IMPLEMENTING A
PAILLIER THRESHOLD CRYPTOGRAPHY SCHEME
AS A WEB SERVICE
by
Brett Wilson
A Master’s project
submitted in partial fulfillment
of the requirements
for the degree of
Master of Science in Computer Science
University of Colorado – Colorado Springs
Fall 2006
Approved by __________________________________________________________
Dr. C. Edward Chow, Advisor
__________________________________________________________
Dr. Jugal Kalita, Advisory Committee Member
__________________________________________________________
Dr. Joe Zhou, Advisory Committee Member
__________________________________________________________
Dr. Seung H. Son, Advisory Committee Member
Date ________________________________________________________________
ABSTRACT
IMPLEMENTING A PAILLIER
THRESHOLD CRYPTOGRAPHY
SCHEME AS A WEB SERVICE
by
Brett Wilson
Chairperson of the Supervisory Committee
Professor C. Edward Chow
Department of Computer Science
Much recent cryptography research has been focused on solving problems that pertain
to secure electronic voting. Specifically, different forms of cryptography are being
applied in order to achieve the all-important voting system properties of
privacy/anonymity, accuracy, verifiability, receipt-freeness, in-coercibility, and
robustness.
The Paillier cryptosystem [1] based on the composite residuosity class
problem has received much of the research interest due to possessing some important
properties useful in the e-voting application. In [2], the authors present a verifiably
secure threshold variant of Paillier’s scheme, which enables the distribution of trust
across a number of system participants, further enabling it’s use in the e-voting
application. This paper presents an implementation of that scheme as a web service.
In this implementation, the web service serves as the “trusted dealer” and thus
removes the requirement for picking the “trusted dealer” from among the users of the
cryptosystem. It is envisioned that such a web service could be commercially operated
by a universally trusted organization similar to modern public key certificate
authorities. A complete user’s guide for installing and using the PTC Web Service is
given in [8].
ii
TABLE OF CONTENTS
INTRODUCTION .................................................................................................................... 1
1.1 Paillier Cryptosystem ...................................................................................................... 1
1.2 Threshold Cryptography ............................................................................................... 3
1.3 Paillier Threshold Cryptography .................................................................................. 4
PAILLIER THRESHOLD CRYPTOSERVICE IMPLEMENTATION .................... 6
2.1 General Architecture ...................................................................................................... 6
2.2 Tools.................................................................................................................................. 6
2.3 General Use Case Descriptions.................................................................................... 7
2.3.1 PaillierThresholdCryptoServiceProvider (PTCSP) ........................................ 7
2.3.1.1 PTCSP constructor .................................................................................... 8
2.3.1.2 Encrypting a Message ................................................................................ 8
2.3.1.3 Decrypting a Message................................................................................ 8
2.3.1.4 Combining Decryption Shares ................................................................ 9
2.3.2 Paillier Threshold Cryptography Web Service (PTC Web Service) ............ 9
2.3.2.1 Obtaining Paillier Threshold Cryptography Parameters................... 10
2.3.2.2 Using the Paillier Threshold Parameters ............................................. 10
DESIGN DISCUSSION ........................................................................................................ 12
3.1 Problems and Solutions ............................................................................................... 12
3.1.1 Generating Large Safe Primes .......................................................................... 12
3.1.2 Implementing Shamir Secret Sharing ............................................................. 13
3.2 Software Choices .......................................................................................................... 14
3.2.1 Big Integer Support ............................................................................................ 14
3.2.1.1 C# BigInteger Class................................................................................. 14
3.2.1.2 Microsoft Visual J# BigInteger Class ................................................... 14
3.2.1.3 GNU Multiprecision Library (GMP) ................................................... 15
3.2.1.3.1 Compiling GMP under Visual Studio 2005 .............................. 15
3.2.1.3.2 Accessing GMP from Visual Basic .NET ................................. 15
3.2.2 Generating Test X.509 Certificates ................................................................. 16
3.3 PTC Web Service Performance Evaluation............................................................. 16
3.3.1 Method of Performance Evaluation ............................................................... 16
3.3.2 Results of Performance Evaluation ................................................................ 17
3.3.3 Performance Evaluation Conclusion .............................................................. 18
3.4 Suggested Enhancements and Optimizations ......................................................... 19
3.4.1 Generating Large Safe Primes .......................................................................... 19
3.4.2 Web Service Signing of Parameter Data ........................................................ 19
3.4.3 Cryptosystem Optimizations ............................................................................ 19
3.4.4 A Non-Interactive Proof That a Given Cipher Encrypts One of a Set of
Messages ........................................................................................................................ 20
APPENDIX ............................................................................................................................... 21
4.1 Data Type Descriptions............................................................................................... 21
iii
4.2 Sample SOAP Request/Response Messages........................................................... 25
4.3 Demonstration E-Voting Application ...................................................................... 28
4.3.1 E-Voting System Overview .............................................................................. 28
4.3.1.1 Election Set-Up ........................................................................................ 29
4.3.1.1.1 Obtaining Election Encryption Parameters ............................. 29
4.3.1.1.2 Creating Ballots .............................................................................. 29
4.3.1.1.3 Adding Ballots to an Election ..................................................... 30
4.3.1.1.4 Vote Format.................................................................................... 30
4.3.1.1.4.1 Cheating the Vote............................................................................ 31
4.3.1.1.5 Saving and Posting an Election ................................................... 31
4.3.1.2 Voting ......................................................................................................... 32
4.3.1.2.1 Creating the Vote ........................................................................... 32
4.3.1.2.2 Submitting the Vote ...................................................................... 32
4.3.1.2.3 Checking the Vote ......................................................................... 32
4.3.1.3 Tally The Vote .......................................................................................... 33
iv
Chapter 1
INTRODUCTION
1.1 Paillier Cryptosystem
Pascal Paillier first introduced a cryptosystem based on the composite residuosity class
problem in [1]. Paillier described 3 different cryptosystems in [1]. We will only discuss
the first of these here. The cryptosystem public parameters n and g, and the secret key
SK are obtained as follows: Let n be a modulus n=pq where p and q are prime
integers. Let g be an integer of order nα mod n2. The public key is PK=(n,g) and the
secret key is SK = λ(n) = lcm((p-1),(q-1)).
To encrypt a message m Є Zn , randomly choose r in Zn* and compute
C=gMrn mod n2
To decrypt c, compute
L(c  ( n ) mod n 2 )
M
mod n
g  ( n ) mod n 2
Since 1999 there has been much research into investigating the scheme’s security,
extending it’s functionality, and improving it’s efficiency. This interest is due primarily
to some unique properties of Paillier’s scheme.
Namely, Paillier’s scheme is
“homomorphic”, “self-blinding”, and “probabilistic”. The scheme is classified as a
homomorphic scheme which in general means that there are certain arithmetic
operations that when carried out in the ciphertext space correspond to a known
1
arithmetic operation occurring in the cleartext space.
In the case of Paillier’s
cryptosystem, the following homomorphisms apply [1]:
For m1; m2 Є Zn and k Є N
D(E(m1) E(m2) mod n2) = m1 +m2 mod n
D(E(m)k mod n2) = km mod n
D(E(m1) gm2 mod n2) = m1 +m2 mod n
D(E(m1)m2 mod n2) = m1m2 mod n
D(E(m2)m1 mod n2)) = m1m2 mod n
Paillier’s cryptosystem also possesses a unique property termed “self-blinding”. This
property is defined as follows:
For m Є Zn and r Є N
D(E(m) rn mod n2) = m or D(E(m) gnr mod n2) = m
Essentially what this means is that a ciphertext can be “re-encrypted” with a random
parameter without changing the underlying cleartext and without changing the ability
to decrypt the ciphertext using the original keypair. This property could be very useful
in implementing mix-nets.
Cryptosystems in general fall into two categories: deterministic or probabilistic. A
deterministic system produces the same ciphertext for the same cleartext message and
key every time. A probabilistic system, such as Paillier’s, includes a randomizing
element so that a given cleartext message can be encrypted into any one of a very large
set of possible encryption values.
All of these properties could be very useful in implementing secure electronic voting
applications. The homomorphic property could help in securing a voter’s privacy as
well as greatly reducing the number of decryption operations required in the system
(essentially to 1). The property of “self-blinding” could be applied in a mix-net
2
scheme to protect a voter’s anonymity. And finally, the probabilistic property is very
important for protecting a voter’s privacy, since no two votes, even if they are identical
votes, will be encrypted to an identical ciphertext.
1.2 Threshold Cryptography
Threshold Cryptography is a term used to describe a cryptosystem in which the ability
to perform a cryptographic function can be distributed amongst several participants in
such a way that only through cooperation of a specified subset of the participants can
the operation be performed.
In addition, if less than the required number of
participants attempt to perform the action, no useful information can be constructed
or obtained. The threshold value is typically denoted by the letter t. In a threshold
system as defined here, only t+1 cooperating authorities can perform the desired
cryptographic operation.
[2] describes the essential components of a threshold cryptography system as a key
generation algorithm, an encryption algorithm, a share decryption algorithm, and a combining
algorithm. First, the key generation algorithm generates the public key parameters, a set
of secret key “shares”, and a set of “verifier keys”.
The secret key shares are
distributed to the participants in a secure manner. The encryption algorithm provides
encryption services for an appropriately-sized message m by applying the public key
parameters and an encryption algorithm to generate the ciphertext c. The share
decryption algorithm is used by each participant with a secret key share to “partially
decrypt” the encrypted message c.
Each participant also uses the verifier key
corresponding to the secret key share to generate a proof of correct encryption. The
combining algorithm takes all of the “partial decryptions” or “decryption shares”,
verifies their corresponding proofs, and combines the decryption shares to reveal the
original message m. The combining step only succeeds if t+1 valid decryption shares
are used.
3
1.3 Paillier Threshold Cryptography
To further enable Paillier’s system for use in scenarios where it is desirable to distribute
trust between numerous interested parties, other parties have developed threshold
versions of Paillier’s system [2][3]. This paper documents an implementation of the
scheme as described in [2]. Both [2] and [3] use Shamir’s Secret Sharing Scheme [4] as
the method of sharing the secret key among the system participants. They only differ
slightly in their encryption and decryption algorithms, in that [3] restricts the selection
of the original system parameter g in order to achieve slightly more efficient
algorithms. [3] also defines a generalization of Paillier’s scheme which also increases
the efficiency of encryption and decryption, especially for larger-sized messages.
The system parameters are chosen in [2]as follows: Choose an integer n, product of
two strong primes p and q, such that p=2p’ + 1 and q=2q’ + 1. Let m = p’q’. Let Β be
random element in Zn*. Also choose (a,b) Є Zn* x Zn*. Set g = (1 + n)a x bn mod n2.
The secret key Β x m is shared with the Shamir scheme: Let a0 = Bm, and randomly
choose ai in {0,…n x m – 1} and set f(X) = Σti=0 aiXi. The share si of the ith server Pi is
f(i) mod nm. The public key PK consists of g, n, and the value θ = L(gmB) = amB mod n.
The verification keys are generated by first finding VK = v as a square that generates
the cyclic group of squares in Zn2*. The verification keys VKi can then be obtained
with the formulat vsi mod n2.
To encrypt a message M Є Zn , randomly choose r Є Zn* and compute
C=gMrn mod n2
To decrypt C, each participant Pi computes the decryption share ci=c2si mod n2 using
his/her secret share si. He/she also makes a proof of correct decryption which assures
that c4 mod n2 and v have been raised to the same power si in order to obtain ci2 and
VKi.
4
And finally, if t + 1 valid shares are available, they can be combined to recover M as
follows: Let S be a set of t + 1 valid shares. Compute


1
2s
M  L  c j 0 , j mod n 2  x 2 mod n
 jS
 4 
where 0s, j  x j 'S \{ j}
j'

j ' j
See [2] for more details on the correctness of the scheme and for proofs of security.
5
Chapter 2
PAILLIER THRESHOLD CRYPTOSERVICE IMPLEMENTATION
2.1 General Architecture
The threshold implementation of Paillier’s cryptosystem uses the algorithms proposed
in [2] and described above. It consists of two main software pieces and many small
supporting pieces. The main piece is named PaillierThresholdCryptoServiceProvider
(PTCSP). It contains all of the algorithms and data handling methods required to fully
implement the Paillier Threshold scheme. The second piece is the Paillier Threshold
Cryptography Web Service (PTC Web Service) itself, which uses the PTCSP to
generate the requested system parameters, properly secure them, and return them to
the requestor.
A simple voting application which demonstrates the critical pieces of the threshold
cryptosystem was also created. It provides a simple user interface in which to set up
and conduct an election using the PTC Web Service to provide the cryptosystem
parameters. The voting application as presented here is not fully complete or secure,
but what is not should be trivial to complete and is left to an interested reader.
2.2 Tools
The entire implementation was written in Microsoft VB.NET using Microsoft Visual
Studio 2005 as the development environment. The PTC Web Service is implemented
as an ASP.NET 2.0 Web Service hosted on Microsoft Internet Information Services
(IIS) Server. And lastly, the simple e-voting application is a Windows .NET Forms
application that also uses an ASP.NET Web Service as a front-end to a Microsoft SQL
Server database that is used to store and retrieve election data.
6
2.3 General Use Case Descriptions
The Paillier Threshold Cryptosystem consists of two main pieces. In addition, there
are numerous other classes and structures primarily used for organizing and passing
data. This section will discuss each piece in more detail and provide more information
on the services each piece can provide. All data structures and their constituent types
are defined in an XSD format in the Appendix.
2.3.1 PaillierThresholdCryptoServiceProvider (PTCSP)
The class PaillierThresholdCryptoServiceProvider implements all of the necessary
algorithms to generate the threshold parameters, encrypt a message, partially decrypt a
ciphertext, and combine the partial decryptions to reveal the original message. It is
modeled after the Microsoft .NET cryptoserviceprovider model and implements the
.NET ICSPAsymmetricAlgorithm interface. Even though it does implement this
interface, not all of the required methods are functionally implemented, as they were
not required for implementing the desired functionality for this project.
The PTCSP holds two main pieces of data. The first of these is the cryptosystem
parameters of type PaillierThresholdParameters. This structure can hold both the
public and private keys, the secret key shares, and the system parameters specifying the
number of shares and the threshold value. This structure can be initialized with new
parameters upon instantiation of the PTCSP class, or imported into an existing PTCSP
object.
Most of the methods of the PTCSP class utilize various values in the
parameters structure, so it is important that these parameters are initialized correctly.
The other main data piece is the DecryptionShares list, exposed as a public property of
generic type List(Of ThresholdDecryptionShares). This list can either be populated by
“importing” decryption shares obtained from participants in the system, or generated
from a given ciphertext and the secret key shares present in the parameters data
7
structure. This list is used by the combining algorithm when reconstructing the
original message.
2.3.1.1 PTCSP constructor
The PTCSP constructor is overloaded four times. The default constructor does not
initialize any parameters. A second constructor takes a keysize parameter and creates a
public key of the specified size and a private set of parameters with only one
decryption share and the threshold value set at 0. This provides for a standard (nonthreshold) Paillier cryptosystem.
A third constructor takes as a parameter a
PaillierThresholdParameters structure and initializes the PTCSP with this set of
parameters. A fourth constructor takes as parameters the keysize, the desired number
of keyshares, and the desired threshold value and creates a set of parameters with these
values.
2.3.1.2 Encrypting a Message
To encrypt a message, two overloads of the Encrypt method are provided. The first
method takes as input a message consisting of an array of bytes, and a random array of
bytes. It uses the internal public key and the provided random value to encrypt the
given message. The second overload is also “shared” or “static” and takes as input a
message array of bytes, a public key of type PaillierPublicKey, and a random array of
bytes. It is provided as a shortcut method to encrypt a message without having to fully
instantiate the class, load a public key, and then call the encrypt method.
2.3.1.3 Decrypting a Message
Once again, there are two overloads provided for decrypting a message. In the
decryption case, however, the results are significantly different. The first overload
takes as input only the ciphertext as an array of bytes. It then uses the internally stored
secret key shares to generate decryption shares and validity proofs, and then combines
these shares to recover the cleartext message. This process assumes that a sufficient
8
number of secret key shares are available in the PaillierThresholdParameters structure,
or it will fail with an exception.
The other overload would be used by an individual system participant to partially
decrypt a ciphertext using his/her decryption share. This overloaded method takes as
parameters the ciphertext as a byte array, the secret key share as type
PaillierThresholdKeyShare, and a verifier key as type PaillierThresholdVerificationKey.
It computes the partial decryption, the proof of correct decryption, inserts them both
into a PaillierThresholdDecryptionShare type and then inserts it into the
DecryptionShares list. A user application could then serialize the resulting decryption
share to an XML file for transmission to a combining authority who would combine
shares from all the participants to recover the message.
2.3.1.4 Combining Decryption Shares
Once the DecryptionShares list has been populated with the required t + 1 decryption
shares, calling the CombineDecryptionShares method with return a byte array with the
reconstructed cleartext message.
CombineDecryptionShares iterates through the
DecryptionShares list, verifies each proof of correct decryption, and then combines the
valid shares to obtain the unencrypted message. The combining algorithm will throw
an exception if t + 1 valid decryption shares are not available.
2.3.2 Paillier Threshold Cryptography Web Service (PTC Web Service)
The PTCSP class by itself can be used by a trusted party to generate the Paillier
threshold parameters and distribute them to the participants. In many situations, it
may be difficult to identify a “trusted dealer” acceptable to all parties. For this reason,
the Paillier Threshold Cryptography Web Service was created. The idea is that the
Web Service would be run (and secured) by a completely disinterested party in much
the same manner as a Certification Authority in the modern Public Key Infrastructure.
9
In this manner, the trust is removed from the “participants” and placed with the web
service.
2.3.2.1 Obtaining Paillier Threshold Cryptography Parameters
A
user
wishing
to
obtain
Paillier
threshold
parameters
calls
the
GeneratePaillierThresholdParameters method of the web service with the appropriate
parameters.
The parameters are supplied as an XML serialization of the
ThresholdParameterRequest structure. The parameters include the desired keysize, the
number of key shares to generate, the system threshold value, and an array of
OwnerInfo types that each contain the key share owner’s name and, optionally,
his/her X.509 certificate.
Upon receiving a request, the web service instantiates a PTCSP object by calling the
appropriate constructor to generate the parameters as requested. Once the parameters
are obtained, the web service iterates through the secret key shares and encrypts each
key share with the target owner’s RSA public key and sets the “encrypted” flag true for
each key share (if X.509 certificates are provided).
If no X.509 certificates are
provided, the key shares are returned un-encrypted, which basically invalidates the use
of the web service since the requestor must now be trusted not to access the secret key
shares.
2.3.2.2 Using the Paillier Threshold Parameters
The parameters are returned by the web service as an XML serialization of the
PaillierThresholdParameters structure. The contents of this structure are defined in
the Appendix.
The client must deserialize the returned XML into a
PaillierThresholdParameters structure in order to pass it to the constructor of a
PTCSP object. This is typically done by a web service client in the web service’s proxy
class implementation.
10
Once deserialized, the PaillierThresholdParameters structure is then ready for use by
the client. A PTCSP object can be created by passing the PaillierThresholdParameters
structure to the PTCSP constructor. Once the PTCSP is created, it can be used to
encrypt and decrypt messages.
11
Chapter 3
DESIGN DISCUSSION
3.1 Problems and Solutions
3.1.1 Generating Large Safe Primes
The generation of the Paillier Threshold parameters requires a set of two very large
safe prime numbers.
The GMP software and .NET both provide methods for
generating large random numbers, and the GMP software provides the capability to
check a number for probable prime. However, there are no known fast algorithms for
directly generating large safe primes.
[5] details a basic algorithm that involves
randomly generating numbers of the desired size, testing for primeness, and then, if
prime, testing for safe primality. [6] specifies a simple method to double the speed of
the safe prime generation. The ideas of [6] were used in this implementation to help
speed up the generation of the safe primes. First, a prime number p of the desired size
was found, and then both 2p+1 and (p-1)/2 were checked for primeness. If 2p+1 is
prime, then p is the safe prime. If (p-1)/2 is prime then (p-1)/2 is the safe prime.
Algorithm
2x speed
Regular
128
bit
256
bit
512
bit
1024
bit
Avg
Max
Min
Avg
0.2728
0.885
0.052
0.3247
1.75315
4.025
0.285
2.34
39.85
152.248
1.524
50.3496
705.5993
1754.158
84.523
1950.934
Max
Min
0.949
0.065
11.672
0.08
110.558
2.103
9426.193
57.191
Figure 1 – Speed Comparison of prime number
generation algorithms
Further speed improvements may be achieved by using the ideas of [7], however,
considerabe time will still be required to generate 1024 bit and larger safe primes. An
12
acceptable solution to this problem would be to develop a background application that
constantly generates safe primes and writes them to a file. The web service could then
obtain a safe prime directly from this file, achieving very significant time
improvements. This would be a trivial enhancement to the existing web service. This
solution is of course only acceptable if the rate of requests does not exceed the rate at
which the background application can generate them.
3.1.2 Implementing Shamir Secret Sharing
In the threshold scheme, the secret keys s0,…,si are distributed to the participants with
the Shamir Secret Sharing scheme (see Chapter 1). From the algorithm equations
given in [2], it is not immediately clear that the share index is a critical piece of the reassembly of the secret. In other words, the index of each of the secret shares
generated during the parameter creation must be saved with the secret share. For
example, suppose 5 secret key shares s0,…s4 are generated during the parameter
creation and distributed to the 5 participants p0,…p4. Suppose the threshold value of
the cryptosystem is 2 and that only 3 of the 5 participants (p0, p2, p3) provide a
decryption share as part of the decryption process. The combining algorithm requires
knowledge of the share indexes being used (0, 2, 3) in order for the algorithm to work.
These indexes are used as the j and j’ indexes in the combining algorithm.


1
2s
M  L  c j 0 , j mod n 2  x 2 mod n
 jS
 4 
where 0s, j  x j 'S \{ j}
j'

j ' j
In addition, these index values are required to properly index the verification keys in
order for the participants to correctly construct the proof of correct decryption. The
verification keys must be properly matched to the secret key shares for correct
13
generation and verification of the proof. For these reasons, the share index was
included in the data structure defining a secret key share (ShamirShare).
3.2 Software Choices
3.2.1 Big Integer Support
The decision to use Microsoft technology was made based on the extensive support
for handling XML data, creating web services, and debugging as well as the author’s
familiarity with the VB.NET language and Visual Studio environment. In general, the
platform allowed rapid development of the project.
However, the one critical
component not addressed in the VB.NET language is the support for handling very
large integers necessary for implementing the encryption algorithms. A significant
effort was undertaken researching the available solutions, downloading them, and
getting them to work in the .NET environment. A short description of each of these
options is presented in the next paragraphs.
3.2.1.1 C# BigInteger Class
This software is a freeware library written by Chew Keong Tan and is available for
download at http://www.codeproject.com/csharp/biginteger.asp. The goal of the
software is to implement the JAVA BigInteger class in C# so as to make it available to
other .NET projects. Since it was found that Microsoft has provided this same class
functionality this software was never downloaded or used.
3.2.1.2 Microsoft Visual J# BigInteger Class
Microsoft has implemented the Java BigInteger Class as part of it’s Visual J# runtime
library. It can be downloaded from Microsoft. This class provides much of the
functionality needed in this implementation. However, one drawback is that many of
the methods use the SByte data type as parameters or return types. Since this is not a
14
.NET CLR data type, it is more difficult to work with during conversions for I/O and
when using with other .NET classes.
3.2.1.3 GNU Multiprecision Library (GMP)
The GMP library is a full-featured multiprecision library supporting big integers as well
as unlimited precision rational and floating point numbers. It has the most extensive
support for different operations on big integers of the solutions researched. It also
claims to be the fastest library available.
The drawback to using it in a .NET
environment is that it is written exclusively in ANSI C, making it impossible to call
directly from .NET managed code. Fortunately, two other solutions were found that
allowed compiling of GMP in Visual Studio and exposing it to VB.NET. GMP can be
downloaded at http://www.swox.com/gmp/.
3.2.1.3.1 Compiling GMP under Visual Studio 2005
Research revealed that Brian Gladman has made available a Visual Studio 2005
solution
for
compiling
GMP.
It
can
http://fp.gladman.plus.com/computing/gmp4win.htm.
The
be
found
at
solution involves
downloading GMP, and then extracting Brian’s solution into the GMP source code
directory structure. A few configuration steps, and you can then compile directly from
within VS2005 for several different optimizations of GMP.
3.2.1.3.2 Accessing GMP from Visual Basic .NET
A VB.NET binding of the Win32 GMP .dll has been made available as the NGMP
project.
NGMP
can
be
downloaded
from
SourceForge
at
http://sourceforge.net/projects/ngmp. Much thanks to the author for making this
available. It was somewhat more difficult to set up, as it requires modifying several
include and reference paths in the C# and C++ projects. Once compiled, however,
the library made it extremely easy to access the GMP functionality from VB.NET.
15
3.2.2 Generating Test X.509 Certificates
In order to test the web service’s ability to properly encrypt the secret key shares, a set
of test X.509 certificates needed to be generated.
At first, a Win32 version of
OpenSSL was downloaded and tested. The tool properly generated a test certificate
with no problems. However, the process of inserting the certificates into a Windows
certificate store is manual and cumbersome. While researching how to do this using
.NET, the Visual Studio tool makecert.exe was discovered. This tool will quickly and
easily generate X.509 certificates and install them into a designated certificate store
(machine or user) automatically. OpenSSL was then abandoned and the MakeCert.exe
tool was used to generate the required test certificates.
3.3 PTC Web Service Performance Evaluation
3.3.1 Method of Performance Evaluation
It is suggested that the PTC Web Service could provide cryptosystem parameter
generation for simultaneous multiple users. In order to test the robustness of the web
service under high loads, the commercial application WebPartner Test and
Performance Center was used. This application provides the capability to run “stress
tests” on a web service. The user provides the SOAP request message which is
submitted to the web service when the test is run. The test is configurable to simulate
any number of simultaneous requests to the web service.
Up to 100 simultaneous requests were configured for the PTC Web Service tests. The
tests were done on a machine with a Pentium IV 3.0GHz dual-threaded processor
with 1.0GB RAM. The web service and testing software were running on the same
machine during the tests. The SOAP request message consisted of a Paillier Threshold
Parameter request for a secret key with a size of 256 bits, 5 encrypted keyshares, and a
threshold value of 3.
16
3.3.2 Results of Performance Evaluation
The test as configured was run in 15 discrete steps (each step at a 1 minute interval).
Each minute of the test, the number of simultaneous requests is increased until it
reaches the maximum configured for the test. The test with the maximum number of
simultaneous requests is repeated 5 times. The testing software outputs a plot of the
test results. The plot shows the average response time in seconds for for each of the
15 steps of the test. Figure 2 is a plot of the first stress test run. It shows that the PTC
Web Service successfully handled 50 simultaneous requests on 5 separate occasions,
with the average response time slightly greater than 2 seconds. This plot does not
show any easily-identified correlation between the number of simultaneous requests
and the average response time. This is likely due to the random nature of the search
for safe prime numbers used in the secret key generation algorithm.
Figure 2 – Performance Testing Results With Up To 50
Simultaneous Users
17
Figure 3 is a plot of the stress test results with up to 100 simultaneous requests. It can
be seen in this plot that the web service returned a “busy” error on several occasions.
These errors occurred at 21, 41, and 100 simultaneous requests. This may be due to
numerous extended times to find safe prime numbers.
Figure 3 – Performance Testing Results With Up To 100
Simultaneous Users
3.3.3 Performance Evaluation Conclusion
The stress tests performed indicate that the PTC Web Service is capable of handling at
least 100 simultaneous requests for PTC threshold parameter generation. However, it
also clearly shows the random inefficiency of the safe prime generation algorithm.
This algorithm appears to be the major performance limiter of the PTC Web Service.
18
3.4 Suggested Enhancements and Optimizations
3.4.1 Generating Large Safe Primes
The generation of the Paillier Threshold parameters requires a set of two very large
safe prime numbers. As discussed above, this process is very slow for 512 bit and
larger safe primes. A definite improvement would be to implement the background
safe prime generator as recommended.
3.4.2 Web Service Signing of Parameter Data
In the current implementation, it would be easy for a corrupt individual to use the
PTCSP to create a set of Paillier threshold cryptosystem parameters, properly encrypt
the resulting secret key shares with the participants’ public keys, and distribute the
parameters as if they were generated by the web service. The corrupted party would
then have the ability to decrypt messages as desired. In order to prevent this, there
must be a way for a participant to verify that the web service did indeed generate the
parameters in question. This could be accomplished by adding the web service’s digital
signature to the returned parameters. Further work could be undertaken to determine
exactly what data should be signed and where the signatures should be included in the
PaillierThresholdParameters structure. It will be necessary to include in the signed
data a unique value differentiating the parameter set from others generated by the web
service.
3.4.3 Cryptosystem Optimizations
In [3], the authors present an extended version of Paillier’s cryptosystem that allows
for larger message block sizes and greatly increases the efficiency of encrypting and
decrypting larger messages.
In addition, several optimizations are presented to
improve the efficiency of the original system. The implementation presented in this
paper could be enhanced by incorporating some of these ideas.
19
3.4.4 A Non-Interactive Proof That a Given Cipher Encrypts One of a Set of
Messages
In order for the implementation presented here to be useful in the secure e-voting
application, there must be a method to prove the validity of a vote. Otherwise, a
corrupt voter could create an invalid vote that corrupts the final tally. In [3], the
authors present an interactive proof that proves a given ciphertext encrypts one of a
set of plaintexts. This proof could be verified prior to including a vote in the final tally
to ensure that the encrypted vote is a valid one. However, a non-interactive version of
the proof would decrease the communication complexity, as a voter could construct
the proof, and the tallier could verify the proof without requiring interaction between
them.
Generation of such a non-interactive proof could be accomplished by applying the
Fiat-Shamir heuristic in which the random challenge by the verifier is replaced by a
hashed commitment generated by the prover. The details of this proof would be a
useful addition to the e-voting application.
20
APPENDIX
4.1 Data Type Descriptions
In
addition
to
the
two
main
software
elements
(the
PaillierThresholdCryptoServiceProvider class and the Paillier Threshold Cryptography
Web Service), the implementation includes many other classes and structures primarily
used for organizing and transferring data. The following is an XSD schema of those
classes
generated
by
running
the
Visual
Studio
tool
xsd.exe
on
ThresholdCryptoLibrary.dll assembly.
<?xml version="1.0" encoding="utf-8"?>
<xs:schema elementFormDefault="qualified"
xmlns:xs="http://www.w3.org/2001/XMLSchema">
<xs:element name="DiscreteLogEqualityProof" nillable="true"
type="DiscreteLogEqualityProof" />
<xs:complexType name="DiscreteLogEqualityProof">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1" name="e"
type="xs:base64Binary" />
<xs:element minOccurs="0" maxOccurs="1" name="z"
type="xs:base64Binary" />
</xs:sequence>
</xs:complexType>
<xs:element name="OwnerInfo" nillable="true" type="OwnerInfo"
/>
<xs:complexType name="OwnerInfo">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1" name="OwnerName"
type="xs:string" />
<xs:element minOccurs="0" maxOccurs="1" name="OwnerX509"
type="xs:base64Binary" />
</xs:sequence>
</xs:complexType>
<xs:element name="PaillierPublicKey" type="PaillierPublicKey"
/>
<xs:complexType name="PaillierPublicKey">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1" name="N"
type="xs:base64Binary" />
<xs:element minOccurs="0" maxOccurs="1" name="G"
type="xs:base64Binary" />
21
the
<xs:element minOccurs="0" maxOccurs="1" name="Theta"
type="xs:base64Binary" />
</xs:sequence>
</xs:complexType>
<xs:element name="PaillierThreshold" nillable="true"
type="PaillierThreshold" />
<xs:complexType name="PaillierThreshold" abstract="true">
<xs:complexContent mixed="false">
<xs:extension base="AsymmetricAlgorithm" />
</xs:complexContent>
</xs:complexType>
<xs:complexType name="AsymmetricAlgorithm" abstract="true">
<xs:sequence>
<xs:element minOccurs="1" maxOccurs="1" name="KeySize"
type="xs:int" />
</xs:sequence>
</xs:complexType>
<xs:element name="PaillierThresholdCryptoServiceProvider"
nillable="true" type="PaillierThresholdCryptoServiceProvider" />
<xs:complexType name="PaillierThresholdCryptoServiceProvider">
<xs:complexContent mixed="false">
<xs:extension base="PaillierThreshold">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1"
name="DecryptionShares" type="ArrayOfThresholdDecryptionShare" />
</xs:sequence>
</xs:extension>
</xs:complexContent>
</xs:complexType>
<xs:complexType name="ArrayOfThresholdDecryptionShare">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="unbounded"
name="ThresholdDecryptionShare" nillable="true"
type="ThresholdDecryptionShare" />
</xs:sequence>
</xs:complexType>
<xs:complexType name="ThresholdDecryptionShare">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1" name="C"
type="xs:base64Binary" />
<xs:element minOccurs="0" maxOccurs="1"
name="DecryptionShare" type="xs:base64Binary" />
<xs:element minOccurs="1" maxOccurs="1" name="ShareIndex"
type="xs:int" />
<xs:element minOccurs="0" maxOccurs="1" name="Proof"
type="DiscreteLogEqualityProof" />
</xs:sequence>
</xs:complexType>
<xs:element name="PaillierThresholdKeyShare" nillable="true"
type="PaillierThresholdKeyShare" />
22
<xs:complexType name="PaillierThresholdKeyShare">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1" name="OwnerInfo"
type="OwnerInfo" />
<xs:element minOccurs="1" maxOccurs="1"
name="SecretKeyShare" type="ShamirShare" />
</xs:sequence>
</xs:complexType>
<xs:complexType name="ShamirShare">
<xs:sequence>
<xs:element minOccurs="1" maxOccurs="1" name="ShareIndex"
type="xs:int" />
<xs:element minOccurs="0" maxOccurs="1" name="SecretShare"
type="xs:base64Binary" />
<xs:element minOccurs="1" maxOccurs="1" name="Encrypted"
type="xs:boolean" />
</xs:sequence>
</xs:complexType>
<xs:element name="PaillierThresholdParameters"
type="PaillierThresholdParameters" />
<xs:complexType name="PaillierThresholdParameters">
<xs:sequence>
<xs:element minOccurs="1" maxOccurs="1" name="PublicKey"
type="PaillierPublicKey" />
<xs:element minOccurs="0" maxOccurs="1" name="SecretKey"
type="xs:base64Binary" />
<xs:element minOccurs="0" maxOccurs="1"
name="VerificationKeys"
type="ArrayOfPaillierThresholdVerificationKey" />
<xs:element minOccurs="0" maxOccurs="1"
name="ThresholdKeyShares" type="ArrayOfPaillierThresholdKeyShare"
/>
<xs:element minOccurs="1" maxOccurs="1" name="T"
type="xs:int" />
<xs:element minOccurs="1" maxOccurs="1" name="Delta"
type="xs:double" />
</xs:sequence>
</xs:complexType>
<xs:complexType name="ArrayOfPaillierThresholdVerificationKey">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="unbounded"
name="PaillierThresholdVerificationKey"
type="PaillierThresholdVerificationKey" />
</xs:sequence>
</xs:complexType>
<xs:complexType name="PaillierThresholdVerificationKey">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="1"
name="VerificationKeyBase" type="xs:base64Binary" />
23
<xs:element minOccurs="0" maxOccurs="1"
name="VerificationKey" type="xs:base64Binary" />
</xs:sequence>
</xs:complexType>
<xs:complexType name="ArrayOfPaillierThresholdKeyShare">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="unbounded"
name="PaillierThresholdKeyShare" nillable="true"
type="PaillierThresholdKeyShare" />
</xs:sequence>
</xs:complexType>
<xs:element name="PaillierThresholdVerificationKey"
type="PaillierThresholdVerificationKey" />
<xs:element name="ShamirShare" type="ShamirShare" />
<xs:element name="ThresholdDecryptionShare" nillable="true"
type="ThresholdDecryptionShare" />
<xs:element name="ThresholdParameterRequest"
type="ThresholdParameterRequest" />
<xs:complexType name="ThresholdParameterRequest">
<xs:sequence>
<xs:element minOccurs="1" maxOccurs="1" name="KeySize"
type="xs:int" />
<xs:element minOccurs="1" maxOccurs="1" name="NumShares"
type="xs:int" />
<xs:element minOccurs="1" maxOccurs="1"
name="DecryptionThreshold" type="xs:int" />
<xs:element minOccurs="0" maxOccurs="1" name="OwnerInfos"
type="ArrayOfOwnerInfo" />
</xs:sequence>
</xs:complexType>
<xs:complexType name="ArrayOfOwnerInfo">
<xs:sequence>
<xs:element minOccurs="0" maxOccurs="unbounded"
name="OwnerInfo" nillable="true" type="OwnerInfo" />
</xs:sequence>
</xs:complexType>
<xs:element name="Utilities" nillable="true" type="Utilities"/>
<xs:complexType name="Utilities" />
</xs:schema>
24
4.2 Sample SOAP Request/Response Messages
The following are example SOAP request and response messages used for requesting
the Paillier Threshold Parameters and Receiving the results. These examples are
directly obtained from browsing to the web service.
POST /ThresholdCryptoService/ThresholdCryptoService.asmx HTTP/1.1
Host: localhost
Content-Type: text/xml; charset=utf-8
Content-Length: length
SOAPAction:
"Barleydog.ThresholdCryptography/GeneratePaillierThresholdParameters
"
<?xml version="1.0" encoding="utf-8"?>
<soap:Envelope xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xmlns:xsd="http://www.w3.org/2001/XMLSchema"
xmlns:soap="http://schemas.xmlsoap.org/soap/envelope/">
<soap:Body>
<GeneratePaillierThresholdParameters
xmlns="Barleydog.ThresholdCryptography">
<request>
<KeySize>int</KeySize>
<NumShares>int</NumShares>
<DecryptionThreshold>int</DecryptionThreshold>
<OwnerInfos>
<OwnerInfo>
<OwnerName>string</OwnerName>
<OwnerX509>base64Binary</OwnerX509>
</OwnerInfo>
<OwnerInfo>
<OwnerName>string</OwnerName>
<OwnerX509>base64Binary</OwnerX509>
</OwnerInfo>
</OwnerInfos>
</request>
</GeneratePaillierThresholdParameters>
</soap:Body>
</soap:Envelope>
Figure 4 – Example SOAP Request for PTC Web Service
GeneratePaillierThresholdParameters
25
HTTP/1.1 200 OK
Content-Type: text/xml; charset=utf-8
Content-Length: length
<?xml version="1.0" encoding="utf-8"?>
<soap:Envelope
xmlns:xsi="http://www.w3.org/2001/XMLSchemainstance"
xmlns:xsd="http://www.w3.org/2001/XMLSchema"
xmlns:soap="http://schemas.xmlsoap.org/soap/envelope/">
<soap:Body>
<GeneratePaillierThresholdParametersResponse
xmlns="Barleydog.ThresholdCryptography">
<GeneratePaillierThresholdParametersResult>
<PublicKey>
<N>base64Binary</N>
<G>base64Binary</G>
<Theta>base64Binary</Theta>
</PublicKey>
<SecretKey>base64Binary</SecretKey>
<VerificationKeys>
<PaillierThresholdVerificationKey>
<VerificationKeyBase>base64Binary</VerificationKeyBase>
<VerificationKey>base64Binary</VerificationKey>
</PaillierThresholdVerificationKey>
<PaillierThresholdVerificationKey>
<VerificationKeyBase>base64Binary</VerificationKeyBase>
<VerificationKey>base64Binary</VerificationKey>
</PaillierThresholdVerificationKey>
</VerificationKeys>
<ThresholdKeyShares>
<PaillierThresholdKeyShare>
<OwnerInfo xsi:nil="true" />
<SecretKeyShare xsi:nil="true" />
</PaillierThresholdKeyShare>
<PaillierThresholdKeyShare>
<OwnerInfo xsi:nil="true" />
<SecretKeyShare xsi:nil="true" />
</PaillierThresholdKeyShare>
</ThresholdKeyShares>
<T>int</T>
<Delta>double</Delta>
</GeneratePaillierThresholdParametersResult>
</GeneratePaillierThresholdParametersResponse>
</soap:Body>
</soap:Envelope>
Figure 5 – Example SOAP Response for PTC Web
Service GeneratePaillierThresholdParameters
26
27
4.3 Demonstration E-Voting Application
In order to demonstrate the capabilities of the Paillier Threshold Cryptography system,
a simple e-voting application was created. The voting system implemented is not fully
secure and should not be used for a real voting scenario.
However, it does
demonstrate the use of the Paillier Threshold Cryptography Web Service and Paillier
Threshold CryptoServiceProvider in such a scenario, and how these services can
provide the critical components for securing such an application.
A complete user’s guide for the Paillier Threshold Cryptography Web Service is
provided in [8].
4.3.1 E-Voting System Overview
The e-voting system allows for 1 out of L candidate ballots. No options are provided
for n out of L ballots or write-in ballots. An “election” may consist of more than one
ballot. An election administrator creates the ballots and other election parameters.
The administrator requests the Paillier threshold encryption parameters from the PTC
Web Service during the initial election set-up. The administrator submits the election
parameters to a VotingService web service, and saves the election parameters
(including the cryptosystem parameters) to an XML file. Voters then load the election
parameters by opening the XML file, make their selection(s), and cast their encrypted
vote(s) to the VotingService web service.
During the tally phase, the votes are
multiplied together, and, due to the homomorphic properties of the Paillier
cryptosystem, the product can be decrypted to reveal the sum total of all the votes.
All of the e-voting system functionality can be accessed through the main form of the
application by clicking on the desired function.
28
4.3.1.1 Election Set-Up
The election administrator uses the Election Builder form to create or modify an
election (before the election is posted to the voting web service). To create a new
election, the administrator selects File->New. A new election is created and a unique
election id is assigned.
The administrator must then enter his/her name and a
descriptive title for the election.
4.3.1.1.1 Obtaining Election Encryption Parameters
Before ballots can be added to the election, the encryption parameters must be
specified and retrieved from the web service. This must occur before the ballots are
added or created, since the vote format is dependent on the specified key size. The
administrator selects Election->Get Encryption Parameters which opens the
Encryption Parameter Request form. Using this form, the administrator specifies the
key size and whether or not to encrypt the returned key shares. The administrator can
then add the key share owner information for each owner that is to receive a secret key
share. If the key shares will be encrypted, the administrator will be required to select
the owner’s public X.509 certificate when the owner is added. Once all owners have
been added, the administrator selects the cryptosystem threshold value and then clicks
“Send Request”, which sends the request to the web service.
In the current
configuration, a key size of larger than 512 bits will result in such a delay that a
“timeout” error is caused, so it is not recommended that key sizes greater than 512 be
used. The web service will generate the requested parameters, encrypt the key shares
(if specified), and return them. The Encryption Parameter Request form will transfer
the returned parameters to the Election Builder form and close automatically. The
election crypto parameters are displayed at the bottom of the Election Builder form.
4.3.1.1.2 Creating Ballots
Existing ballots can now be added to the election or new ballots can be created using
the options under the Election menu. To create a new ballot, the administrator selects
29
Election->Create Ballot to open the Ballot Builder form. A new ballot will be created
and the random ballot id displayed in the form. The ballot creator must type in the
ballot issue/question, and then enter all of the available choices, one at a time. Each
choice is entered by typing the appropriate text, and clicking the “Add” button. A
choice can be deleted by selecting the choice in the list, and clicking Ballot->Delete
Candidate. When the ballot is complete, the ballot should be saved by clicking File>Save and choosing a known location and filename to save the ballot. The Ballot
Builder Form must be closed and then re-opened in order to create another ballot.
4.3.1.1.3 Adding Ballots to an Election
Once all desired ballot(s) have been created and saved, it/they can be added to any
election by selecting Election->Add Ballot from the Election Builder form. The user
will then be prompted to select the ballot file. If the ballot is valid, it will be imported
into the election and displayed in the Election Builder form.
4.3.1.1.4 Vote Format
When a ballot is added to an election, the format of the vote for that ballot is derived
from the key size chosen for the election and the number of “candidate” choices on
the ballot. These two values determine the maximum number of voters allowed. The
total size of the vote is limited to the key size k (in bits). The vote is split into c bit
fields where c is the number of candidates. The size of the bit fields vc= k/c.
However, vc is limited to 32 bits so that each candidate’s field will fit into a 32-bit
integer (for ease of extraction only). Therefore, if k/c > 32, vc=32 andonly the first
32*c bits of the vote will be used. To cast a vote, a voter votes the value 2^(ic*vc)
where ic is the desired candidates ballot index (0,…,c-1). By using votes of this format,
the tally can be computed by multiplying all of the votes together and decrypting the
product.
Due to the homomorphic property of the Paillier cryptosystem, the
multiplication carried out in the ciphertext space corresponds to addition in the
cleartext space, and thus the decryption of the product will contain the summed votes
30
for each candidate. Each candidate’s bit field can then be extracted and evaluated to
determine the total number of votes for that candidate.
4.3.1.1.4.1 Cheating the Vote
The vote format discussed above would be very easy target for a malicious voter. For
example, instead of voting the value 2^(ic*vc) for the desired candidate, a malicious
voter could simply vote 2^(ic*vc + 5) to vote 2^5 times for the desired candidate.
When such an encrypted vote is multiplied with others, it would simply be summed to
the target candidate’s decrypted vote total. In order to prevent such a scam, the vote
must be checked for validity. This can be done if the voter includes a proof of validity
with his/her vote that proves that the vote is one of the c valid vote values. In [3], the
authors present an interactive proof that proves a given ciphertext encrypts one of a
set of plaintexts. This proof could be made non-interactive using the Fiat-Shamir
heuristic, and is left as an exercise for the reader.
4.3.1.1.5 Saving and Posting an Election
Once the election has been created, the encryption parameters obtained, and the
necessary ballots created and added to the election, the election must be saved as an
XML file and it must be posted to the voting web service. First save the election by
selecting File->Save in the Election Builder form. Select the desired location and
filename and click OK. Posting the election to the voting web service is a nonreversible operation in the demo application. Be sure that the election is complete and
saved before posting. To post the election, select Election->Post Election To Voting
Service. A web service call will be made that posts the election data to the web service,
which then creates the appropriate database entries that are used to manage the
election.
31
4.3.1.2 Voting
4.3.1.2.1 Creating the Vote
Once an election has been created, saved, and posted to the election web service,
voters can create and cast votes. To do so, open the VoterForm by clicking the
VOTE button on the main menu. Type the voter’s name in the VoterForm, and then
open the desired election by selecting Election->Open from the form’s menu. The
ballots from the election will be loaded, with each issue being loaded into the issue text
box, and it’s corresponding choices loaded into the textbox to the right (the choices
textbox). The voter can make his/her choice simply by clicking on the desired choice.
To vote on a different ballot issue, simply select the desired issue. That issue’s choices
will then be displayed in the choices textbox to the right. Again, select the desired
choice by clicking on it in the choices textbox. Once a choice has been selected, the
ballot issue and the selected choice will appear in the “Current Votes” textbox. To the
right of the issue question and the selected choice is the hex value of the vote to be
cast.
4.3.1.2.2 Submitting the Vote
Once all choices have been made, the voter can submit his/her vote by selecting Vote>Submit Vote. Doing so posts the vote information to the voting web service, which
saves the vote information to the voting database. Once the vote is submitted, no
changes can be made. However, if the vote was incomplete (i.e. choices for all ballots
in the election were not made), the voter can clear the vote by selecting Vote->Clear,
make choices for the unsubmitted ballots, and then submit only those votes. When
submitting the vote, a dialog box will appear indicating how many votes were
successfully submitted to the voting web service.
4.3.1.2.3 Checking the Vote
At any time after submitting his/her vote, a voter can check the posted values of
his/her vote by selecting Vote->Check Submitted Vote. This invokes a web service
32
call to the voting web service which retrieves the encrypted vote values posted for that
election.
4.3.1.3 Tally The Vote
At any time during the election process, the demo voting application allows you to tally
the vote. To do so, click the “Tally/Decrypt Vote” button on the main menu. The
Tally form will open. Select Election->Open, and then browse to and open the
election file for which you wish to tally the vote. If the secret key shares were
encrypted, you will be prompted to select the owner’s secret RSA key in order to
decrypt the owner’s Paillier secret key share. The demo application currently prompts
for the owner’s secret RSA key each time it is required (for each ballot). Once all of
the owner’s secret RSA keys have been provided for each ballot in the election, the
votes are retrieved from the voting web service. The product of the votes for each
ballot is then calculated and displayed both encrypted and decrypted, and the
candidate’s tallies are extracted from the decrypted bit field and displayed. Select a
different ballot issue to view the results for that ballot.
33
REFERENCES
[1] P. Paillier, Public-Key Cryptosystems Based on Composite Degree Residuosity Classes,
Eurocrypt ‘99
[2] P. Fouque, G. Poupard, J.Stern, Sharing Decryption in the Context of Voting or Lotteries,
Financial Cryptography 2000 Proceedings
[3] I. Damgard, M. Jurik, J. Nielson, A Generalization of Paillier’s Public-Key System with
Applications to Electronic Voting, Aarhus University, Dept. of Computer Science
[4] A. Shamir, How to Share a Secret, Communications of the ACM 1979
[5] A.J. Menezes, P. C. van Oorschot, and S.A. Vanstone, Handbook of Applied
Cryptography, CRC Press, 1997
[6] D. Naccache, Double-Speed Safe Prime Generation, Gemplus Card International
[7] M. Wiener, Safe Prime Generation with a Combined Sieve, Cryptographic Clarity
[8] B. Wilson, C. E. Chow, Paillier Threshold Cryptography Web Service User’s Guide,
University of Colorado – Colorado Springs Master’s Project, 2006
5