The One-time-pad (OTP) was mathematically proven to be perfectly secure by Shannon in 1949. We propose to extend the classical OTP from an n-bit finite field to the entire symmetric group over the finite field. Within this context the symmetric group can be represented by a discrete Hilbert sphere (DHS) over an n-bit computational basis. Unlike the continuous Hilbert space defined over a complex field in quantum computing, a DHS is defined over the finite field GF(2). Within this DHS, the entire symmetric group can be completely described by the complete set of n-bit binary permutation matrices. Encoding of a plaintext can be done by randomly selecting a permutation matrix from the symmetric group to multiply with the computational basis vector associated with the state corresponding to the data to be encoded. Then, the resulting vector is converted to an output state as the ciphertext. The decoding is the same procedure but with the transpose of the pre-shared permutation matrix. We demonstrate that under this extension, the 1-to-1 mapping in the classical OTP is equally likely decoupled in Discrete Hilbert Space. The uncertainty relationship between permutation matrices protects the selected pad, consisting of M permutation matrices (also called Quantum permutation pad, or QPP). QPP not only maintains the perfect secrecy feature of the classical formulation but is also reusable without invalidating the perfect secrecy property. The extended Shannon perfect secrecy is then stated such that the ciphertext C gives absolutely no information about the plaintext P and the pad.

翻译：一次性密码本（OTP）于1949年由香农在数学上证明具有完美保密性。我们提议将经典OTP从n位有限域扩展到该有限域上的整个对称群。在此语境下，对称群可由基于n位计算基的离散希尔伯特球面（DHS）表示。与量子计算中基于复数域的连续希尔伯特空间不同，DHS定义于有限域GF(2)上。在此DHS内，整个对称群可通过n位二进制置换矩阵的完备集完全描述。明文编码可通过从对称群中随机选取一个置换矩阵，将其与待编码数据状态对应的计算基向量相乘实现。所得向量随后被转换为输出状态作为密文。解码过程相同，但使用预共享置换矩阵的转置。我们证明在此扩展下，经典OTP中的一一映射在离散希尔伯特空间中具有等概率解耦特性。置换矩阵之间的不确定性关系保护由M个置换矩阵构成的选定密码本（亦称量子置换密码本QPP）。QPP不仅保持了经典形式的完美保密性特征，且可重复使用而不破坏完美保密性。扩展后的香农完美保密性可表述为：密文C绝不泄露关于明文P及密码本的任何信息。