- Active work is being done to create and develop quantum computers. Google Corporation, NASA and the Universities Space Research Association (USRA) have teamed up with DWAFE, the manufacturer of quantum processors. D-Wave 2X is a quantum processor that contains 2,048 physical qubits. 1152 qubits from the whole number of qubits are used to perform the calculations. As we see, quantum computers can easily solve the problem of calculating the discrete logarithm used in DiffieHellman algorithm. So it can break Diffie-Hellman algorithm. When quantum computers are released all existing crypto systems will be useless, because there will be no way to transfer the key securely. In the article is proposed the new key exchange method using high dimensional matrix, this method is safe against attacks implemented using quantum computers. The case concerns the matrix function and algorithm for cryptographic keys exchange with open channel. For the algorithm is offered the method of building a high dimensional matrix multiplicative group. The arising of this goal is that traditional key exchange methods are vulnerable to quantum computer attacks.
But this problem can easily be solved by quantum computers using Shor algorithm [1,2].
The security of RSA algorithm relies on factorization problem, but this problem can be easily solved using quantum computers [3].
Active work is being done to create and develop quantum computers. Google Corporation, NASA and the Universities Space Research Association (USRA) have teamed up with DWAFE, the manufacturer of quantum processors. D-Wave 2X is a quantum processor that contains 2,048 physical qubits. 1152 qubits from the whole number of qubits are used to perform the calculations. As we see, quantum computers can easily solve the problem of calculating the discrete logarithm used in DiffieHellman algorithm. So it can break Diffie-Hellman algorithm. When quantum computers are released all existing crypto systems will be useless, because there will be no way to transfer the key securely [4,5].
In the article is proposed the new key exchange algorithm using high dimensional matrix. This algorithm is safe against quantum computer attacks.
The case concerns the matrix function and algorithm for cryptographic keys exchange with open channel.
methods are vulnerable to quantum computer attacks.
to calculate for any argument, but it is “difficult” to find an argument for the given value of the function. The word "difficult" is to understand the complexity of the computation. In other words, finding the relevant argument of the given function in real time is difficult even with the modern computing techniques. The irreversibility of function does not mean that the function is one-way [6,7].
The existence of one-way functions is the basis for the idea of asymmetric cryptography. It (one-way function) is the foundation of asymmetric cryptography, personal identification, authentication, and other fields of information protection. Although there is no theoretical proof of the existence of one-way functions in general, there are several “possible pretendents” (eg, multiplication and factorization, squaring and module rooting, discreet exponent and logarithmization), whose one-wayity (ie the difficulty of finding the argument for the value of function) at this time real and is actively used in information exchange protocols.
in the algorithms for developing a cryptographic open key. The initial idea (1976) belongs to Whitfield Duffie and Martin Helman. Based of their idea was established the first practical wel-known Diffie-Helman-Merkel method, which enabled the development of a common cryptographic key using the open (unprotected) channel. A year later, the first RSA algorithm of asymmetric encryption was formed. The RSA in fact, resolved the problem of exchange information with open channel. Both algorithms are not safe against quantum computers attacks. Are proposed quantum key exchange protocols, but quantum computers are needed to implement them [8,9].
The new one-way function for the development of common
cryptographic keys is based on high order cyclic matrix groups,
with the power = 2 − 1 , where the n is row dimension of
the square matrix. Let's assume that "A" is the above matrix
group, while A is the initial × matrix, then "A" = A={ , 2,
3, … , 2 −1 = } (
One-way function and algorithm for common key development are as follows: The sender chooses 1 ∈ A secret matrix to send to the receiving party via open channel the
u 1 = 1 (
2 = 2 (
= 1 2 = 2 1 (
Let = ( 1, 2, 3, ⋯ , )∈ (
One fact must be taken into consideration if the 1 matrix contains the internal recurrence, or if each of its rows are in a certain recurrence with the previous row, then the task of 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 oneway character of our function and requires the existence of Abelian multiplicative matrix group with a high order, that is free from the recurrence of the inside.
Let's consider (1 + ) , where j = 0,1,2, ⋯, and α represents the root of primitive polynomial in the (2 ) field odule with the module p (x).
(1 + )0 = 1 1 (1 + )1 = 1 + 11 (1 + )2 = 1 + 2 101 (1 + )3 = 1 + + 2 + 3 1111 (1 + )4 = 1 + 4 10001 (1 + )5 = 1 + + 4 + 5 110011 The polynomial coefficients generated the structure known as Serpinsky Triangle. The derived structure contains a number of sub-structures that can be used as a generator (generating matrix) for multiplicative groups, ie primitive elements. Such is, for example, Lets keep the structure of 3 matrix and extend it by elements 32( , )= ( 3
0 ) = Fore example, when = 5 and = 6, we have 0 ), where i,j=0..6. (13) 0 36 36 0 take into consideration that the set each sub-matrix of the
matrix is in the the same set:
0, 30, 31, 32, 33, 34, 35, 36 is a field, it is easy to assure that
1,1 = 35 × 35 + 36 × 36 + 36 × 36 = 33,
1,2 = 35 × 36 + 36 × 0+ 36 × 36 = 32,
1,3 = 35 × 36 + 36 × 0+ 36 × 0 = 34,
2,1 = 36 × 35 + 0× 36 + 0× 36 = 34,
2,2 = 36 × 36 + 0× 0+ 0× 36 = 35,
2,3 = 36 × 36 + 0× 0+ 0× 0 = 35,
3,1 = 36 × 35 + 36 × 36 + 0× 36 = 32,
3,2 = 36 × 36 + 36 × 0+ 0× 36 = 35,
365, 438, 511 are diagonal matrices (see pic. 3):
0 0 33
0
0
0 ),( 0
0 ),( 0
0
30
0
0
0
30
0 36
0 30
0
0 ),
0
0 )
Pic.3 : [ 32(
Also 32(
0
0
30
0 )
0
0 )
0 30
pic.4: [ 32(
Set of non-zero elements of the diagonal matrices represents the perturbation of the group 30, 31, 32, 33, 34, 35, 36 (called as Finally, we can conclude that empirically we proved the following fact:
of the matrix 3 is a primitive element and creates the Abelian
multiplicative finite group ( 32( , + 1)), with the power
Below we can see other primitive elements that are results of
0 ),
0 ),
0 )
0
0 )
0 )
0 )
0
32(
retain the structure of 3 matrix and put into the elements of the 2( , + 1)). We get 33 order matrix (call it a third following matrix (pic. 5):
For example, if we use the elements of group ( 32(
0 ) = 0 ) 0 ) 0 ) 0
0 ( 31 0 ) 0 0 ) 0 ( 31 31 0 31 0 0 0 ) 0 ) (18)
Let's consider [ 33(
For the set
32 − 1, we get the final element of the set
finite group with power 233 − 1.
Definition: We call the following matrix 3 ( , + 1)= ( 3+1 −1 )
(20) 3 −1 3+−11 3+−11
0 3+−11 3+−11 0 0 where
3 −1 ∈ ( 3 −1( , + 1)), as the kth order ( , + 1) expansion of the 3 matrix.
Theorem: 3 ( , + 1)is a primitive element and creates the multiplicative finite group ( 3 ( , + 1)) with abelian power 2 elements are one of the permutations of the elements of the (22∙3 −1+23 −1+1)= [ 3 ( , + 1)] = [ 3 ( , + 1)] = [ 3 ( , + 1)]
[ 3 –1( , + 1)] = ( (21)
primitive matrices obtained have an interesting fractal structure
(see pic. 6). Abelian multiplicative groups adopted by the above
mentioned method represent sufficient sets for realizing our
one-way matrix functions
pic.6. 34(
matrices they generate abelian multiplicative matrix groups.
An interesting trend of research results in the idea: use the elements of the primary field as the first and second expanding 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 one way function, that we offer. So the key exchange method is got and it is secure against quantum computers attacks.