with wildcard characters (*, ?) and
relational operators (< >).
Statistical operations like COUNT and AVG.
Figure: Client side of the application

23 January 2020
PERFORMANCE ANALYSIS 20
Table lists the number of arithmetic operations required to execute some basic SQL statements
over an encrypted database of 10 records. From this table we can see that processing data in
encrypted form creates a substantial computation overhead.
TABLE: NUMBER OF ARITHMETIC OPERATIONS

23 January 2020
PERFORMANCE ANALYSIS 21
Figure: Processing time required to perform
the product of two n-bits integers
As we can see in Figure, it takes 23 minutes to
compute the product of two 16-bit integers.
The implementation of the system proves that the
execution of SQL statements over encrypted data is
feasible.
The time required to execute these statements is very
high and therefore is not suitable for real-time
transactions that involve a large database.
 This drawback is mainly due to the homomorphic
encryption scheme.
In fact, there might be more efficient techniques to
optimize the implementation, that is, one could
perform recryption only when it is necessary, since the
noise value can be bounded.

23 January 2020
CONCLUSION & FUTURE DIRECTIONS 22
This concept has many direct applications in cloud computing environments, banking,
electronic voting and many other applications.
In this paper we developed the first secure database system based on a fully homomorphic
encryption scheme.
Presented the circuits to implement SQL statements over encrypted data.
Built a prototype of a database system where data is stored & processed in encrypted form.
 Conducted performance analysis to measure the time needed to execute a simple query on
the database.
As future work, here planning to work on the optimization of the efficiency of the system.
 Also investigate how to reduce the number of recryptions needed.

23 January 2020
REFERENCES 23
[1] R. Rivest, A. Shamir, and L. Adleman, A Method for Obtaining Digital Signatures and Public-Key Cryptosystems, Communications of the ACM 21 (2): pp.
120–126, 1978.
[2] T. ElGamal, A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms, IEEE Transactions on Information Theory, pp. 469–472,
1985.
[3] S. Goldwasser and S. Micali, Probabilistic Encryption. Journal of Computer and System Sciences, 28(2): pp. 270-299, April 1984.
[4] P. Paillier, Public-Key Cryptosystems Based on Composite Degree Residuosity Classes, Advances in Cryptology — EUROCRYPT ’99 In Advances in
Cryptology — EUROCRYPT ’99 , Vol. 1592 (1999), pp. 223-238, 1999.
[5] M. V. Dijk, C. Gentry, S. Halevi, and V. Vaikuntanathan, Fully Homomorphic Encryption over the Integers. EUROCRYPT 2010: pp. 24-43, June 2010.
[6] C. Gentry, Computing arbitrary functions of encrypted data, Commun. ACM, Vol. 53, No. 3., pp. 97-105, March 2010.
[7] C. Gentry, A fully homomorphic encryption scheme. PhD thesis, Stanford University, 2009.
[8] R. Chow, P. Golle, M. Jakobsson, R. Masuoka, and J. Molina, Controlling Data in the Cloud : Outsourcing Computation without Outsourcing Control.
CCSW’09, pp. 85-90, Chicago, Illinois, USA, November 13, 2009.
[9] B. Chor, E. Kushilevitz, O. Goldreich, and M. Sudan, Private Information Retrieval, Journal of the ACM, 45(6): pp. 965-982, 1998.
[10] M. Raykova, B. Vo, and S. Bellovin, Secure Anonymous Database Search, CCSW’09, pp. 115-126, Chicago, Illinois, USA, November 13, 2009.
[11] N. Shang, G. Ghinita, Y. Zhou, and E. Bertino, Controlling Data Disclosure in Computational PIR Protocols. ASIACCS’10, pp. 310-313, Beijing, China, April
13–16, 2010.
[12] T. Nakamura, S. Inenaga, D. Ikeda, K. Baba, H. Yasuura, Anonymous Authentication Systems Based on Private Information Retrieval. Networked Digital
Technologies. NDT '09, pp.53-58, 28-31 July 2009.
[13] S. Yinan and Z. Cao, Extended Attribute Based Encryption for Private Information Retrieval. Mobile Adhoc and Sensor Systems, 2009. MASS '09, pp. 702-
707, 12-15 Oct. 2009.
[14] C. Gentry and Z. Ramzan, Single-Database Private Information Retrieval with Constant Communication Rate. ICALP 2005, LNCS 3580, pp. 803–815, 2005.
[15] C. A. Melchor and P. Gaborit, A Fast Private Information Retrieval Protocol. ISIT 2008, pp. 1848-1852, Toronto, Canada, July 6 - 11, 2008.

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Homomorphic encryption scheme

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