RU2793408C1 - Block cipher method using kronecker product of involutive matrices - Google Patents

Abstract

FIELD: data encryption.

SUBSTANCE: cryptoalgorithm, involutive matrix, Kronecker product, computational complexity. Modifying the cipher with an involutive key matrix of the plaintext size, represented as the Kronecker product of K elementary involutive matrices, which avoids reversing the key matrix for decryption.

EFFECT: increasing the cryptographic strength of the encryption algorithm.

1 cl, 1 dwg

Description

The invention relates to the field of information encryption, namely to a block encryption method, in which the square key matrix is not actually calculated. The method can be used in a one-time key mode. We propose a modification of the cipher with an involutive key matrix of plaintext size T , represented as the Kronecker product of K elementary involutive matrices, which avoids reversing the key matrix for decryption.

Various block cipher techniques are known and are described in the following patents:

RU 2728527 C1 "Method and device for encoding", which presents a method and device for encoding information in a communication system during data transmission.

RU 2738321 C1 "Cryptographic conversion method and device for its implementation", which discloses a cryptographic conversion method for processing a large array, packets or data streams based on hardware and software implementation in the microprocessor of the AES block cipher algorithm, including converting the initial data into the S structure, consisting of 16-byte blocks.

RU 2748964 C2 "Method for secure transmission of requested data and system implementing it", which describes a method for secure transmission of requested data from a device of a trusted party to a device of a third party.

Also articles:

“Dukhnich, E.I., Chefranov A.G. Using the Kronecker product of matrices to modify cryptographic algorithms / E.I. Dukhnich, A.G. Chefranov // State Medical University. adm. F.F. Ushakov. Operation of maritime transport - 2020. - No. GMU im. adm. F.F. Ushakov. Operation of Maritime Transport - 2020. - No. 2 (95) 2020 - P. 45-52. - DOI: 10.34046/aumsuomt95/23"

“Dukhnich, E.I. Hardware-oriented algorithm for fast multiplication of the Kronecker product of matrices by a vector / E.I. Dukhnich, A.G. Chefranov // Izvestiya SFedU. Technical science. - 2020. - No. 7(217). - S. 134-138. - DOI 10.18522/2311-3103-2020-7-45-52.”

The closest technical solution is the method adopted for the prototype:

RU 2715411 C2 "Digital complex of a satellite communication system", which uses one-time block ciphers, i.e. ciphers whose key operator explicitly depends on the time parameter t. The nature of changes in this parameter determines the time intervals of "one-time use" of key material, using the L. Hill cipher as an example.

A one-time pad, a cryptosystem using "one-time" keys, is an absolutely strong cipher if the following properties are met:

First property: a key is generated for each message, each key is used only once;

Second property: the size of the key matrix is equal to or greater than the length of the message T ;

The third property: the key is statistically reliable, that is, the probabilities of occurrence of each of the possible characters are equal, the characters in the key sequence are independent and random.

The method described in the prototype has the following disadvantages:

The first drawback: the need to reverse the key matrix during decryption;

на вектор имеет вычислительную сложность O(n ² ).The second drawback: the quadratic complexity of calculations and the amount of memory for storing the key matrix (in relation to the size of the message), because multiplying a square size matrix

per vector has computational complexity O(n ² ) .

To eliminate the first drawback in patent No. RU 2728527 C1 "Method and device for encoding", it was proposed to use involutive (inverse to itself) matrices in block ciphers as key ones, in order to eliminate the need to calculate the matrix, the inverse of the key, for decryption.

However, unfortunately, both analogs and the prototype do not provide sufficient encryption efficiency, since redundant arithmetic operations must be performed to encrypt and decrypt a message.

The technical objective of the present invention is to improve the efficiency and cryptographic strength of the encryption algorithm, eliminate the shortcomings of analogues and the prototype by modifying the method proposed by L. Hill.

где K=

, при этом генерируемая матрица ключа имеет выражение:The technical problem is achieved by the fact that in the claimed method of block encryption and decryption of data using the L. Hill cipher, the input message vector is set, the key matrix is generated in the form of an involutive matrix, characterized in that the generation of the key matrix A of size T using the Kronecker product of K elementary involutive 2×2 matrices

where K =

, while the generated key matrix has the expression:

,

,

encryption is performed by using a special algorithm for combining the calculation of the involutive key matrix A and multiplying it by the vector of the input message, and the cipher is given the properties of an absolutely secure cipher operating in the "one-time" mode, for which the key is generated for each message and each key is used only once, in this case, the size of the key matrix is equal to or greater than the size of the message, and the statistical reliability of the key is ensured by the fact that the characters in the key sequence are independent and random, the probabilities of occurrence of each of the possible characters are equal, and the "one-time" mode is provided by a sufficiently large variable block size, performed as follows that one block covers the entire plaintext, which excludes attacks based on the use of already known “plaintext-ciphertext” pairs obtained using the same key, while the block size is set by the size of the key matrix A , decryption is performed by using a special algorithm for combining the calculation of the involutive matrix of key A and multiplying it with the vector of the encrypted input message, and the Kronecker product of involutive matrices is also an involutive matrix, which excludes the stage of calculating the inverse matrix during decryption.

The technical result consists in the fact that in the proposed algorithm there is an increase in cryptographic strength by giving the cipher the properties of an absolutely secure cipher using the One-Time Pad method as an example.

) при сложности памяти

).The technical result is achieved by the fact that instead of actually calculating the key matrix, the involutive elementary matrices are iteratively multiplied by the plaintext of length T in time

) with memory complexity

) .

In the claimed method, the first property of the "one-time pad" is provided with a sufficiently large variable block size, such that one block covers the entire plaintext, which excludes attacks based on the use of already known "plaintext-ciphertext" pairs obtained using one and the same same key.

The second property of the One-Time Pad is provided by the fact that the size of the block is set by the size of the key matrix, which is formed as the Kronecker product (KP) of elementary (2×2) matrices.

The fulfillment of the above two properties ensures the statistical strength of the key (the third property of the One-Time Pad).

The claimed technical solution proposes the formation of a key matrix as a CP of involutive elementary matrices (2×2) in the form:

, (1)

, (1)

, a и b - произвольные числа из Z_N (берутся из ключа свои для каждого j ), причем b -нечетное, если N=2 ^d , аwhere j=1,2,…

, a and b are arbitrary numbers from Z _N (they are taken from the key for each j ), and b is odd if N=2 ^d , and

.

.

There is a well-known property of KP that the result of KP of involutive matrices is also an involutive matrix.

To eliminate the second drawback, it is proposed to construct an algorithm that accelerates the processes of forming a CP and multiplying a vector by it. The algorithm allows you to combine these procedures in time. Thus, the KP matrix is not actually calculated explicitly. Instead, the KP factor matrices are iteratively multiplied by the vector components in O(nlog ₂ n) time, where n = T.

Thus, the formation of the key matrix as a CP of elementary involutive matrices significantly increases the efficiency of the cipher by eliminating the preliminary separate calculation of the CP and its inversion, as well as the absence of the need to store the key matrix (CP) in memory. In this case, the computational complexity and memory costs will be respectively equal to O(nlogn) and O(n), in contrast to the quadratic complexity of other modifications of the Hill cipher, where n is the message length.

, причем:The essence of the invention is illustrated by graphic material, where in Fig. 1 shows the calculation scheme for message n=8, order CP

, and:

X _1-8 - bytes of the input message vector;

C, B, A- matricesP ₁ ,P ₂ ,P ₃ , obtained by formula (1);

- результат произведения частей векторов на матрицы С (k=3), В (k=2), А (k=1);

- the result of the product of parts of vectors by matricesC (k=3), B (k=2), A (k=1);

Y _1-8 - bytes of the encrypted message.

Encryption is performed as the calculation of the product of KP by the vector X , which is mathematically described as :

(2)

(2)

However, instead of calculating the CP and expression (2), the following sequence of steps is performed:

, где i=1,3,5,7, j=2,4,6,8.1. Vector X is divided into n/2 parts - 2-dimensional vectors

, where i=1,3,5,7, j=2,4,6,8.

2. These vectors are multiplied by the matrixC=P ₁ (1):

так, чтобы i=1,2,5,6, j=3,4,7,8:3. The elements of the resulting vectors are reindexed by multiplying them by the permutation matrix

so that i=1,2,5,6, j=3,4,7,8:

4. These vectors are multiplied by the matrix B=P ₂ :

так, чтобы i=1,3,2,4, j=5,7,6,8:5. The elements of the resulting vectors are reindexed by multiplying them by the permutation matrix

so that i=1,3,2,4, j=5,7,6,8:

6. These vectors are multiplied by the matrix A=P ₃ :

так, чтобы i=1,5,2,6, j=3,7,4,8:7. The elements of the resulting vectors are reindexed by multiplying them by the permutation matrix

so that i=1,5,2,6, j=3,7,4,8:

конкатенацией формируется результирующий вектор Y выражения (2).8. From the obtained two-dimensional vectors

the resulting vector Y of expression (2) is formed by concatenation.

Thus, in FIG. 1, transformations (2) in steps 1-8 (from top to bottom) are presented in the form of a diagram with a hypercube topology, where all matrices A, B, C are indicated by rectangles and for each of the four instances of each of these matrices, the corresponding input and output data are indicated, and multiplications by permutation matrices are shown by connecting lines between levels (rows of rectangles). It should be noted that decryption is described by exactly the same calculation scheme.

Claims (3)

Block encryption method using the Kronecker product of involutive matrices, based on data encryption and decryption using the L. Hill cipher, in which the input message vector is specified, the key matrix is generated in the form of an involutive matrix, characterized in that an involutive key matrix A of size T is generated using Kronecker product multiplication of K elementary involutive 2×2 matrices

, where K=log ₂ T, while the generated key matrix has the expression:

, encryption is performed by using an algorithm for combining the calculation of the involutive key matrix A and multiplying it by the vector of the input message, and the cipher is given the properties of an absolutely secure cipher operating in the "one-time" mode, for which the key is generated for each message and each key is used only once, with In this case, the key length is determined by the value K , and the statistical reliability of the key is ensured by the fact that the characters in the key sequence are independent and random, the probabilities of occurrence of each of the possible characters are equal, moreover, the "one-time" mode is provided with a sufficiently large variable block size, made so that one block covers the entire plaintext, which excludes attacks based on the use of already known "plaintext-ciphertext" pairs obtained using the same key, while the block size is set by the size of the key matrix A the key matrix A and multiplying it with the vector of the encrypted input message, and the Kronecker product of involutive matrices is also an involutive matrix, which eliminates the step of calculating the inverse matrix during decryption.

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