A Pairing-Based Blind Signature E-Voting Scheme LOURDES L O PEZ-GARC I A, LUIS J. DOMINGUEZ PEREZ, FRANCISCO RODR I GUEZ-HENR I QUEZ The Computer Journal July 2013 Presenter:陳昱安 Date:2013/10/14

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 2

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 3

Introduction(1/2) 4

Introduction(2/2) Eligibility Uniqueness No-coercion Accuracy Receipt-freeness Variability 5

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 6

Mathematical Background Elliptic curves Bilinear pairings over Barreto-Naehig curves Security assumptions 7

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 8

Digital Signatures(1/4) The Boneh-Lynn-Shacham short signature scheme Let (𝔾1 =〈P〉 , 𝔾2 =〈Q〉) : additive groups of order r P , Q : points over an elliptic curve r : a prime number H1 : the map-to-point function H1 : 0,1 ∗ → 𝔾1 9

Digital Signatures(2/4) Key generation Pick a random integer d ∈ ℤr and compute V = dQ. V ∈ 𝔾2 : public key , d : private key. Signing Given a private key d, a message m ∈ 0,1 ∗ . Compute M = H1(m) and S = dM. The signature of m is S ∈ 𝔾1. Verification Given a public key V ∈ 𝔾2, a message m ∈ 0,1 ∗ , and a signature S ∈ 𝔾1. 𝑒 𝑄, 𝑆 = 𝑒 (V,H1(m)) ? 10

Digital Signatures(3/4) Blind signatures (𝔾1 , 𝔾2, P , Q , r , H1) Key generation Pick a random integer d ∈ ℤr and compute V = dQ. V ∈ 𝔾2 : public key , d : private key. Blinding (user) Given a message m , calculate M = H1(m) , randomly find b ∈ ℤ 𝑟 ∗ ; compute 𝑀 =𝑏𝑀. 11

Digital Signatures(4/4) Signature (signer) Given a blind message 𝑀 ; d : private key of the signer, compute S =𝑑 𝑀 ; Unblinding (user) Given a blind signature S and a blind factor b, calculate 𝑆=𝑏 −1 𝑆 . Then S is the signature of the message m. Signature Verification (third party) Given a message m, a signature S ; V : public key of the signer , check 𝑒 𝑄, 𝑆 = 𝑒 (V,H1(m)) ? 12

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 13

The Proposed E-Voting Scheme(1/4) Registration Authentication Voting Counting 14

The Proposed E-Voting Scheme(2/4) Protocol dataflow Notation Authentication Server (AS) ; Voting Server (VS) {dAS , VAS}: private/public key pair of AS. {dVS , VVS}: private/public key pair of VS. {IDV , dV , VV}: identifier and private/public key 15

The Proposed E-Voting Scheme(3/4) 16

The Proposed E-Voting Scheme(4/4) 17

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 18

Security Analysis (1/6) Voter privacy randomly generated. The pseudonym private key dt and public key Vt are randomly generated. Knowing the message m implies finding b in the equation. 19

Security Analysis (2/6) Eligibility will be accepted as legitimate. The voter requests from the AS a blind signed ballot that will be accepted as legitimate. Before producing the blind signature , the AS must authenticate the voter by reviewing the nominal list, S M using the public key of the voter who is requesting the blank ballot. 20

Security Analysis (3/6) Uniqueness During the authentication phase the AS marks the voter record in the nominal list. In the voting phase, the VS checks the ballots, if both signatures are valid, then the ballot is stored as valid or invalid otherwise. In the counting phase, the VS verifies the signatures with which was generated for the ballot. 21

Security Analysis (4/6) No-coercion ; Receipt-freeness When the results are published after the counting phase, the voter cannot prove who she voted for. This is because of the generation of a random value a that adds randomness to the hash message used as a receipt. The ACK has the goal to show to the voter that the ballot was received by the VS. 22

Security Analysis (5/6) Accuracy To identify a fraudulent ballot means to find a pair that uses the same value for Vt . If when comparing two ballots, both have the same Vt , then the VS discards the second ballot as fraudulent/repeated and counts only the first one. 23

Security Analysis (6/6) Verifiability The ACK guarantees two things : a. The voter can verify if her ACK is found in the list of valid votes , no chance to extract the value of the vote, due to the random number a and the hash of all values mentioned. b. The VS can prove the accuracy of the results to show that all ACK are unique. 24

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 25

Implementation Aspects (1/4) 26

Implementation Aspects (2/4) 27

Implementation Aspects (3/4) 28

Implementation Aspects (4/4) 29

Outline Introduction Mathematical Background Digital Signatures The Proposed E-Voting Scheme Security Analysis Implementation Aspects Conclusions 30

Conclusions An electronic voting scheme based on blind signature is proposed which meets the necessary requirements to guarantee a reliable election. This proposal requires a minimal number of interactions with electoral entities and more efficient than other e-voting schemes based on RSA or DSA crypto schemes. 31