The system is based on generating a binary CDMA set of codes, in which the crosscorrelation spectrum is the same for all pairs of codes belonging to the set.
If one of these codes is assigned to the election committee (person pt ) and each eligible voter is assigned a distinct code of this set (other than that of the committee), we in the election committee center, can make sure that the received code belongs to the set by checking its crosscorrelation spectrum, without the need to reveal the identity of the voter.
The set of people who participate in this decision making process are in fact a subset E of size z , belonging to a mother set P of size n . The set P is comprised of all persons eligible (and registered) for voting.
The size of P is larger than or equal to the size of E, because some persons who have the right to participate in the process of voting, may not do that.
The identity of the person who voted for a certain choice is not going to be revealed.
There is a record, at the election center (election committee), indicates whether certain person has voted or not. This could be very important from the audit trail point of view.
II- Prevent any eligible voter from examining the code assigned to him, generating a new proper code, to be used by a person who is not eligible for voting, or even by the eligible person himself again. Solution is achieved by encrypting the code assigned to each voter, using the public key of the election committee (person pt).
The proposed e-voting system is described as follows:
Suppose that there is a set P of size n of geographically separated persons,
P = {p 0 , p 1 , p 2 ,……, p i ,……, p n-1 }.
Those people are wishing to electronically elect one of them. This can be expanded to electing a certain committee or list. Another possible expansion is to take a decision based on a certain threshold number of persons agreeing with a certain proposal.
This set of persons asks a Highly Trusted Organization (company or third party) HTO , to arrange for this election. HTO will assign each person p i belonging to P a binary sequence (code) s i which belongs to a set of sequences S of size n such that:
The set S has an interesting property: the spectrum of the periodic crosscorrelation function for a pair of sequences u and v belonging to S is the same as that for any other pair of sequences belonging to S . There are many CDMA sequence sets classes which possess this property, such as:
To illustrate the crosscorrelation spectrum, let us consider the case of Gold sequence sets. A Gold sequence set is comprised of D+2 sequences, each of length (period) D, such that:
D= 2 q - 1,
Where q is a positive integer not a multiple of 4.
The periodic crosscorrelation function for any pair belonging to the set, takes only one of the following three values:
{-1, -t(q), t(q)-2}
for all relative shifts between this pair of sequences, where
t(q) = 1 + 2 (q+2)/2 ;
where: x denotes the integer part of the real number x.
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accomplished by HTO w >> n accomplished by HTO one – to – one correspondence voted n> =z Set E Set P Set S Set C c 0 e 0 p 0 s 0 c 1 e 1 p 1 s 1 c 2 | | | | | | | | e z-1 p n-1 s n-1 | | c w-1
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Combined message ms i MS i = encrypted ms i using the public key of pt Unsecure channel Vote message m i of e i Code s i assigned to e i (originally as p i ) Encryption of ms i using the public key of pt Decryption of MS i using the private key of pt Vote message m i Code s i assigned to e i
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Comparing s i with the already voted codes (people) mismatch repeated Not repeated checking the crosscorrelation properties between s i and st (the code of pt) Vote rejection
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m o m 1 m i m z-1 Fig. 3 Steps used to accomplish voting in the proposed system Accepting the vote message m i of e i Taking decision according to the specified threshold Yes or No (decision)
Implementing the proposed system is quite feasible. This is because all of the processes required for realizing such a system are quite feasible. Those processes are:
Comparative study of different sequence sets of unified crosscorrelation spectrum, especially those of very big size and of very long sequences..
Writing a software package for generating those sequence sets and examining their crosscorrelation spectra.
Writing a software package for examining the best subsets of sequences from the randomness point of view, taking into consideration the required size of those subsets.
Designing a model for the system and conducting simulation processes to check its strengths against different types of attacks, including impersonation.
Comparing the strengths of the proposed system with the existing e-voting systems from different points of view.
It could be better to present an example, applying Gold sequence sets. Suppose that the system will be used for electing the Mayor of a county of 100,000 eligible voters . Let F denote the number of all possible preferred pairs of m-sequences of degree 33 (required to generate the Gold set), and let q = 33, then the sequence length:
D = 2 33 – 1 = 8 589 934 592 – 1, and the set size w = 8 589 934 592 + 1.
Suppose that F was around 120000; then the key space will be:
120 000*858 993=103 079 160 000
This means that there are around one hundred trillion candidates for each assigned code (sequence). Remembering that each code is very very long, then the brute force attack on this system is very very hard, (nearly impossible).
I believe that the idea of the proposed system is original and it will be easy to realize such a system. Consequently CDMA can be used in e-voting as an alternative to the existing systems. However, the proposed system needs more investigation, especially from the audit trail point of view. It is expected that publishing this paper will trigger such investigations.
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