Perfectly Secure Shannon Cipher Construction Based on the Matrix Power Function

Abstract

A Shannon cipher can be used as a building block for the block cipher construction if it is considered as one data block cipher. It has been proved that a Shannon cipher based on a matrix power function (MPF) is perfectly secure. This property was obtained by the special selection of algebraic structures to define the MPF. In an earlier paper we demonstrated, that certain MPF can be treated as a conjectured one-way function. This property is important since finding the inverse of a one-way function is related to an N P -complete problem. The obtained results of perfect security on a theoretical level coincide with the N P -completeness notion due to the well known Yao theorem. The proposed cipher does not need multiple rounds for the encryption of one data block and hence can be effectively parallelized since operations with matrices allow this effective parallelization.

Matrix Power Function Based Block Cipher Operating in CBC Mode

“…For this reason, our cipher lacked the flexibility necessary for the implementation of our scheme in practice. Furthermore, our investigation in [12] has shown that the statistical properties of the proposed scheme leave much to be desired for the parameters introduced in [15]. However, that very same investigation revealed that extra flexibility in the main parameters significantly improves the statistical properties of our scheme.…”

“…Recently in our paper [15] we introduced a new block cipher and proposed a concept of single round symmetric encryption based on the MPF mapping. However, there we used low cardinality algebraic structures.…”

Avalanche effect and bit independence criterion of perfectly secure Shannon cipher based on matrix power

“…where 𝑊, 𝐸 ∈ 𝑀𝑎𝑡 (𝕊) are matrices with entries from semigroup 𝕊 and 𝑋, 𝑌 ∈ 𝑀𝑎𝑡 (ℝ) are matrices with entries from a finite ring of integers ℝ. This mapping allows us to raise the base matrix 𝑊 to the so-called power matrices 𝑋 and 𝑌. E. Sakalauskas with co-authors used the MPF in 2020 to propose a perfectly secure Shannon cipher defined over ℤ [1]. This cipher uses a plaintext matrix 𝑀, private keys 𝑋 and 𝑌 along with a function 𝑓: ℤ ↦ 𝔾 , which maps elements of ℤ to elements of the multiplicative Sylow group 𝔾 = {1,2,4}, which is a subgroup of ℤ * .…”

“…Upon receiving the ciphertext its decryption is performed in the reverse order and can be summarized by the following expression: Perfect secrecy of the presented block cipher and the statistical independency of the ciphertext 𝐶 from the plaintext 𝑀 is proven in [1].…”

“…

In 2020 E. Sakalauskas with coauthors published a paper defining perfectly secure Shannon cipher based on matrix power function, proposing effective parallelization, and ensuring no need for multiple rounds encrypting one data block [1]. In this paper we present computational results with the avalanche effect and bit independence criterion (BIC).

…”