Advanced Topics in Security

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Presentation transcript:

Advanced Topics in Security Lecture ID: ET-IDA -044 Section-B: Tutorial-2 Voting Systems over GF(p) 22.01.2011 V-2 MSc. A. M. Basil , Prof. Wael Adi Institute for Computer and Network Engineering Technical University of Braunschweig Braunschweig, Germany Technische Universitaet Braunschweig

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

Technische Universitaet Braunschweig Applying Homomorphic mapping for Voting using a group in a ring Zm Find a strong prime: Strong prime p = 2 . q + 1 (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=2 . 11 +1=23 Check by applying Pocklington’s theorem! P2=2 . 23+1=47 Check by applying Pocklington’s theorem! Take ⟹ m = 23 . 47 Technische Universitaet Braunschweig

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

Technische Universitaet Braunschweig

Technische Universitaet Braunschweig

Technische Universitaet Braunschweig