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Date Submitted: 02/10/2014 10:28 AM
Use of Linear Algebra in Cryptography
Hong-Jian Lai Department of Mathematics West Virginia University Morgantown, WV
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Creating a Matrix
Problem: To create a matrix A = 4 5 6 . 7 8 10
1 2
3
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Creating a Matrix
Problem: To create a matrix A = 4 5 6 . 7 8 10 Matlab:
1 2
3
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Creating a Matrix
Problem: To create a matrix A = 4 5 6 . 7 8 10 Matlab: A = [1 2 3; 4 5 6; 7 8 10]
1 2
3
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Compute the inverse of A mod m
Output setting: Need to set the output as "rational" using a matlab comment format rat;
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Compute the inverse of A mod m
Output setting: Need to set the output as "rational" using a matlab comment format rat; 1 2 3 4 5 6 , find A−1 mod 26. Problem: Given A = 7 8 10
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Step 1: Compute inverse over the reals
Matlab commends:
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Step 1: Compute inverse over the reals
Matlab commends: format rat;
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Step 1: Compute inverse over the reals
Matlab commends: format rat; Ainv = inv(A)
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Step 1: Compute inverse over the reals
Matlab commends: format rat; Ainv = inv(A) −2/3 −4/3 1 Ainv = −2/3 11/3 −2 1 −2 1
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Step 1: Compute inverse over the reals
Matlab commends: format rat; Ainv = inv(A) −2/3 −4/3 1 Ainv = −2/3 11/3 −2 1 −2 1 Observation: 3 is a common denominator.
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Step 2: Making it all integral
Problem: Need to rationalize this matrix before we take modulo m. As every entry of Ainv has a common denominator 3, multiply by 3 to make it an integer valued matrix.
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Step 2: Making it all integral
Problem: Need to rationalize this matrix before we take modulo m. As every entry of Ainv has a common denominator 3, multiply by 3 to make it an integer valued matrix. A1=(Ainv*3)
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Step 2: Making it all integral
Problem: Need...