This problem concerns the Galois field GF (2) with the irreducible polynomial P(x) = x+x+x+x+1. Let...

 

 

  

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This problem concerns the Galois field GF (2³) with the irreducible polynomial P(x) = x³+x+x³+x+1. Let A(x) = x+x³ and B(x) = x³+x³+1. (a) How many elements are in the field GF (2³)? (b) What is the maximum degree of any polynomials in GF(2³) (c) List all elements (i.e. polynomials) in GF (2³) with degree 3. (d) Compute A(x) + B(x) mod P(x). (e) Compute A(x) * B(x) mod P(x). (f) Find A¹(x) and B-¹(x) by using Table 4.2. (g) Verify your answer in (f) by performing GF (2³) multiplication. Your task is to compute the S-Box, i.e., the ByteSub, values for the input bytes 29, F3, and 01, where each byte is given in hexadecimal notation. (a) First, look up the inverse using Table 4.2 to obtain values B. (b) Now, perform the affine mapping by computing the matrix-vextor multiplication and addition. (c) Verify your result using the S-Box Table 4.3. (d) What is the value of S(0)? Table 4.2 Multiplicative inverse table in GF (28) for bytes xy used within the AES S-Box Y 0 1 2 3 4 5 6 7 89 A B C DE F 0 00 01 8D F6 CB 52 7B D1 E8 4F 29 C0 B0 E1 E5 C7 174 B4 AA 4B 99 2B 60 5F 58 3F FD CC FF 40 EE B2 23A 6E 5A F1 55 4D A8 C9 C1 0A 98 15 30 44 A2 C2 3 2c 45 92 6C F3 39 66 42 F2 35 20 6F 77 BB 59 19 4 1D FE 37 67 2D 31 F5 69 A7 64 AB 13 54 25 E9 09 5 ED 5C 05 CA 4C 24 87 BF 18 3E 22 F0 51 EC 61 17. 6 16 5E AF D3 49 A6 36 43 F4 47 91 DF 33 93 21 3B 7 79 B7 97 85 10 B5 BA 3C B6 70 DO 06 A1 FA 81 82 X 8 83 7E 7F 80 96 73 BE 56 9B 9E 95 D9 F7 02 B9 A4 9 DE 6A 32 6D D8 8A 84 72 2A 14 9F 88 F9 DC 89 9A A FB 7C 2E C3 8F B8 65 48 26 C8 12 4A CE E7 D2 62 BOC EO 1F EF 11 75 78 71 A5 8E 76 3D BD BC 86 57 COB 28 2F A3 DA D4 E4 OF A9 27 53 04 1B FC AC E6 D7A 07 AE 63 C5 DB E2 EA 94 8B C4 D5 9D F8 90 6B EB1 OD D6 EB C6 0E CF AD 08 4E D7 E3 5D 50 1E B3 F5B 23 38 34 68 46 03 8C DD 9C 7D A0 CD 1A 41 1c Table 4.3 AES S-Box: Substitution values in hexadecimal notation for input byte (xy) y 3 5 6 7 4 9 A B C D E F F2 6B 6F C5 30 01 67 2B FE D7 AB 76 7B 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 CO 3 04 C7 23 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84 6A CB BE 39 4A 4C 58 CF 4 5 53 D1 00 ED 20 FC B1 5B 6 DO EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8 7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2 X 8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73 9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE SE OB DB A EO 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79 BE7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08 CBA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A D 70 3E B5 66 48 03 F6 OE 61 35 57 B9 86 C1 ID 9E E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF F 8C AI 89 OD BF E6 42 68 41 99 2D OF B0 54 BB 16 0 1 2 63 7C 77 1 CA 82 C9 2 B7 FD 93 0 8 This problem concerns the Galois field GF (2³) with the irreducible polynomial P(x) = x³+x+x³+x+1. Let A(x) = x+x³ and B(x) = x³+x³+1. (a) How many elements are in the field GF (2³)? (b) What is the maximum degree of any polynomials in GF(2³) (c) List all elements (i.e. polynomials) in GF (2³) with degree 3. (d) Compute A(x) + B(x) mod P(x). (e) Compute A(x) * B(x) mod P(x). (f) Find A¹(x) and B-¹(x) by using Table 4.2. (g) Verify your answer in (f) by performing GF (2³) multiplication. Your task is to compute the S-Box, i.e., the ByteSub, values for the input bytes 29, F3, and 01, where each byte is given in hexadecimal notation. (a) First, look up the inverse using Table 4.2 to obtain values B. (b) Now, perform the affine mapping by computing the matrix-vextor multiplication and addition. (c) Verify your result using the S-Box Table 4.3. (d) What is the value of S(0)? Table 4.2 Multiplicative inverse table in GF (28) for bytes xy used within the AES S-Box Y 0 1 2 3 4 5 6 7 89 A B C DE F 0 00 01 8D F6 CB 52 7B D1 E8 4F 29 C0 B0 E1 E5 C7 174 B4 AA 4B 99 2B 60 5F 58 3F FD CC FF 40 EE B2 23A 6E 5A F1 55 4D A8 C9 C1 0A 98 15 30 44 A2 C2 3 2c 45 92 6C F3 39 66 42 F2 35 20 6F 77 BB 59 19 4 1D FE 37 67 2D 31 F5 69 A7 64 AB 13 54 25 E9 09 5 ED 5C 05 CA 4C 24 87 BF 18 3E 22 F0 51 EC 61 17. 6 16 5E AF D3 49 A6 36 43 F4 47 91 DF 33 93 21 3B 7 79 B7 97 85 10 B5 BA 3C B6 70 DO 06 A1 FA 81 82 X 8 83 7E 7F 80 96 73 BE 56 9B 9E 95 D9 F7 02 B9 A4 9 DE 6A 32 6D D8 8A 84 72 2A 14 9F 88 F9 DC 89 9A A FB 7C 2E C3 8F B8 65 48 26 C8 12 4A CE E7 D2 62 BOC EO 1F EF 11 75 78 71 A5 8E 76 3D BD BC 86 57 COB 28 2F A3 DA D4 E4 OF A9 27 53 04 1B FC AC E6 D7A 07 AE 63 C5 DB E2 EA 94 8B C4 D5 9D F8 90 6B EB1 OD D6 EB C6 0E CF AD 08 4E D7 E3 5D 50 1E B3 F5B 23 38 34 68 46 03 8C DD 9C 7D A0 CD 1A 41 1c Table 4.3 AES S-Box: Substitution values in hexadecimal notation for input byte (xy) y 3 5 6 7 4 9 A B C D E F F2 6B 6F C5 30 01 67 2B FE D7 AB 76 7B 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 CO 3 04 C7 23 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84 6A CB BE 39 4A 4C 58 CF 4 5 53 D1 00 ED 20 FC B1 5B 6 DO EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8 7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2 X 8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73 9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE SE OB DB A EO 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79 BE7 C8 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08 CBA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A D 70 3E B5 66 48 03 F6 OE 61 35 57 B9 86 C1 ID 9E E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF F 8C AI 89 OD BF E6 42 68 41 99 2D OF B0 54 BB 16 0 1 2 63 7C 77 1 CA 82 C9 2 B7 FD 93 0 8