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2 (Cryptographic Hash Functions) H : {0, 1} {0, 1} l (Unkeyed hash function) (MDC: Manipulation Detection Code) (Keyed hash function) (MAC: Message Authentication Code) 2 / 37

3 OAEP (One-wayness) (Preimage resistance) (Second-preimage resistance) (Collision resistance) 3 / 37

4 (Preimage resistance) h H(M) = h M (Second-preimage resistance) M M M H(M ) = H(M) M (Collision resistance) M M H(M) = H(M ) M, M PR 2ndPR CR? M \=?? \=? H H H H H h h h PR 2ndPR CR O(2 l ) O(2 l ) O(2 l/2 ) 4 / 37

5 N 1 q 2 P 1 P = N 1 N N 2 N N (q 1) = N x q 1 i=1 ( 1 i ) N e x = 1 x + x2 2! x3 3! + 1 x 1 P q 1 e i N i=1 P 1 e q2 2N = e q 1 i=1 i N = e q(q 1) 2N e q2 2N 5 / 37

6 e q2 2N q2 1 P 2N ln(1 P ) 1 q 2N ln 1 P P = 1/2 q (2 ln 2) N 1.17 N 1.17 N 2 1/2 23 1/ / 37

7 l l/2 1/ / 37

8 = + (compression function) > (iterated hash function) F : {0, 1} n {0, 1} b {0, 1} n IV {0, 1} n b M padding IV F F F F H(M) 8 / 37

9 F CR H CR Length-Extension H(M 1 M 2 ) H(M 1 ) M 2 M 1 M 1 M 2 IV F F F F H(M 1 M 2 ) H(M 1 ) 9 / 37

10 (padding) M = (M 1, M 2,..., M m ), M i {0, 1} b M 1 M m 1 M m 10 0 MD-strengthening (named after Merkle & Damgård) M 1 M m 1 M m 00 0 M 10 / 37

11 MDx MD4, MD5; RIPEMD-160; SHA-1, SHA-224, SHA-256, SHA-384, SHA-512 (single block length) Preneel-Govaerts-Vandewalle (double block length) MDC-2, MDC-4, abreast/tandem Davies-Meyer 11 / 37

12 Secure Hash Standard (SHS) FIPS 180 (Federal Information Processing Standards) ( ) SHA (Secure Hash Algorithm, SHA-0 ) FIPS ( ) SHA-1 ( 1 ) FIPS ( ) SHA-1, SHA-256/384/512 FIPS 180-2, Change Notice ( ) SHA-224 FIPS ( ) SHA-1, SHA-224/256/384/ / 37

13 Secure Hash Standard (SHS) SHA-1 < SHA-224 < SHA-256 < SHA-384 < SHA-512 < SHA-256 SHA-224 SHA-512 SHA-384 IV F F F F 13 / 37

14 SHS M j... v j 1 v j 14 / 37

15 A B C D SHA-1 (80 ) SHA-256 (64 ) f j E ROL 5 ROL 30 A B C D E F G H Σ 1 Ch Σ 0 Maj Kj Wj K j W j K j f j 20 Ch, Parity, Maj, Parity Σ 0 (x) = ROR 2 (x) ROR 13 (x) ROR 22 (x) Σ 1 (x) = ROR 6 (x) ROR 11 (x) ROR 25 (x) 15 / 37

16 SHA-1 f j f j (u, v, w) = K j = Ch(u, v, w) = u v u w (0 j 19) Parity(u, v, w) = u v w (20 j 39) Maj(u, v, w) = u v u w v w (40 j 59) Parity(u, v, w) (60 j 79) 5a for 0 j 19 6ed9eba1 for 20 j 39 8f1bbcdc for 40 j 59 ca62c1d6 for 60 j efcdab89 98badcfe c3d2e1f0 16 / 37

17 M i = (M i,0, M i,1,..., M i,15 ), M i,j {0, 1} 32 (W 0, W 1,..., W r ) (M i,0, M i,1,..., M i,15 ) W j {0, 1} 32 ROL 1 SHA-1 W j M i,j SHA-256 W j σ 0 σ 1 Mi,j σ 0 (x) = ROR 7 (x) ROR 18 (x) SHR 3 (x) σ 1 (x) = ROR 17 (x) ROR 19 (x) SHR 10 (x) 17 / 37

18 SHA-1, SHA-224/256 M 1 z = M 10 d d M d z = z α α M 2 α = 64 M M / 37

19 SHA-0/1 H H(M) = H(M ) M, M Wang, et. al. (1997, 1998, 2004 ) ( ) SHA SHA SHA-224/256/384/ / 37

20 SHS NIST s Policy on Hash Functions (3/15/2006) SHA-2 (SHA-224/256/384/512) 2010 SHA-1 SHA-1 20 / 37

21 NIST Cryptographic Hash Algorithm Competition ( ) ( ) ( ) 224, 256, 384, / 37

22 NIST Hash Competition: (FIPS 186-2) (NIST SP A) HMAC (FIPS 198) DRBG (NIST SP ) ( ) Length-extension HMAC Joux Kelsey-Schneier 22 / 37

23 NIST Hash Competition: HMAC n/2 n k n/2 n n k s 2 s k 2 k 23 / 37

24 NIST Hash Competition: 64 (2008/10/31) 1 (51 ) (2008/12/10) 2 (14 ) (2009/07/24) 3 (5 ) (2010/12/09) winner (2012/10/02) 3 BLAKE (CHE) Grøstl (DNK) JH (SGP) Keccak (CHE) winner! Skein (USA) AURORA 1 Lesamnta 1 Luffa 2 24 / 37

25 F : {0, 1} n+t {0, 1} n t 2 H : {0, 1} {0, 1} n H x 1 x > 0 x = (x 1, x 2,..., x m ) x i = { t 1 for 1 i m 1 t 1 d for i = m and 0 d t 2 x i for 1 i m 1 z i = x i 0 d for i = m d 2 for i = m x = 0 m = 0 z 1 = 0 t 1 (0 2 ) 25 / 37

26 H(x) v 1 = F (0 n 0 z 1 ), v i = F (v i 1 1 z i ) for 2 i m + 1 H(x) = v m+1 0 z 1 1 z 2 1 z m 1 z m+1 0 n F v F F F 1 v 2 v v m m+1 1 F H 26 / 37

27 1 (1/3) H F x x, H(x) = H(x ), x x (z 1,..., z m+1 ), (z 1,..., z m +1 ) x, x 0 z 1 1 z 2 1 z m 1 z m+1 0 n F F F F H(x) 0 z 1 1 z 2 1 z m 1 z m +1 0 n F F F F H(x ) 27 / 37

28 1 (2/3) x = 0 x > 0 F (v m 1 z m+1 ) = F (0 n 0 z 1) F 28 / 37

29 1 (3/3) x > 0 Case 1. x x (mod t 1) z m+1 z m +1 F (v m 1 z m+1 ) = F (v m 1 z m +1 ) F Case 2. x x (mod t 1) Case 2a. x = x m = m z m+1 = z m+1 v m v m F (v m 1 z m+1 ) = F (v m 1 z m+1 ) F v m = v m z m z m v m 1 v m 1 F (v m 1 1 z m ) = F (v m 1 1 z m) F 2 i m + 1 z i = z i v i 1 = v i 1 z 1 z 1 F (0n 0 z 1 ) = F (0 n 0 z 1 ) F z 1 = z 1 x = x Case 2b. x x 29 / 37

30 MAC MAC MAC F : {0, 1} k {0, 1} {0, 1} n F Alice Bob K {0, 1} k Alice M {0, 1} tag = F (K, M) (M, tag) Bob tag = F (K, M) tag = tag? tag tag 30 / 37

31 MAC MAC = M 1 A tag 1. O M i {0, 1} (M, tag) M q tag q K {0, 1} k tag i = F (K, M i ) tag = F (K, M) M {M 1,..., M q } A A F MAC 31 / 37

32 MAC CBC-MAC... HMAC 32 / 37

33 CBC-MAC E K : {0, 1} n {0, 1} n K M = (M 1,..., M m ) M i {0, 1} n v 0 = 0 n v i = E K (M i v i 1 ) for 1 i m tag = v m M 1 M 2 M m 1 M m E K E K E K E K tag Length-Extension 33 / 37

34 Two Key CBC-MAC M 1 M 2 M m 1 M m E K E K E K E K E K v i = { E K (M i ) (i = 1) tag = E K (v m ) Length-Extension tag E K (M i v i 1 ) (2 i m) 2 E Two Key CBC-MAC 34 / 37

35 HMAC (MAC) M ipad=0x K H opad=0x5c5c5c... ipad opad H Length-Extension 35 / 37

36 NIST SP Hash DRBG V Hash Hash Hash HMAC DRBG Key V HMAC HMAC HMAC Key, V 36 / 37

37 1 1 2 MAC 37 / 37

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