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电子投票协议隐私类型属性验证
1 隐私与投票协议基础概念
在电子投票协议的研究中，有两个重要的概念，分别是“无收据性”（receipt - freeness）和“投票隐私性”（vote - privacy）。“无收据性”并不排除向选民提供某种无法用于证明其投票方式的收据的投票系统。并且，若一个协议具有无收据性（对于给定的一组诚实权威机构而言），那么它也会尊重投票隐私性（对于同一组机构）。
1.1 定义的探讨
定义中的第一个等价关系表明，如果忽略 $V’$ 在胁迫者通道 $chc$ 上的输出，那么 $V’$ 看起来就像一个投了 $a$ 票的选民进程 $V_A$。不过，这个等价关系可能被认为要求过高。有人可能觉得，这种等价性只应在特定的 $S$ 上下文中要求，而非在任何上下文（可访问协议所有私有通道）中都要求。这样会得到一个更弱的定义，但使用起来也更困难。实际上，这种变体定义只是稍弱一些，很难构造出能区分这两种可能性的自然示例，并且对后续章节的案例研究没有影响，所以我们还是采用定义 6。
2 Fujioka、Okamoto 和 Ohta 协议
2.1 协议描述
该协议涉及选民、管理员和收集者。管理员负责验证只有合格选民才能投票，收集者负责收集和公布选票。与认证协议相比，该协议使用了一些不常见的加密原语，如安全位承诺和盲签名，并且依赖匿名通道。匿名通道的处理方式未作具体规定，像 Chaum 引入的 MIX - nets（主要思想是对一系列对象进行置换和修改，以隐藏原始序列和最终序列元素之间的对应关系）或洋葱路由等机制都可能适用。
2.1.1 加密原语介绍
- 位承诺方案 ：允许代理 $A$ 向代理 $B$ 承诺一个值 $v$ 而不立即透露它。$A$ 以某种方式加密值 $v$ 并发送给 $B$，$B$ 直到 $A$ 发送密钥才能恢复 $v$，且能确保 $A$ 之后不能改变主意。
- 盲签名方案 ：允许请求者在不向任何人（包括签名者）透露消息 $m$ 的情况下获得消息 $m$ 的签名。在电子投票协议中，它允许选民在不向权威机构透露投票内容的情况下，获得权威机构对其投票的签名。
2.1.2 协议阶段
-
第一阶段：获取签名
- 选民 $V$ 选择投票 $v$，使用随机密钥 $r$ 计算承诺 $x = ξ(v,r)$，使用盲化函数 $χ$ 和随机盲化因子 $b$ 计算消息 $e = χ(x, b)$，最后对 $e$ 进行数字签名并将签名 $σ_V (e)$ 与身份一起发送给管理员 $A$。
- 管理员 $A$ 验证 $V$ 是否有投票权、是否未投票以及签名是否有效。若都通过，$A$ 对 $e$ 进行数字签名并将签名 $σ_A(e)$ 发送给 $V$。
- $V$ 对 $σ_A(e)$ 进行去盲处理，得到 $y = σ_A(x)$，即对 $V$ 投票的签名承诺。
-
第二阶段：实际投票
- $V$ 使用匿名通道将 $y$（$A$ 对投票承诺的签名）发送给收集者 $C$。
- $C$ 检查签名 $y$ 的正确性，若通过则将 $(ℓ, x, y)$ 作为第 $ℓ$ 项录入列表。
-
第三阶段：公布选票
- 收集者 $C$ 决定收到所有选票后（如在固定截止日期后），公布获得的承诺列表 $(ℓ_i, x_i, y_i)$。
- 选民 $V$ 验证其承诺是否在列表中，然后通过匿名通道将 $ℓ$ 和 $r$ 发送给 $C$。
- $C$ 使用随机数 $r$ 打开第 $ℓ$ 张选票并公布投票 $v$。
为避免公布选举的部分结果并确保隐私，投票阶段需分为承诺阶段和开放阶段，这通过要求选举中涉及的不同代理进行同步来实现。
2.2 应用 pi 演算模型
2.2.1 加密原语的等式理论
密码学采用 Dolev - Yao 风格建模为完美的，相关等式如下：
open(commit(m, r), r) = m
checksign(sign(m, sk), pk(sk)) = m
unblind(blind(m, r), r) = m
unblind(sign(blind(m, r), sk), r) = sign(m, sk)
在这个模型中，位承诺（由函数 $commit$ 和 $open$ 建模）与经典对称密钥加密相同。为简单起见，我们将主机名和公钥等同起来。我们对加密原语的模型是一种抽象，例如位承诺具有完美的绑定和隐藏特性。数字签名建模为带消息恢复的签名，即签名本身包含可使用 $checksign$ 函数提取的签名消息。为建模盲签名，我们添加了一对函数 $blind$ 和 $unblind$，它们类似于完美对称密钥加密和位承诺。需要注意的是，当 $r_1 \neq r_2$ 时，建模承诺的等式不能应用于 $open(commit(m, r_1), r_2)$ 项。
2.2.2 流程同步
协议分为三个阶段，每个选民必须在完成第一阶段后才能进入第二阶段，完成第二阶段后才能进入第三阶段。在模型中，我们通过关键字 $synch$ 来强制执行同步。当一个进程遇到 $synch n$ 时，它会等待所有可能遇到 $synch n$ 的其他进程到达该点，然后所有进程才能继续。如果有 $k$ 个进程可能遇到 $synch n$，我们可以按以下方式实现同步：将命令 $synch n$ 替换为 $out(n, 0); in(n, =1)$，其中 $n$ 是全局声明的私有通道。此外，我们假设存在一个额外的进程 $in(n, =0);…; in(n, =0); out(n, 1);…; out(n, 1)$，它有 $k$ 个输入和 $k$ 个输出。这个简单的编码对于我们的目的是可行的，因为 $k$ 的值可以通过检查代码来推断，但如果创建了新的进程（如使用“!”），它就不起作用了。
2.2.3 主要进程
主要进程设置私有通道并指定进程如何并行组合。大多数私有通道用于密钥分发，我们仅对两个选民的协议进行建模，并为每个选民启动两份管理员和收集者进程。以下是主要进程的代码：
(* private channels *)
ν privCh . ν pkaCh1 . ν pkaCh2 . ν skaCh . ν skvaCh . ν skvbCh .
(* administrators *)
( processK
| processA
| processA
| processC
| processC
| (* voters *)
( let skvCh = skvaCh in let v = a in
processV )
| ( let skvCh = skvbCh in let v = b in
processV )
)
2.2.4 选民进程
每个选民首先从 PKI 获取其秘密密钥以及管理员的公钥，其余规范直接遵循 2.1 节中的非正式描述。选民进程代码如下：
processV =
(* parameters : skvCh , v *)
(* her private key *)
in ( skvCh , skv ) .
(* public keys of the administrator *)
in ( pkaCh1 , pubka ) .
ν blinder .
ν r .
let committedvote = commit ( v , r )
in
let blindedcommittedvote=blind ( committedvote , blinder )
in
out ( ch1 , ( pk ( skv ) , sign ( blindedcommittedvote , skv ) ) ) .
in ( ch2 ,m2) .
let result = checksign (m2, pubka ) in
if result = blindedcommittedvote then
let signedcommittedvote=unblind (m2, blinder )
in
synch 1 .
out ( ch3 , ( committedvote , signedcommittedvote )) .
synch 2 .
in ( ch4 , ( l ,= committedvote ,= signedcommittedvote )) .
out ( ch5 , ( l , r ))
我们的模型还包括一个用于生成和分发密钥材料的专用进程（模拟 PKI，即 $processK$）、一个管理员进程和一个收集者进程（这些进程的具体代码未给出）。
2.3 协议分析
2.3.1 投票隐私性
根据我们的定义，要证明该协议尊重隐私，需要证明：
$S[V_A{a/v} | V_B{b/v}] \approx S[V_A{b/v} | V_B{a/v}]$
其中 $V_A = processV{skvaCh/skvCh}$，$V_B = processV{skvbCh/skvCh}$。我们不要求任何权威机构是诚实的，所以它们不在 $S$ 中建模，而是作为攻击者上下文的一部分。但我们必须确保两个选民使用相同的管理员公钥，因此我们通过私有通道 $pkaCh1$ 发送这个公钥，尽管公钥及其对应密钥攻击者是知道的。实际上，我们证明：
$νpkaCh1.(V_A{a/v} | V_B{b/v}| processK) \approx νpkaCh1.(V_A{b/v} | V_B{a/v}| processK)$
证明使用了标记双模拟的等价定义而非观察等价性。我们使用 ProVerif 工具自动完成了部分证明（静态等价关系），其余的双模拟部分通过考虑所有情况手动完成。虽然 ProVerif 提供观察等价性检查，但在这种情况下无法完成证明，因为观察等价性检查是不可判定的，ProVerif 旨在证明一个更精细的关系，该关系依赖于轻松匹配两个进程的执行路径，而这个关系对于证明上述等价性来说过于精细。我们的证明依赖于在第一阶段（同步 1 之前）将左侧的 $V_A{a/v}$ 与右侧的 $V_A{b/v}$ 匹配，在后续阶段（同步 1 之后）将左侧的 $V_A{a/v}$ 与右侧的 $V_B{a/v}$ 匹配。协议阶段的使用对于尊重隐私至关重要，当省略与管理员注册阶段后的同步时，隐私会被侵犯。例如，选民 $V_A$ 联系管理员后，由于没有同步，$V_A$ 可能在 $V_B$ 联系管理员之前将其承诺的投票发送给收集者，攻击者就能将这个承诺与第一个选民的身份关联起来。
2.3.2 无收据性
该方案不是无收据的，实际上它在设计时并未考虑无收据性。如果选民泄露了用于盲化和承诺的随机数（$b_A$ 和 $r_A$），胁迫者可以验证承诺的投票是否符合其意愿，并且通过对第一条消息去盲化，胁迫者可以追踪哪个投票对应这个特定的选民。而且，选民不能对这些值说谎，因为胁迫者会立即检测到。在我们的框架中，不存在满足以下条件的 $V’$：
- $V’ \setminus out(chc,·) \approx V_A{a/v}$
- $S[V_A{c/v}
{chc} | V_B{a/v}] \approx S[V’ | V_B{c/v}]$
$V_A{c/v}
{chc}$ 在通道 $chc$ 上输出值 $r_A$ 和 $b_A$，这会在框架中生成条目，所以 $V’$ 需要在框架中生成类似条目。胁迫者可以验证 $r_A$ 和 $b_A$ 是否用于对发送给管理员的消息中的投票 $c$ 进行编码，因此 $V’$ 为了满足第二个等价性，无法承诺与 $c$ 不同的值。但这样 $V’$ 就不满足第一个等价性，因为管理员已经对 $c$ 的承诺进行了签名，$V’$ 之后无法改变其投票。所以，对 $V’$ 的要求无法满足。需要注意的是，无收据性的失败不是由于管理员或收集者可能的不诚实，即使将它们视为诚实方，该协议仍然不能保证无收据性，因此也不具备抗胁迫性。
协议流程 mermaid 流程图
graph LR
classDef startend fill:#F5EBFF,stroke:#BE8FED,stroke-width:2px
classDef process fill:#E5F6FF,stroke:#73A6FF,stroke-width:2px
classDef decision fill:#FFF6CC,stroke:#FFBC52,stroke-width:2px
A([开始]):::startend --> B(第一阶段: 获取签名):::process
B --> C(选民选择投票v):::process
C --> D(计算承诺x = ξ(v,r)):::process
D --> E(计算消息e = χ(x, b)):::process
E --> F(选民对e签名并发送给管理员):::process
F --> G{管理员验证}:::decision
G -- 通过 --> H(管理员对e签名并发送给选民):::process
G -- 未通过 --> I(结束流程):::process
H --> J(选民去盲处理得到y = σ_A(x)):::process
J --> K(第二阶段: 实际投票):::process
K --> L(选民使用匿名通道将y发送给收集者):::process
L --> M{收集者检查签名}:::decision
M -- 通过 --> N(收集者将(ℓ, x, y)录入列表):::process
M -- 未通过 --> I
N --> O(第三阶段: 公布选票):::process
O --> P(收集者公布承诺列表):::process
P --> Q(选民验证承诺是否在列表中):::process
Q --> R(选民发送ℓ, r给收集者):::process
R --> S(收集者打开选票并公布投票v):::process
S --> T([结束]):::startend
协议阶段表格
| 阶段 | 步骤 | 操作内容 |
|---|---|---|
| 第一阶段：获取签名 | 1 | 选民选择投票 $v$，计算承诺 $x = ξ(v,r)$，消息 $e = χ(x, b)$，对 $e$ 签名并发送给管理员 |
| 2 | 管理员验证选民资格和签名有效性，若通过则对 $e$ 签名并发送给选民 | |
| 3 | 选民去盲处理得到 $y = σ_A(x)$ | |
| 第二阶段：实际投票 | 1 | 选民使用匿名通道将 $y$ 发送给收集者 |
| 2 | 收集者检查签名，若通过则将 $(ℓ, x, y)$ 录入列表 | |
| 第三阶段：公布选票 | 1 | 收集者公布承诺列表 |
| 2 | 选民验证承诺是否在列表中，发送 $ℓ, r$ 给收集者 | |
| 3 | 收集者打开选票并公布投票 $v$ |
3 Okamoto 协议
3.1 协议描述
该协议的选举管理机构包括负责注册的管理员、收集令牌的收集者以及公布最终票数的及时性成员（用 $T$ 表示）。与 Fujioka 等人提出的协议的主要区别在于使用了陷门位承诺方案，以实现无收据性。这种承诺方案允许进行承诺的代理以多种方式打开承诺，即陷门位承诺不会将选民绑定到投票 $v$。为确保选民在最后（开放阶段）不改变主意，选民必须在投票阶段说明如何打开其承诺，这通过不可窃听的匿名通道将所需信息发送给 $T$ 来实现。
3.1.1 协议阶段
- 第一阶段 ：与 Fujioka 等人的协议类似，唯一的变化是使用了陷门位承诺方案 $ξ$。
-
第二阶段：实际投票
- 选民 $V$ 使用匿名通道将 $y$（$A$ 对选民投票的陷门承诺的签名）发送给收集者 $C$。
- 收集者 $C$ 检查签名 $y$ 的正确性，若通过则将 $(x, y)$ 录入列表。
- 选民 $V$ 通过不可窃听的匿名通道将 $(v, r, x)$ 发送给及时性成员 $T$。
-
第三阶段
：收集者 $C$ 决定收到所有选票后（如在固定截止日期后）开始。
- 收集者 $C$ 公布获得的陷门承诺列表 $(x_i, y_i)$。
- 选民 $V$ 验证其承诺是否在列表中。
- 及时性成员 $T$ 以随机顺序公布投票 $v_i$ 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