Use of Linear Algebra in Cryptography
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1 Use of Linear Algebra in Cryptography Hong-Jian Lai Department of Mathematics West Virginia University Morgantown, WV p. 1/??
2 Creating a Matrix Problem: To create a matrix A = p. 2/??
3 Creating a Matrix Problem: To create a matrix A = Matlab: p. 2/??
4 Creating a Matrix Problem: To create a matrix A = Matlab: A = [1 2 3;4 5 6;7 8 10] p. 2/??
5 Compute the inverse of A mod m Output setting: Need to set the output as "rational" using a matlab comment format rat; p. 3/??
6 Compute the inverse of A mod m Output setting: Need to set the output as "rational" using a matlab comment format rat; Problem: Given A = 4 5 6, find A 1 mod p. 3/??
7 Step 1: Compute inverse over the reals Matlab commends: p. 4/??
8 Step 1: Compute inverse over the reals Matlab commends: format rat; p. 4/??
9 Step 1: Compute inverse over the reals Matlab commends: format rat; Ainv = inv(a) p. 4/??
10 Step 1: Compute inverse over the reals Matlab commends: format rat; Ainv = inv(a) 2/3 4/3 1 Ainv = 2/3 11/ p. 4/??
11 Step 1: Compute inverse over the reals Matlab commends: format rat; Ainv = inv(a) 2/3 4/3 1 Ainv = 2/3 11/ Observation: 3 is a common denominator. p. 4/??
12 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/??
13 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/??
14 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) A1 = p. 5/??
15 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/??
16 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 (mod 26). p. 6/??
17 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 (mod 26). Use matlab commend: M2 = round(mod(m1 9,26)) to find A 1 = A2 p. 6/??
18 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 (mod 26). Use matlab commend: M2 = round(mod(m1 9,26)) to find A 1 = A A 1 := A2 = p. 6/??
19 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/??
20 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/??
21 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/??
22 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/??
23 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/??
24 Example 1: Compute deciphering matrix, Step 2 Compute (mod 26) (using powermod or gcd). p. 9/??
25 Example 1: Compute deciphering matrix, Step 2 Compute (mod 26) (using powermod or gcd). C4 = C3 189;C5 = round(mod(c4 15,26)) p. 9/??
26 Example 1: Compute deciphering matrix, Step 2 Compute (mod 26) (using powermod or gcd). C4 = C3 189;C5 = round(mod(c4 15,26)) C5 = p. 9/??
27 Example 1: Compute deciphering matrix Ainv, Step 3 Strategy review: A 1 = P 1 C 1 1, and C5 = C 1 1. p. 10/??
28 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/??
29 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) Ainv = p. 10/??
30 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) Ainv = p. 10/??
31 Example 1: Decoding Decoding computation: P = A 1 C p. 11/??
32 Example 1: Decoding Decoding computation: P = A 1 C C = [ ; ];P = mod(ainv C,26) p. 11/??
33 Example 1: Decoding Decoding computation: P = A 1 C C = [ ; ];P = mod(ainv C,26) P = p. 11/??
34 Example 1: Decoding Decoding computation: P = A 1 C C = [ ; ];P = mod(ainv C,26) P = Back to English: the message is THISISVERYEASY. p. 11/??
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