Logical Analysis and Verification of Cryptographic Protocols - Loria

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184 CHAPTER 7. VOTER VERIFIABILITY FOR E-VOTING PROTOCOLS and Nσ for the result of applying σ to N. We write ˜m i for the tuple of names obtained from ˜m by annotating with subscript i. For example given ˜m = (n, n ′ ) we write ˜m 1 for (n1, n ′ 1). Finally, we write a(x1, ..., xn).P as a convenient shorthand for a(x).let x1 = nth n 1(x) in . . . let xn = nth n n(x) in P . 7.3 Formalising electronic voting protocols We refine our sort system with the new base type Candidate. A voting process is characterised with respect to a voter process V and administrators processes. We capture administrators who are assumed to be honest by the process P . Dishonest administrators do not need to be modelled as they are directly part of the adversarial environment in which the protocol is executed. Processes V, P will each be instantiated n times where n is the number of voters. We assume secrets ˜s only occur in process V ; similarly ˜t only occur in process P ; additionally, shared secrets ˜m appear in both V, P . Definition 58 formalises a voting process specification accordingly. Definition 58 (Voting process specification) A voting process specification is a tuple 〈V, P, ˜s, ˜t, ˜m〉 where plain processes V, P do not contain any name restrictions, the tuples of names ˜s, ˜t, ˜m are disjoint and (fn(V ) ∪ fn(P )) ∩ � j∈N (˜sj ∪ ˜t j ∪ ˜m j ) = fn(V ) ∩ ˜t = fn(P ) ∩ ˜s = ∅. We write V P ({¯v1/u}, . . . , {¯vn/u}) for: ν ˜m 1 , . . . , ˜m n , ˜s 1 , . . . , ˜s n , ˜t 1 , . . . , ˜t n � V {˜s 1 /˜s, ˜m 1 / ˜m, ¯v1/u} | P { ˜t 1 /˜t, ˜m 1 / ˜m} | . . . | V {˜s n /˜s, ˜m n / ˜m, ¯vn/u} | P { ˜t n /˜t, ˜m n / ˜m} � where ¯vi of type Candidate is the ith voter’s vote. As an example consider the simple “postal ballot” voting protocol (Figure 7.1). A voter V receives her private signing key skV from a keying authority P , and then sends her signed vote to the bulletin board. The keying authority also sends the voter’s public verification key P k(skV ) to the bulletin board. By convention we use the channel bb (referring to the bulletin board). We consider the equational theory H defined by getmsg(sign(x, y)) = x and the predicate checksign defined as checksign(M1, M2) if and only if M1 =H sign(N1, N2)∧M2 =H P k(N2). To analyse verifiability, we consider a modified version of the voting process, in which all inputs that a voter receives are stored on the frame (perhaps in encoded form). To achieve this we extend the equational theory to include functions senc/2, sdec/2, the associated equation sdec(senc(x, y), y) = x and define the process modification (Definition 59).
7.4. FORMALISING VOTER VERIFIABILITY PROPERTY 185 Definition 59 (Input-storing process) Given channel c and process ν ñ.P where P has no restrictions, the input-storing process is defined as ν k, ñ.P c,ñ,k where: • 0 c,ñ,k �= 0 • (P | Q) c,ñ,k �= P c,ñ,k | Q c,ñ,k • (ν m.P ) c,ñ,k �= ν m.P c,ñ,k • (a(x).P ) c,ñ,k �= a(x).c〈senc(x, k)〉.P c,ñ,k if a ∈ ñ • (a(x).P ) c,ñ,k �= a(x).c〈x〉.P c,ñ,k otherwise • (a〈M〉.P ) c,ñ,k �= a〈M〉.P c,ñ,k • (if ψ then P else Q) c,ñ,k �=if ψ then P c,ñ,k else Q c,ñ,k Given a voting process specification 〈V, P, ˜s, ˜t, ˜m〉 and votes ¯v1, . . . , ¯vn we write �V P ({¯v1/u}, . . . , {¯vn/u}) for the special input-storing process defined as follows: ν ñ.(V {˜s 1 /˜s, ˜m 1 / ˜m, ¯v1/u} bb,ñ,kpc1 | P { ˜t 1 /˜t, ˜m 1 / ˜m} | . . . | V {˜s n /˜s, ˜m n / ˜m, ¯vn/u} bb,ñ,kpcn | P { ˜t n /˜t, ˜m n / ˜m}) where ñ = � n j=1 ( ˜mj ∪ ˜s j ∪ ˜t j ∪ {kpcj}). The definition is illustrated on the postal ballot example in Figure 7.1. 7.4 Formalising voter verifiability property Now we introduce voter verifiability. As mentioned, there are two parts, corresponding to individual verifiability and universal verifiability. A voting process satisfies voter verifiability if there are two tests it can apply to check these two items. Each test is a predicate which, after substitutions from the bulletin board and elsewhere, evaluates to true or false. Individual verifiability: The test R IV is performed by a voter, and has parameters u (the vote cast by the voter), x1, . . . , xk (the items on the bulletin board corresponding to that vote), and ˜z (the secrets of the voter). The test is required to return true if and only if the correct items are given. In the definition below, the functions f1, . . . , fk pick out the k bulletin board items corresponding to the voter. Universal verifiability: The test R UV is performed by an observer, and has parameters ũ (the declared outcome), ˜x1, . . . , ˜xk (the items on the bulletin board corresponding to all the voters), and ˜y (the items on the bulletin
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Page 214 and 215: 202 BIBLIOGRAPHY [12] M. Abadi and
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