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Shannon Perfect Secrecy in a Discrete Hilbert Space

机译：离散希尔伯特空间中的Shannon完美保密

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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.
机译：1949年，香农（Shannon）在数学上证明了一次性垫（OTP）是完全安全的。我们建议将经典的OTP从n位有限域扩展到整个有限域上的整个对称组。在这种情况下，对称组可以在n位计算的基础上由离散的希尔伯特球（DHS）表示。与在量子计算中在复杂域上定义的连续希尔伯特空间不同，在有限域GF（2）上定义了DHS。在此DHS中，整个对称组可以由完整的n位二进制置换矩阵集完全描述。可以通过从对称组中随机选择一个置换矩阵，并与与要编码的数据相对应的状态相关的计算基础向量相乘，来对明文进行编码。然后，将所得的矢量转换为密文的输出状态。解码是相同的过程，但具有预共享置换矩阵的转置。我们证明了在这种扩展下，经典OTP中的1对1映射同样可能在离散希尔伯特空间中解耦。排列矩阵之间的不确定性关系保护了选定的填充，该填充由M个排列矩阵（也称为“量子排列填充”或QPP）组成。 QPP不仅保持了经典配方的完美保密性，而且在不破坏完美保密性的情况下也可以重复使用。然后声明扩展的Shannon完全保密性，以使密文C绝对不提供有关明文P和填充的信息。

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