(Daemen and Rijmen (1999); National Institute of Standards and Technology (NIST) (2001). Matrix operations utilize different methods of multiplication in the finite field (Chen and Huang (2020); Stepanov and Rose (2015); Wang et al. (1983); Mahboob and Ikram (2006); Reed and Chen (1999)). Thus, the methods are used in the encryption and decryption, such as the Rijndael method and the Twofish method (Schneier et al. (1998). In addition, due to attacks (Biryukov Khovratovich (2009) is a computation complexity 2 method by a known-key distinguishing attack in AES-128. Therefore, that can use the diversty of the matrix to enhance security (Wang et al. (2020). In this paper, the idea is using the diversity 8´8 circulant matrix, it has large branch number for diffusion more than 4´4 involutory matrix. Therefore, we propose an enhancement security method for AES encryption/decryption transformations with the Elliptic Curve Diffie–Hellman key exchange (ECDH) using elliptic curve in ANSI X9.62. Therefore, the AES key and first rows of the matrix can use ECDH method for exchanging both. The method will be more difficult for attacking which is to avoid the key by pre-computing the possible output to attacks. We propose method can be designed for a circuit (Selimis et al. (2006); Jing et al. (2007); Wang et al. (2016); Maximov (2019); Langenberg et al. (2020); Yang and Chien (2020), the circuit can decrease logic gates to use. Finally, using 8´8 circulant matrix is running for AES key of 128 bits on Intel(R) Core(TM) i7-8700 CPU. The reducing encryption time is above 79% faster than the 8´8 involutory matrix multiplication. Finally, the paper is in combining diversity AES and ECDH methods for security enhancement approaches to protect against new threats. The flowchart is as shown in Figure 1.
2. PRELIMINARIES 2.1. FINITE FIELD MULTIPLICATION: Let
and be two polynomials of
degree
Where
is a polynomial. The
equation (1) written in the program by
C language form as: unsigned char a, b, c; c=a ^ b; In this
paper, the symbol of is an XOR bitwise operation that is an
instruction of the CPU. It is not required an extra function to programming.
The multiplication modulo an irreducible polynomial given as,
Where
an irreducible polynomial is . In (2), using the Russian Peasant
multiplication, the powers of two in the decomposition of the multiplicand that
it on the left shift to discarding any remainder. The multiplication is writing
a programming code in the C language as follows:
In (2), using Horner rule,
the multiplication is writing a programming in the C language as follows:
2.2. MATRIX
MULTIPLICATION Let and be the polynomial equation of
degree
Where the polynomial
The
It can be
rewritten as from as follows:
Where the
The 2´2 circulant matrix is defined as , where . The 8´8 circulant matrix is defined as follows:
2.3. CIRCULANT MATRIX MULTIPLICATION BY 4´4 In Scheme 1, it computes 4´4 circulant
matrix method, there are
three items which can be precomputed in the program, so that the method only used 5 multiplications
and 15 additions, namely
3.
MATRIX
MULTIPLICATION IN AES MIXCOLUMNS OVER The polynomial product is accomplished with the polynomial in AES standard, where , . In MixColumns encoding, the matrix
it needs running four times
in AES MixColumns steps. The , the . In this
paper The circulant matrix
3.1. DECREASING MULTIPLICATIONS FOR 8´8 CIRCULANT MATRIX The 8´8 matrix
multiplication is defined as follows:
where , , , , , . Using the two-points convolution method instead of
8´8 matrix multiplication, it
can be shown a matrix and multiply the matrix .
3.2. REDUCING MULTIPLICATIONS BY MULTIPLY 2 According to (8), and . In the
finite field addition property is , where , then the
property can be also the same using in the matrix from additions. For example,
if adds , it is equal to because is zero and is also the same property.
Therefore, we rewrite in the equation (8) into the equation (9).
Where and . Let…., , and . Both the
matrix
Where , , , , and . Adding two entries of and are into matrix product
In (11), the matrix product can be also further simplified
by properties of addition over
Where , , , , and . Both the
matrix , and . The matrix
Where ,,,, , , , , , , , , , , ,, and . The matrix
Finally, the matrix
Therefore, Scheme 2, (i.e., MF()), for matrix . . .
That meaning require 8 additions, the total number
of the matrix 3.3. THE INVERSE OF 8´8 MATRIX FOR AES INVMIXCOLUMNS The others method, the inversion matrix
is given by mathematics,
Gaussian elimination is a method
for finding inverse of the matrix . The transpose of the matrix Therefore,
if the matrix Therefore, the first rows of matrix also has as shown below: The sum of the
coefficients of the polynomial is one, means . For example has property, the inverse
of matrix
4.
RESULT
AND DISCUSSIONS Using the multiplication based on several
algorithms in
In the
paper, using
For example,
how to share key got it. If Alice wants to establish a shared key with Bob, the
only channel available for them may be eavesdropped on by a hacker. Publicly,
the parameters; that is, the binary case in
5. CONCLUSIONS AND RECOMMENDATIONS In
summary, it is demonstrated herein that the computational complexity matrix
multiplication over ACKNOWLEDGEMENTS This study was supported in part by Taiwan’s Ministry
of Science and Technology MOST
110-2813-C-214-019-E and
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