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.Comment: 7 pages, 1 figure, presented and published by QCE202