1. Announcements: Quizzes graded, but not in gradebook. (Current grade gives 0 on the parts you shouldn’t have done .) Quizzes graded, but not in gradebook. (Current grade gives 0 on the parts you shouldn’t have done .) No partial credit on decryptions No partial credit on decryptionsToday: Wrap up meet in the middle attacks Wrap up meet in the middle attacks DES Modes (5 min) DES Modes (5 min) Prep. for Rijndael and Discrete Logs: GF(2 8 ) Prep. for Rijndael and Discrete Logs: GF(2 8 ) Questions, like on DES? DTTF/NB479: DszquphsbqizDay 17

2. Triple-DES? Type DES computations ComparisonsMemory Brute force DES (3) Double C=E k2 (E k1 (P)) O(N) O(N 2 ) O(N) (1) Triple1 C=E k3 (E k2 (E k1 (P))) O(N 2 ) O(N 3 ) O(N 2 ) O(N 3 ) (2) Triple2 C=E k1 (E k2 (E k1 (P))) O(N 2 ) (3) Triple3 C=E k2 (E k1 ((E k1 (P)) O(N) O(N 2 ) O(N) Describe attacks on triple 1-3, fill out chart, and order by level of security

3. DES Modes of Operation Electronic codebook: Each block is encoded independently TextASCII bit vector Block 1 (64 bits) DES Encoded 1 (64 bits) Encoded bit vector Block 2 (64 bits) DES Encoded 2 (64 bits) …

4. DES Modes of Operation Cipher-block chaining: Each plaintext block is XOR’ed with the previous ciphertext before going into DES TextASCII bit vector Block 1 (64 bits) DES Encoded 1 (64 bits) Encoded bit vector Block 2 (64 bits) DES Encoded 2 (64 bits) + … ++ C0C0 (random)

5. DES Modes of Operation Others: Cipher feedback: similar, but 64-bit blocks overlap, giving k bits at a time (like 8 for 1 character at a time) Cipher feedback: similar, but 64-bit blocks overlap, giving k bits at a time (like 8 for 1 character at a time) Uses pseudorandom bits like LFSR Output feedback: similar but helps catch errors before propagate. Output feedback: similar but helps catch errors before propagate. Counter: Some output can be computed independently, so better for parallelizing Counter: Some output can be computed independently, so better for parallelizing I trust you could implement these if needed. Not part of HW4…

6. Rijndael 128-bit blocks Encrypted using functions of 128-bit key for 10 rounds Versions exist for keys with 192 bits (12 rounds), 256 bits (14 rounds) Versions exist for keys with 192 bits (12 rounds), 256 bits (14 rounds) The S-boxes, round keys, and MixColumn functions require the use of GF(2 8 ), so…

7. Lots of calcs coming up… Will want to follow in notebook

8. Fields (T&W, 3.11) A field is a set of numbers with the following properties: Addition, with identity: a + 0 = a and inverse a+(-a)=0 Addition, with identity: a + 0 = a and inverse a+(-a)=0 Multiplication with identity: a*1=a, and inverse (a * a -1 = 1 for all a != 0) Multiplication with identity: a*1=a, and inverse (a * a -1 = 1 for all a != 0) Subtraction and division (using inverses) Subtraction and division (using inverses) Commutative, associative, and distributive properties Commutative, associative, and distributive properties Closure over all four operations Closure over all four operationsExamples: Real numbers Real numbers GF(4) = {0, 1, ,  2 } with these additional laws: x + x = 0 for all x and  + 1 =  2. GF(4) = {0, 1, ,  2 } with these additional laws: x + x = 0 for all x and  + 1 =  2. GF(p n ) for prime p is called a Galois Field. GF(p n ) for prime p is called a Galois Field.

9. Are these fields? A field is a set of numbers with the following properties: Addition, with identity: a + 0 = a and inverse a+(-a)=0 Addition, with identity: a + 0 = a and inverse a+(-a)=0 Multiplication with identity: a*1=a, and inverse (a * a -1 = 1 for all a != 0) Multiplication with identity: a*1=a, and inverse (a * a -1 = 1 for all a != 0) Subtraction and division (using inverses) Subtraction and division (using inverses) Commutative, associative, and distributive properties Commutative, associative, and distributive properties Closure over all four operations Closure over all four operationsExamples: Real numbers Real numbers GF(4) = {0, 1, ,  2 } with these additional laws: x + x = 0 for all x and  + 1 =  2. GF(4) = {0, 1, ,  2 } with these additional laws: x + x = 0 for all x and  + 1 =  2. GF(p n ) for prime p is called a Galois Field. GF(p n ) for prime p is called a Galois Field. 1.Positive integers 2.Integers 3.Rational numbers 4.Complex numbers 5.The set of 2x2 matrices of real numbers? 6.Integers mod n (be careful here)

10. Galois fields For every power of n with prime p, there is only 1 finite field with p n elements. The integers (mod p n ) aren’t a field. (Why not?) Another way to get GF(4) = GF(2 2 ) using polynomials Technique extends to GF(2 8 ) Technique extends to GF(2 8 )

11. Galois fields If Z p [X] is set of polynomials with coefficients (mod p) …and P(X) is degree n and irreducible (mod p) Then GF(p n ) = Z p [X] (mod P(X) is a field with p n elements. Consider GF(2 n ) with P(X) = X 8 + X 4 + X 3 + X + 1 Rijndael uses this!