Post-quantum Key Exchange Protocol Using High
Dimensional Matrix
Richard Megrelishvili Melkisadeg Jinjikhadze Maksim Iavich
I. J. Tbilisi State University Akaki Tsereteli State University Caucasus University
Tbilisi, Georgia Kutaisi, Georgia Tbilisi, Georgia
richard.megrelishvili@tsu.ge mjinji@yahoo.com m.iavich@scsa.ge
Avtandil Gagnidze Giorgi Iashvili
Georia University Georia University
Tbilisi, Georgia Tbilisi, Georgia
gagnidzeavto@yahoo.com g.iashvili@scsa.ge
Abstract— Active work is being done to create and develop But this problem can easily be solved by quantum computers
quantum computers. Google Corporation, NASA and the using Shor algorithm [1,2].
Universities Space Research Association (USRA) have teamed up The security of RSA algorithm relies on factorization problem,
with DWAFE, the manufacturer of quantum processors. D-Wave but this problem can be easily solved using quantum computers
2X is a quantum processor that contains 2,048 physical qubits.
1152 qubits from the whole number of qubits are used to perform
[3].
the calculations. As we see, quantum computers can easily solve Active work is being done to create and develop quantum
the problem of calculating the discrete logarithm used in Diffie- computers. Google Corporation, NASA and the Universities
Hellman algorithm. So it can break Diffie-Hellman algorithm. Space Research Association (USRA) have teamed up with
When quantum computers are released all existing crypto DWAFE, the manufacturer of quantum processors. D-Wave 2X
systems will be useless, because there will be no way to transfer the is a quantum processor that contains 2,048 physical qubits. 1152
key securely. qubits from the whole number of qubits are used to perform the
In the article is proposed the new key exchange method using calculations. As we see, quantum computers can easily solve the
high dimensional matrix, this method is safe against attacks problem of calculating the discrete logarithm used in Diffie-
implemented using quantum computers. The case concerns the
matrix function and algorithm for cryptographic keys exchange
Hellman algorithm. So it can break Diffie-Hellman algorithm.
with open channel. For the algorithm is offered the method of When quantum computers are released all existing crypto
building a high dimensional matrix multiplicative group. systems will be useless, because there will be no way to transfer
The arising of this goal is that traditional key exchange the key securely [4,5].
methods are vulnerable to quantum computer attacks. In the article is proposed the new key exchange algorithm using
Keywords— post-quantum cryptography, attacks, a matrix one- high dimensional matrix. This algorithm is safe against
way function, Abelian multiplicative group, asymmetric quantum computer attacks.
cryptography, high dimensional matrix finite field The case concerns the matrix function and algorithm for
I. INTRODUCTION cryptographic keys exchange with open channel.
One of the fundamental problems of cryptography is the safe For this is offered the method building a high dimensional
communication over the listening channel. Messages need to be matrix multiplicative group.
encrypted and decrypted, but for this, both parties need to have The arising of this goal is that traditional key exchange
a common key. If this key is transmitted via the same channel, methods are vulnerable to quantum computer attacks.
then the listening side will also receive it, and the meaning of One-way function (OWF) is a function whose value is easy
the encryption will disappear. to calculate for any argument, but it is “difficult” to find an
Diffie-Hellman algorithm allows the two parties to obtain a argument for the given value of the function. The word
common secret key using an unprotected, but spoofed, "difficult" is to understand the complexity of the computation.
communication channel. The received key can be used to In other words, finding the relevant argument of the given
exchange messages using symmetric encryption. function in real time is difficult even with the modern
The security of forming a common key in the Diffie-Hellman computing techniques. The irreversibility of function does not
algorithm follows from the fact that, although it is relatively mean that the function is one-way [6,7].
easy to calculate exponents modulo a prime number, it is very The existence of one-way functions is the basis for the idea
difficult to calculate discrete logarithms. For large prime of asymmetric cryptography. It (one-way function) is the
numbers of hundreds and thousands of bits, the task is foundation of asymmetric cryptography, personal
considered unsolvable, since it requires a tremendous amount identification, authentication, and other fields of information
of computational resources. protection. Although there is no theoretical proof of the
existence of one-way functions in general, there are several
Copyright held by the author(s).
83
“possible pretendents” (eg, multiplication and factorization, 𝑣1 𝑎11 + 𝑣2 𝑎21 + ⋯ + 𝑣𝑛 𝑎𝑛1 𝑢1
squaring and module rooting, discreet exponent and 𝑣1 𝑎12 + 𝑣2 𝑎22 + ⋯ + 𝑣𝑛 𝑎𝑛2 𝑢2
𝑣𝐴1 = ( ⋮ )=( ⋮ ) (10)
logarithmization), whose one-wayity (ie the difficulty of
finding the argument for the value of function) at this time real 𝑣1 𝑎𝑛1 + 𝑣2 𝑎𝑛2 + ⋯ + 𝑣3 𝑎𝑛3 𝑢𝑛
and is actively used in information exchange protocols. The number of unknowns in the system of linear equations is
As we have mentioned, one-way functions are actively used the square of number of equations. Obviously, the system can
in the algorithms for developing a cryptographic open key. The not be solved in limited time, if the size of the matrix is large
initial idea (1976) belongs to Whitfield Duffie and Martin enough. Size of the matrix must be chosen considering Grover’s
Helman. Based of their idea was established the first practical algorithm. Classically, searching requires a linear search, which
wel-known Diffie-Helman-Merkel method, which enabled the is O(N) in time. Grover's algorithm needs O(N1/2) time, it is
development of a common cryptographic key using the open considered as fastest quantum algorithm for searching an
(unprotected) channel. A year later, the first RSA algorithm of unsorted information. This algorithm provides a quadratic
asymmetric encryption was formed. The RSA in fact, resolved speedup [10,11].
the problem of exchange information with open channel. Both One fact must be taken into consideration if the 𝐴1 matrix
algorithms are not safe against quantum computers attacks. Are contains the internal recurrence, or if each of its rows are in a
proposed quantum key exchange protocols, but quantum certain recurrence with the previous row, then the task of
computers are needed to implement them [8,9]. solving the system will be replaced by a simpler task that is easy
to solve. It is so important that it puts itself in doubt the one-
II. ONE-WAY MATRIX FUNCTION way character of our function and requires the existence of
Abelian multiplicative matrix group with a high order, that is
The new one-way function for the development of common free from the recurrence of the inside.
cryptographic keys is based on high order cyclic matrix groups, III. FINITE MATRIX GROUPS CONSTRUCTION
with the power 𝑒 = 2𝑛 − 1 , where the n is row dimension of Let's consider (1 + 𝛼)𝑗 , where j = 0,1,2, ⋯, and α represents
the square matrix. Let's assume that "A" is the above matrix the root of primitive polynomial in the 𝐺𝐹(2𝑛 ) field odule with
group, while A is the initial 𝑛 × 𝑛 matrix, then "A" = A={𝐴, 𝐴2 , the module p (x).
𝑛
𝐴3 , … , 𝐴2 −1 = 𝐼} (1) (1 + 𝛼)0 = 1 1
where I represents an identity matrix. (1 + 𝛼)1 = 1 + 𝛼 11
One-way function and algorithm for common key development (1 + 𝛼)2 = 1 + 𝛼 2 101
are as follows: (1 + 𝛼)3 = 1 + 𝛼 + 𝛼 2 + 𝛼 3 1111
The sender chooses 𝐴1 ∈ A secret matrix to send to the (1 + 𝛼)4 = 1 + 𝛼 4 10001
receiving party via open channel the (1 + 𝛼)5 = 1 + 𝛼 + 𝛼 4 + 𝛼 5 110011
u𝑢1 = 𝑣𝐴1 (2) The polynomial coefficients generated the structure known as
vector where 𝑣 ∈ 𝑉𝑛 vector is known (𝑉𝑛 – is a vector Serpinsky Triangle. The derived structure contains a number of
space on GF field); sub-structures that can be used as a generator (generating
The receiving party shall, on the other hand, choose matrix) for multiplicative groups, ie primitive elements. Such
𝐴2 ∈ A secret matrix and send to the sender is, for example,
𝑢2 = 𝑣𝐴2 (3) 1 1 1 1 1
vector; 1 1 1 1 0 0 0 0
𝑃3 = (1 0 0) (9), 𝑃5 = 1 1 0 0 0 (11)
Sender calculates 𝑘1 = 𝑢2 𝐴1 (4)
vector; 1 1 0 1 0 1 0 0
Receivier calculates 𝑘2 = 𝑢1 𝐴2 (5) (1 1 1 1 0)
And many more. Their natural powers create Abelian
where 𝑘1 and 𝑘2 – are secret keys; multiplicative cyclic group.
Obviously, 𝑘1 = 𝑘2 = 𝑘, because For example:
𝑘 = 𝑣𝐴1 𝐴2 = 𝑣𝐴2 𝐴1 (6) 1 1 1 1 0 1
because of the commutativeness of the "A" group. The 𝑃31 = (1 0 0) , 𝑃32 = (1 1 1),
𝑣𝐴𝑖 = 𝑢 (7) 1 1 0 0 1 1
is one-way fast function. 0 0 1 1 1 0
Let 𝑣 = (𝑣1 , 𝑣2 , 𝑣3 , ⋯ , 𝑣𝑛 ) ∈ 𝑉𝑛 (8) 𝑃33 = (1 0 1) , 𝑃34 = (0 0 1),
and 0 1 0 1 0 0
𝑢 = (𝑢1 , 𝑢2 , 𝑢3 , ⋯ , 𝑢𝑛 ) ∈ 𝑉𝑛 are non-secret vectors from 0 1 1 0 1 0
the above algorithm and 𝑃35 = (1 1 0) , 𝑃36 = (0 1 1),
𝑎11 ⋯ 𝑎1𝑛 1 1 1 1 0 1
𝐴1 = ( ⋮ ⋱ ⋮ )∈A (9) 1 0 0
0
𝑎𝑛1 … 𝑎𝑛𝑛 𝑃37 = 𝑃3 = (0 1 0) (12)
is a secret matrix. Then, according to algorithm the following 0 0 1
It’s easy to confirm that
system is formed:
𝑃30 , 𝑃31 , 𝑃32 , 𝑃33 , 𝑃34 , 𝑃35 , 𝑃36
84
is an Abelian multiplicative group. 𝑃3,2 = 𝑃36 × 𝑃36 + 𝑃36 × 0 + 0 × 𝑃36 = 𝑃35 ,
Lets keep the structure of 𝑃3 matrix and extend it by elements 𝑃3,3 = 𝑃36 × 𝑃36 + 𝑃36 × 0 + 0 × 0 = 𝑃35 .
of set (2) as follows: 𝑃33 𝑃32 𝑃34
𝑗 𝑗
𝑃3𝑖 𝑃3 𝑃3 or 𝑃 = (𝑃34 𝑃35 𝑃35 ) (15) ( see pic. 2).
𝑃32 (𝑖, 𝑗) = (𝑃3𝑗 0 0 ), where i,j=0..6. (13) 𝑃32 𝑃35 𝑃35
𝑗 𝑗
𝑃3 0𝑃3
Fore example, when 𝑖 = 5 and 𝑗 = 6, we have
𝑃35 𝑃36 𝑃36
𝑃32 (5,6) = (𝑃36 0 0)=
𝑃36 𝑃36 0 pic.2: 𝑃 = [𝑃32 (5,6)]2
0 1 1 0 1 0 0 1 0
(1 1 0) (0 1 1) (0 1 1) Using the software package we have developed it has been
1 1 1 1 0 1 1 0 1 confirmed that the 𝑷𝟑𝟐 (𝟓, 𝟔) matrix is a primitive element. Its
0 1 0 natural powers generate Abelian multiliplicative group, whose
(0 1 1) 0 0 ( (14) 𝟐
power is 𝟐𝟑 −𝟏 .
1 0 1
0 1 0 0 1 0 The elements of [𝑃32 (5,6)]𝑘 when k = 73, 146, 219, 292,
(0 1 1) (0 1 1) 0 365, 438, 511 are diagonal matrices (see pic. 3):
( 1 0 1 1 0 1 ) 𝑃34 0 0 𝑃31 0 0 𝑃35 0 0
4 1
When 𝑖 = 0 and 𝑗 = 1, we have (pic. 1): ( 0 𝑃3 0 ) , ( 0 𝑃3 0 ) , ( 0 𝑃35 0 ),
0 0 𝑃34 0 0 𝑃31 0 0 𝑃35
2 6
𝑃3 0 0 𝑃3 0 0
( 0 𝑃32 0 ) , ( 0 𝑃36 0 ),
0 0 𝑃32 0 0 𝑃36
3 0
Pic.1: 𝑃32 (5,6) 𝑎𝑛𝑑 𝑃32 (0,1) 𝑃3 0 0 𝑃3 0 0
( 0 𝑃33 0 ) , ( 0 𝑃30 0 )
𝑃30 𝑃31 𝑃31 0 0 𝑃33 0 0 𝑃30
𝑃32 (0,1) = (𝑃31 0 0)=
𝑃31 𝑃31 0
1 0 0 1 1 1 1 1 1
(0 1 0) (1 0 0) (1 0 0)
0 0 1 1 1 0 1 1 0 Pic.3 : [𝑃32 (5,6)]𝑘 , 𝑘 = 73, 146, 219, 292, 365, 438, 511
1 1 1
(1 0 0) 0 0 (15) Also 𝑃32 (0,1) matrix is a primitive element, and elements
1 1 0 of the set [𝑃32 (0,1)]𝑘 , when k=73, 146, 219, 292, 365, 438,
1 1 1 1 1 1 511, are diagonal matrices(pic 4):
(1 0 0) (1 0 0) 0
( 1 1 0 1 1 0 ) 𝑃33 0 0 𝑃36 0 0 𝑃32 0 0
Consider 𝑃 = [𝑃32 (5,6)]2 = (0 𝑃33 6
0 ) , ( 0 𝑃3 0 ) , ( 0 𝑃32 0 ),
𝑃35 𝑃36 𝑃36 𝑃35 𝑃36 𝑃36
0 0 𝑃33 0 0 𝑃36 0 0 𝑃32
= (𝑃36 0 0 ) × (𝑃36 0 0 ) (16) 5
𝑃3 0 0 𝑃31 0 0
6 6 6 6
𝑃3 𝑃3 0 𝑃3 𝑃3 0 ( 0 𝑃35 0 ) , ( 0 𝑃31 0 )
If we take into consideration that the set 0 0 𝑃35 0 0 𝑃31
4 0
0,𝑃30 , 𝑃31 , 𝑃32 , 𝑃33 , 𝑃34 , 𝑃35 , 𝑃36 is a field, it is easy to assure that 𝑃3 0 0 𝑃3 0 0
each sub-matrix of the 𝑃 matrix is in the the same set: ( 0 𝑃34 0 ) , ( 0 𝑃30 0 )
𝑃1,1 = 𝑃35 × 𝑃35 + 𝑃36 × 𝑃36 + 𝑃36 × 𝑃36 = 𝑃33 , 0 0 𝑃34 0 0 𝑃30
𝑃1,2 = 𝑃35 × 𝑃36 + 𝑃36 × 0 + 𝑃36 × 𝑃36 = 𝑃32 ,
𝑃1,3 = 𝑃35 × 𝑃36 + 𝑃36 × 0 + 𝑃36 × 0 = 𝑃34 ,
𝑃2,1 = 𝑃36 × 𝑃35 + 0 × 𝑃36 + 0 × 𝑃36 = 𝑃34 ,
𝑃2,2 = 𝑃36 × 𝑃36 + 0 × 0 + 0 × 𝑃36 = 𝑃35 ,
6 6 5
𝑃2,3 = 𝑃3 × 𝑃3 + 0 × 0 + 0 × 0 = 𝑃3 ,
𝑃3,1 = 𝑃36 × 𝑃35 + 𝑃36 × 𝑃36 + 0 × 𝑃36 = 𝑃32 , pic.4: [𝑃32 (5,6)]𝑘 , 𝑘 = 73, 146, 219, 292, 365, 438, 511
85
Set of non-zero elements of the diagonal matrices represents the 𝑃302
𝑃312 𝑃312
perturbation of the group 𝑃30 , 𝑃31 , 𝑃32 , 𝑃33 , 𝑃34 , 𝑃35 , 𝑃36 (called as 1
𝑃33 (0,1) = (𝑃32 0 0 )=
primary group) and one of the elements is 𝑃30 . 1 1
Finally, we can conclude that empirically we proved the 𝑃32 𝑃32 0
0
following fact: 𝑃3 0 0 𝑃30 𝑃31 𝑃31 𝑃30 𝑃31 𝑃31
0
The second order (𝑖, 𝑖 + 1) expansion 𝑃32 (𝑖, 𝑖 + 1) , 𝑖 = 0. .5, ( 0 𝑃3 0 ) (𝑃31 0 0 ) (𝑃31 0 0)
of the matrix 𝑃3 is a primitive element and creates the Abelian 0 0 𝑃30 𝑃31 𝑃31 0 𝑃31 𝑃31 0
multiplicative finite group 𝐹(𝑃32 (𝑖, 𝑖 + 1)) , with the power 0 1
𝑃3 𝑃3 𝑃3 1
2
23 – 1. = (𝑃31 0 0) 0 0 (18)
Below we can see other primitive elements that are results of 1 1
𝑃3 𝑃3 0
expansion of 𝑃3 matrix:
𝑃30 𝑃31 𝑃31 𝑃30 𝑃31 𝑃31 𝑃30 𝑃31 𝑃31
𝑃32 (0,1) = (𝑃31 0 0 ), (𝑃31 0 0 ) (𝑃31 0 0) 0
1 1
𝑃31 𝑃31 0 ( 𝑃3 𝑃3 0 𝑃31 𝑃31 0 )
𝑃31 𝑃32 𝑃32
Let's consider [𝑃33 (0,1)]𝑘 set. It has the same basic
𝑃32 (1,2) = (𝑃32 0 0 ),
2 2
structure 𝑃3 as the primary group, as well as the first and second
𝑃3 𝑃3 0 expansion matrices taken from the primary group. It is expected
𝑃32 𝑃33 𝑃33 that this set is characterized by the same properties as the
𝑃32 (2,3) = (𝑃33 0 0) primary group has. Indeed, experimentally, it also has diagonal
𝑃33 𝑃33 0 matrices, whose diagonal elements represent one of the
perturbations of the primary group.
𝑃33 𝑃34 𝑃34
𝑃32 (3,4) = (𝑃34 0 0) For the set [𝑃33 (0,1)]𝑘 diagonal matrices are
2 2
𝑗∙(22∙3 +23 +1) 32
𝑃34 𝑃34 0 [𝑃33 (0,1)] , 𝑗 = 1,2,3, ⋯ , 2 − 1.
𝑃34 𝑃35 𝑃35 32
When 𝑗 = 2 − 1, we get the final element of the set
𝑃32 (4,5) = (𝑃35 0 0) [𝑃33 (0,1)]𝑘 :
𝑃35 𝑃35 0
𝑃35 𝑃36 𝑃36 32 2∙32 32
[𝑃33 (0,1)](2 −1)∙(2 +2 +1) = [𝑃33 (0,1)](2 −1) =
33
𝑃32 (5,6) = (𝑃36 0 0) [𝑃30 (0,1)]0 0 0
𝑃36 6
𝑃3 0 =( 0 [𝑃30 (0,1)] 0
0 ) (19)
𝑃36 𝑃30 𝑃30
0 0 [𝑃30 (0,1)]0
𝑃32 (6,0) = (𝑃30 0 0 ) (17)
𝑃30 0
𝑃3 0
We see that this is an Identity matrix. Therefore 𝑃33 (0,1)
In order to get higher order primitive elements, we still is a primitive element and creates the Abelian multiplicative
retain the structure of 𝑃3 matrix and put into the elements of the 3
finite group with power 23 − 1.
group 𝐹(𝑃32 (𝑖, 𝑖 + 1)) . We get 33 order matrix (call it a third
Definition: We call the following matrix
order expansion).
For example, if we use the elements of group 𝐹(𝑃32 (0,1))
𝑃3𝑖𝑘−1 𝑃3𝑖+1
𝑘−1 𝑃3𝑖+1
𝑘−1
for the first and second expansions of the matrix of 𝑃3 ,
respectively [𝑃32 (0,1)]0 [𝑃32 (0,1)]1 matrices, we get the 𝑃3𝑘 (𝑖, 𝑖 + 1) = (𝑃3𝑖+1
𝑘−1 0 0 ) (20)
following matrix (pic. 5): 𝑃3𝑖+1
𝑘−1 𝑃3𝑖+1
𝑘−1 0
where 𝑃3𝑖𝑘−1 ∈ 𝐹(𝑃3𝑘−1 (𝑖, 𝑖 + 1)), as the kth order (𝑖, 𝑖 + 1)
expansion of the 𝑃3 matrix.
Theorem: 𝑃3𝑘 (𝑖, 𝑖 + 1) is a primitive element and creates the
abelian multiplicative finite group 𝐹(𝑃3𝑘 (𝑖, 𝑖 + 1)) with
𝑘
power 23 – 1.
pic.5. 𝑃33 (0,1) 𝑗(22∙3
𝑘−1 𝑘−1
+23 +1)
In general matrices [𝑃3𝑘 (𝑖, 𝑖 + 1)] , where
𝑘−1
𝑗 = 1,2,3, ⋯ , 23 − 1 are diagonal matrices and diagonal
elements are one of the permutations of the elements of the
primary group.
86
𝑘−1
When 𝑗 = 23 − 1, we get one way function, that we offer. So the key exchange method is
𝑗(22∙3
𝑘−1
+23
𝑘−1
+1)=
got and it is secure against quantum computers attacks.
[𝑃3𝑘 (𝑖, 𝑖 + 1)]
𝑘−1 𝑘−1 𝑘−1
(23 −1)∙(22∙3 +23 +1)= ACKNOWLEDGEMENT
= [𝑃3𝑘 (𝑖, 𝑖 + 1)]
𝑘−1
(23 −1) The Work Was Conducted as a Part of Research Grant of Joint
= [𝑃3𝑘 (𝑖, 𝑖 + 1)]
0 Project of Shota Rustaveli National Science Foundation and
[𝑃3𝑘–1 (𝑖, 𝑖 + 1)] 0 0 Science & Technology Center in Ukraine [№ STCU-2016-08]
0
= 0 [𝑃3𝑘–1 (𝑖, 𝑖 + 1)] 0
0
( 0 0 [𝑃3𝑘–1 (𝑖, 𝑖 + 1)] ) REFERENCES
(21) [1]. Shor, P. Polynomial-time algorithms for prime factorization and
discrete logarithms on a quantum computer. SIAM J. Comput.
26, 1484–1509 (1997)
This means that (3) the structure is a primitive matrix. The [2]. Jones, J. A. NMR quantum computation. Prog. NMR Spectrosc. 38,
primitive matrices obtained have an interesting fractal structure 325–360 (2001)
[3]. Ekert, A. & Jozsa, R. Quantum computation and Shor's factoring
(see pic. 6). Abelian multiplicative groups adopted by the above algorithm. Rev. Mod. Phys. 68(3), 733–753 (1996).
mentioned method represent sufficient sets for realizing our [4]. Avtandil Gagnidze, Maksim Iavich, Giorgi Iashvili// Novel Version
one-way matrix functions of Merkle Cryptosystem// BULLETIN OF THE GEORGIAN
NATIONAL ACADEMY OF SCIENCES, vol. 11, no. 4, 2017, p.
28-3
[5]. Avtandil Gagnidze & Maksim Iavich & Giorgi Iashvili, 2017.
"Some Aspects Of Post-Quantum Cryptosystems," Eurasian Journal
of Business and Management, Eurasian Publications, vol. 5(1),
pages 16-20
[6]. Werner Alexi , Benny Chor , Oded Goldreich , Claus P. Schnorr,
RSA and Rabin functions: certain parts are as hard as the whole,
SIAM Journal on Computing, v.17 n.2, p.194-209, April 1988
[doi>10.1137/0217013]
[7]. R. P. Megrelishvili, Analysis of the matrix one-way function and
two variants of its implementation, International J. of
Multidisciplinary Research And Advances In Engineering
(IJMRAE), v.5, n. IV (October 2013), pp. 99-105.
[8]. G. L. Long, X. S. Liu , Theoretically efficient high-capacity
pic.6. 𝑃34 (0,1) quantum-key-distribution scheme, Phys. Rev. A 65, 2002
[9]. W. Shor, John Preskill, Simple Proof of Security of the BB84
Quantum Key Distribution Protocol Peter Phys. Rev. Lett. 85, 441,
IV. CONCLUSION 2000
[10]. Shu-Shen Li, Gui-Lu Long, Feng-Shan Bai, Song-Lin Feng and
Basic 𝑃3 matrix 𝑃3𝑘 (𝑖, 𝑖 + 1) expansions are primitive Hou-Zhi Zheng, Quantum computing, PNAS 2001 October, 98 (21)
matrices they generate abelian multiplicative matrix groups. 11847-11848. https://doi.org/10.1073/pnas.191373698
An interesting trend of research results in the idea: use the [11]. Marlan O. Scully and M. S. Zubairy, Quantum optical
elements of the primary field as the first and second expanding implementation of Grover's algorithm, PNAS 2001 August, 98 (17)
9490-9493. https://doi.org/10.1073/pnas.171317798
matrices with the same characteristic polynom. It is also
important the use of other baseline matrices, which enlarges a
new type of primitive structures. Elements of abelian
multiplicative matrix groups can be used in implementation of
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