Logical Analysis and Verification of Cryptographic Protocols - Loria

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190 CHAPTER 7. VOTER VERIFIABILITY FOR E-VOTING PROTOCOLS 2. Universal verifiability. We observe that RUV � Φ = n i=1 eq(open(nth3 2(〈li, commit(¯vi, ri), sign(commit(¯vi, ri), skA)〉), ri), v ′ i) =E �n i=1 eq(open(commit(¯vi, ri), ri), v ′ i) =E eq(¯vi, v ′ i) and hence has a single solution for v ′ 1, . . . , v ′ n and namely ˜v ′ = (¯v1, . . . , ¯vn), concluding our proof. Moreover, by inspection of the voting process specification, we have �V P ({¯v1/u}, . . . , {¯vn/u})(−→ ∗ α −→−→ ∗ ) ∗ B and dom(σ) = {x ′ 1, . . . , x ′ 7·n}. 7.5.4 Protocol due to Lee et al. Description The protocol [141] involves an administrator, voters, mixers, talliers and a trusted randomisation service (in practice the randomisation service is implemented as a secure smart card called a tamper resistant randomiser). The voter encrypts her ballot and sends it to the randomisation service using a private channel. The randomisation service re-encrypts the ballot and returns the signed re-encrypted ballot along with a designated verifier proof which demonstrates the re-encrypted ballot is indeed a re-encryption of the voter’s encrypted ballot. The additional randomisation ensures the voter cannot reconstruct her ballot and hence is unable to create a receipt for a potential coercer. The voter signs her ballot and posts it on the bulletin board. The administrator verifies the double signed ballots and publishes valid ballots on the bulletin board. The mixers then perform a secret shuffle. Finally the talliers use signature proofs of knowledge to reveal an (t, n)-threshold decryption key for each ballot. Applied pi formalism The voting specification of this protocol is defined as 〈voter, registrar, ˜s, ˜t, ˜m〉 where ˜s = (r), ˜t = (skV , skR), ˜m = (aV) and processes voter,registrar are defined below. voter � aV(skV , pkR, pkT ) let b = penc(u, r, pkT ) in aVR〈b〉.aVR(sb ′ , dvp) if checkdvp(dvp, b, getmsg(sb ′ ), P k(skV )) then if checksign(sb ′ , pkR) then bb〈sign(sb ′ , skV )〉 registrar � aV〈skV , P k(skR), P k(skT )〉 | aR〈skR〉 | bb〈〈P k(skV ), P k(skR), P k(skT )〉〉 For simplicity we consider (1, n)-threshold decryption. In addition we assume the existence of a secure mixer and hence do not verify the shuffle. In [141] it is suggested that the protocol makes use of the ElGamal encryption scheme which we model by the following equations for decryption and
7.5. CASES STUDIES 191 re-encryption. dec(penc(x, y, P k(z)), z) = x renc(penc(x, y, z), w) = penc(x, f(y, w), z) The ElGamal encryption scheme exhibits the feature expressed by the equation dec(penc(x, y, P k(z)), commit(penc(x, y, P k(z)), z)) = x which is used by the protocol. We also add functions dvp, checkdvp to model designated verifier proofs of the fact that a message is a re-encryption of another one; we adopt the equations checkdvp(dvp(x, renc(x, y), y, pk(z)), x, renc(x, y), pk(z)) = true checkdvp(dvp(x, y, z, w), x, y, pk(w)) = true The second equation models that checkdvp also succeeds for a fake proof constructed using the designated verifier’s private key. By a slight abuse of notation we also interpret checkdvp(t1, t2, t3, t4) as a predicate which evaluates to true whenever checkdvp(t1, t2, t3, t4) =E true. We adopt the formalism for signature proofs of knowledge due to Backes et al. [25]. A signature proof of knowledge is a term SPKi,j( ˜ M, Ñ, F ) where ˜ M = (M1, . . . , Mi) denotes the witness (or private component), Ñ = (N1, . . . , Nj) defines the public parameters and F is a formula over those terms. More precisely F is a term without names or variables, but includes distinguished constants αk, βl where k, l ∈ N. The constants αk, βl in F denote placeholders for the terms Mk ∈ ˜ M, Nl ∈ Ñ used within a signature of knowledge SPKi,j( ˜ M, Ñ, F ). For example the signature proof of knowledge used by the Lee et al. voting protocol [141] demonstrates possession of a secret key skT such that P k(skT ) = pkT and d = commit(b ′ , skT ) i.e. the proof shows the public key pkT is correctly formed and d is a decryption key for the voter’s ballot b ′ . This can be captured as SPK1,3((skT ), (pkT , commit(b ′ , skT ), b ′ ), FL) where FL = eq(β1, P k(α1)) ∧ eq(β2, commit(β3, α1)). A term SPKi,j( ˜ M, Ñ, F ) represents a valid signature if the term obtained by substituting Mk, Nl for the corresponding αk, βl evaluates to true. Verification of such a statement is modelled by the function Veri,j. The equational theory includes the following equations defined over all tuples ˜x = (x1, . . . , xi), ˜y = (y1, . . . , yj) and formula F ∈ TΣ∪{αk,βl|k≤i, l≤j} without names or variables: Publicp(SPKi,j(˜x, ˜y, F )) = nth j p(˜y) where p ∈ [i, j] Formula(SPKi,j(˜x, ˜y, F )) = F We also make use of the predicate Veri,j defined as Veri,j(F, SPKi,j( ˜ M, Ñ, F ′ )) if and only if F =E F ′ and F {M1/α1, . . . , Mi/αi, N1/β1, . . . , Nj/βj} holds where i = | ˜ M|, j = | Ñ| and F, F ′ ∈ TΣ∪{αk,βl|k≤i, l≤j} without names or variables.
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Logical Analysis and Verification o
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Abstract This thesis is developed i
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x CONTENTS 2.1.7 Unification . . .
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xii CONTENTS 5.8 Decidability of re
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2 CHAPTER 1. INTRODUCTION Figure 1.
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Page 214 and 215: 202 BIBLIOGRAPHY [12] M. Abadi and
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Page 220 and 221: 208 BIBLIOGRAPHY [83] H. Comon-Lund
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Page 224 and 225: 212 BIBLIOGRAPHY [132] T. Kohno, A.
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