Use of Linear Algebra in Cryptography

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Transcription:

Use of Linear Algebra in Cryptography Hong-Jian Lai Department of Mathematics West Virginia University Morgantown, WV p. 1/??

Creating a Matrix Problem: To create a matrix A = 1 2 3 4 5 6 7 8 10. p. 2/??

Creating a Matrix Problem: To create a matrix A = 1 2 3 4 5 6 7 8 10. Matlab: p. 2/??

Creating a Matrix Problem: To create a matrix A = 1 2 3 4 5 6 7 8 10. Matlab: A = [1 2 3;4 5 6;7 8 10] p. 2/??

Compute the inverse of A mod m Output setting: Need to set the output as "rational" using a matlab comment format rat; p. 3/??

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 Problem: Given A = 4 5 6, find A 1 mod 26. 7 8 10 p. 3/??

Step 1: Compute inverse over the reals Matlab commends: p. 4/??

Step 1: Compute inverse over the reals Matlab commends: format rat; p. 4/??

Step 1: Compute inverse over the reals Matlab commends: format rat; Ainv = inv(a) p. 4/??

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 p. 4/??

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. p. 4/??

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. p. 5/??

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) p. 5/??

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) 2 4 3 A1 = 2 11 6 3 6 3 p. 5/??

Step 3: Find the inverse of A (mod 26) Problem: In Step 2, we computed A1 = 3A 1 (mod 26). We need to multiply A1 by 3 1 to get A 1. p. 6/??

Step 3: Find the inverse of A (mod 26) Problem: In Step 2, we computed A1 = 3A 1 (mod 26). We need to multiply A1 by 3 1 to get A 1. Compute 3 1 9 (mod 26). p. 6/??

Step 3: Find the inverse of A (mod 26) Problem: In Step 2, we computed A1 = 3A 1 (mod 26). We need to multiply A1 by 3 1 to get A 1. Compute 3 1 9 (mod 26). Use matlab commend: M2 = round(mod(m1 9,26)) to find A 1 = A2 p. 6/??

Step 3: Find the inverse of A (mod 26) Problem: In Step 2, we computed A1 = 3A 1 (mod 26). We need to multiply A1 by 3 1 to get A 1. Compute 3 1 9 (mod 26). Use matlab commend: M2 = round(mod(m1 9,26)) to find A 1 = A2 8 16 1 A 1 := A2 = 8 21 24 1 24 1 p. 6/??

Example 1: The problem Problem Suppose that we have intercepted a cipher text HVSCSCBKRYMIUA, and intelligent agents informed us that this is encrypted by a block cipher using digraphs (size 2 blocks) with a mod 26 alphabet, and that the plain text of THRY has been encrypted as HVRY. Break this code by finding the plain text. p. 7/??

Example 1: Compute deciphering matrix, Step 1 Solution strategy: Compute the decoding matrix A 1 = P 1 C 1 1, where P 1 is given by plain text THRY and C 1 is by cipher text HVRY. p. 8/??

Example 1: Compute deciphering matrix, Step 1 Solution strategy: Compute the decoding matrix A 1 = P 1 C 1 1, where P 1 is given by plain text THRY and C 1 is by cipher text HVRY. format rat; C1 = [7 17;21 24],C3 = inv(c1) p. 8/??

Example 1: Compute deciphering matrix, Step 1 Solution strategy: Compute the decoding matrix A 1 = P 1 C 1 1, where P 1 is given by plain text THRY and C 1 is by cipher text HVRY. format rat; C1 = [7 17;21 24],C3 = inv(c1) 8/63 17/189 C3 = 1/9 1/27 p. 8/??

Example 1: Compute deciphering matrix, Step 1 Solution strategy: Compute the decoding matrix A 1 = P 1 C 1 1, where P 1 is given by plain text THRY and C 1 is by cipher text HVRY. format rat; C1 = [7 17;21 24],C3 = inv(c1) 8/63 17/189 C3 = 1/9 1/27 Observation: 189 is a common denominator. p. 8/??

Example 1: Compute deciphering matrix, Step 2 Compute 189 1 15 (mod 26) (using powermod or gcd). p. 9/??

Example 1: Compute deciphering matrix, Step 2 Compute 189 1 15 (mod 26) (using powermod or gcd). C4 = C3 189;C5 = round(mod(c4 15,26)) p. 9/??

Example 1: Compute deciphering matrix, Step 2 Compute 189 1 15 (mod 26) (using powermod or gcd). C4 = C3 189;C5 = round(mod(c4 15,26)) C5 = 4 21 3 25 p. 9/??

Example 1: Compute deciphering matrix Ainv, Step 3 Strategy review: A 1 = P 1 C 1 1, and C5 = C 1 1. p. 10/??

Example 1: Compute deciphering matrix Ainv, Step 3 Strategy review: A 1 = P 1 C 1 1, and C5 = C 1 1. P1 = [19 17;7 24];Ainv = mod(p1 C5,26) p. 10/??

Example 1: Compute deciphering matrix Ainv, Step 3 Strategy review: A 1 = P 1 C 1 1, and C5 = C 1 1. P1 = [19 17;7 24];Ainv = mod(p1 C5,26) 23 18 Ainv = 22 19 p. 10/??

Example 1: Compute deciphering matrix Ainv, Step 3 Strategy review: A 1 = P 1 C 1 1, and C5 = C 1 1. P1 = [19 17;7 24];Ainv = mod(p1 C5,26) 23 18 Ainv = 22 19 p. 10/??

Example 1: Decoding Decoding computation: P = A 1 C p. 11/??

Example 1: Decoding Decoding computation: P = A 1 C C = [7 18 18 1 17 12 20;21 2 2 10 24 8 0];P = mod(ainv C,26) p. 11/??

Example 1: Decoding Decoding computation: P = A 1 C C = [7 18 18 1 17 12 20;21 2 2 10 24 8 0];P = mod(ainv C,26) 19 8 8 21 17 4 18 P = 7 18 18 4 24 0 24 p. 11/??

Example 1: Decoding Decoding computation: P = A 1 C C = [7 18 18 1 17 12 20;21 2 2 10 24 8 0];P = mod(ainv C,26) 19 8 8 21 17 4 18 P = 7 18 18 4 24 0 24 Back to English: the message is THISISVERYEASY. p. 11/??