1. Advanced Topics in Security
Lecture ID: ET-IDA -044
Section-B: Tutorial-2
Voting Systems over GF(p)
V-2
MSc. A. M. Basil , Prof. Wael Adi
Institute for Computer and Network Engineering
Technical University of Braunschweig
Braunschweig, Germany
Technische Universitaet Braunschweig

2. Homomorphic Encryption
ElGamal Crypto – System Setup (1985)
Teller: α primitive element in GF(p)
Teller secret key = x, Public key y = αx
( Ci = Mi Zi)
n voters 1
( C1 C2 · · Cn , h1 h2 · · hn ) X X X
(M1 · · Mn · Z1 · · Zn , h1 · · hn)
(C1, h1)
(Ci, hi)
(Cn, hn)
Mi = αvi
Mn = αvn
M1 = αv1
( αv1+…+vn · αx(R1+…+Rn) , αR1+…+Rn ) X X X
Encryption of αVs = α v1+v2+..+vn
Problem : getting the sum of the votes Vs. Solution by exhaustive search to get the discrete log as Vs is not cryptographically huge!
Z1 = yR1 = αx.R1
Zi = yRi = αx.Ri
Zn = yRn = αx.Rn
h1 = αR1
hi = αRi
hn = αRn
Technische Universitaet Braunschweig

3. Technische Universitaet Braunschweig
Applying Homomorphic mapping for Voting using a group in a ring Zm
Find a strong prime:
Strong prime p = 2 . q (see Pocklington’s Theorem)
Generating ELGAMAL System:
GF ( p) , α (Generator of a cyclic group) very high order
Alternatively take an element in Zm with a high order
P=2 . q +1
P1= =23 Check by applying Pocklington’s theorem!
P2= =47 Check by applying Pocklington’s theorem!
Take ⟹ m =
Technische Universitaet Braunschweig

4. Technische Universitaet Braunschweig
Highest possible element order = 506 =
All possible elements orders are:
1 , 2 , 11 , 22 , 23 , 46 , 253 , 506
Technische Universitaet Braunschweig

5. Technische Universitaet Braunschweig

6. Technische Universitaet Braunschweig

7. Technische Universitaet Braunschweig