We study retrospective auditing for dynamic ordered sets maintained by an untrusted party. A passive auditor watches insert, delete, membership, predecessor, successor, min, and max operations, stores five machine words and a flag, and receives a constant-size public tally record per operation. At audit time the maintainer discloses the claimed live vacant intervals. The method represents order semantics by maximal gaps: gaps are born, cited, consumed, and timestamped, while two hidden field accumulators test equality of the birth and consumption ledgers. Honest executions are accepted with probability one. If any answer in a T-operation session is wrong, acceptance occurs with probability at most (4T+1)/p over one secret field element, against computationally unbounded maintainers. We prove that deterministic and visible-coin auditors require linear state, and that removing the timestamp rule permits an exact replay forgery. A leaf-oriented (2,4)-tree implements the maintainer in O(log n) worst-case time per operation with one extra word per element, and its rebalancing events admit an auditable O(m) envelope over m updates. Checkpoint audits compose with additive error.

翻译：我们研究由不可信方维护的动态有序集合的事后审计。被动审计者观察插入、删除、成员查询、前驱、后继、最小值及最大值操作，存储五个机器字和一个标志位，并在每次操作时接收一个常量大小的公开计数记录。在审计阶段，维护者披露声称的实时空区间。该方法通过最大间隙表示顺序语义：间隙被生成、引用、消耗并加盖时间戳，而两个隐藏的域累加器用于验证生成账本与消耗账本的等价性。诚实执行会被以概率1接受。若在T次操作的会话中存在任何错误答案，则针对单个秘密域元素，接受概率最多为(4T+1)/p，且该概率对计算能力无界的维护者成立。我们证明确定性及可见硬币的审计者需要线性状态，并指出移除时间戳规则将允许精确重放伪造。基于叶导向(2,4)-树的实现使得每次操作的维护时间达到最坏情况O(log n)，且每个元素仅需额外一个机器字；其再平衡事件可在m次更新上产生可审计的O(m)开销。检查点审计可通过累加误差进行组合。