(* Title: HOL/SPARK/Examples/RIPEMD-160/RMD.thy
Author: Fabian Immler, TU Muenchen
Verification of the RIPEMD-160 hash function
*)
theory RMD
imports Word
begin
(* all operations are defined on 32-bit words *)
type_synonym word32 = "32 word"
type_synonym byte = "8 word"
type_synonym perm = "nat \<Rightarrow> nat"
type_synonym chain = "word32 * word32 * word32 * word32 * word32"
type_synonym block = "nat \<Rightarrow> word32"
type_synonym message = "nat \<Rightarrow> block"
(* nonlinear functions at bit level *)
definition f::"[nat, word32, word32, word32] => word32"
where
"f j x y z =
(if ( 0 <= j & j <= 15) then x XOR y XOR z
else if (16 <= j & j <= 31) then (x AND y) OR (NOT x AND z)
else if (32 <= j & j <= 47) then (x OR NOT y) XOR z
else if (48 <= j & j <= 63) then (x AND z) OR (y AND NOT z)
else if (64 <= j & j <= 79) then x XOR (y OR NOT z)
else 0)"
(* added constants (hexadecimal) *)
definition K::"nat => word32"
where
"K j =
(if ( 0 <= j & j <= 15) then 0x00000000
else if (16 <= j & j <= 31) then 0x5A827999
else if (32 <= j & j <= 47) then 0x6ED9EBA1
else if (48 <= j & j <= 63) then 0x8F1BBCDC
else if (64 <= j & j <= 79) then 0xA953FD4E
else 0)"
definition K'::"nat => word32"
where
"K' j =
(if ( 0 <= j & j <= 15) then 0x50A28BE6
else if (16 <= j & j <= 31) then 0x5C4DD124
else if (32 <= j & j <= 47) then 0x6D703EF3
else if (48 <= j & j <= 63) then 0x7A6D76E9
else if (64 <= j & j <= 79) then 0x00000000
else 0)"
(* selection of message word *)
definition r_list :: "nat list"
where "r_list = [
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
7, 4, 13, 1, 10, 6, 15, 3, 12, 0, 9, 5, 2, 14, 11, 8,
3, 10, 14, 4, 9, 15, 8, 1, 2, 7, 0, 6, 13, 11, 5, 12,
1, 9, 11, 10, 0, 8, 12, 4, 13, 3, 7, 15, 14, 5, 6, 2,
4, 0, 5, 9, 7, 12, 2, 10, 14, 1, 3, 8, 11, 6, 15, 13
]"
definition r'_list :: "nat list"
where "r'_list = [
5, 14, 7, 0, 9, 2, 11, 4, 13, 6, 15, 8, 1, 10, 3, 12,
6, 11, 3, 7, 0, 13, 5, 10, 14, 15, 8, 12, 4, 9, 1, 2,
15, 5, 1, 3, 7, 14, 6, 9, 11, 8, 12, 2, 10, 0, 4, 13,
8, 6, 4, 1, 3, 11, 15, 0, 5, 12, 2, 13, 9, 7, 10, 14,
12, 15, 10, 4, 1, 5, 8, 7, 6, 2, 13, 14, 0, 3, 9, 11
]"
definition r :: perm
where "r j = r_list ! j"
definition r' :: perm
where "r' j = r'_list ! j"
(* amount for rotate left (rol) *)
definition s_list :: "nat list"
where "s_list = [
11, 14, 15, 12, 5, 8, 7, 9, 11, 13, 14, 15, 6, 7, 9, 8,
7, 6, 8, 13, 11, 9, 7, 15, 7, 12, 15, 9, 11, 7, 13, 12,
11, 13, 6, 7, 14, 9, 13, 15, 14, 8, 13, 6, 5, 12, 7, 5,
11, 12, 14, 15, 14, 15, 9, 8, 9, 14, 5, 6, 8, 6, 5, 12,
9, 15, 5, 11, 6, 8, 13, 12, 5, 12, 13, 14, 11, 8, 5, 6
]"
definition s'_list :: "nat list"
where "s'_list = [
8, 9, 9, 11, 13, 15, 15, 5, 7, 7, 8, 11, 14, 14, 12, 6,
9, 13, 15, 7, 12, 8, 9, 11, 7, 7, 12, 7, 6, 15, 13, 11,
9, 7, 15, 11, 8, 6, 6, 14, 12, 13, 5, 14, 13, 13, 7, 5,
15, 5, 8, 11, 14, 14, 6, 14, 6, 9, 12, 9, 12, 5, 15, 8,
8, 5, 12, 9, 12, 5, 14, 6, 8, 13, 6, 5, 15, 13, 11, 11
]"
definition s :: perm
where "s j = s_list ! j"
definition s' :: perm
where "s' j = s'_list ! j"
(* Initial value (hexadecimal *)
definition h0_0::word32 where "h0_0 = 0x67452301"
definition h1_0::word32 where "h1_0 = 0xEFCDAB89"
definition h2_0::word32 where "h2_0 = 0x98BADCFE"
definition h3_0::word32 where "h3_0 = 0x10325476"
definition h4_0::word32 where "h4_0 = 0xC3D2E1F0"
definition h_0::chain where "h_0 = (h0_0, h1_0, h2_0, h3_0, h4_0)"
definition step_l ::
"[ block,
chain,
nat
] => chain"
where
"step_l X c j =
(let (A, B, C, D, E) = c in
((* A *) E,
(* B *) word_rotl (s j) (A + f j B C D + X (r j) + K j) + E,
(* C *) B,
(* D *) word_rotl 10 C,
(* E *) D))"
definition step_r ::
"[ block,
chain,
nat
] \<Rightarrow> chain"
where
"step_r X c' j =
(let (A', B', C', D', E') = c' in
((* A' *) E',
(* B' *) word_rotl (s' j) (A' + f (79 - j) B' C' D' + X (r' j) + K' j) + E',
(* C' *) B',
(* D' *) word_rotl 10 C',
(* E' *) D'))"
definition step_both ::
"[ block, chain * chain, nat ] \<Rightarrow> chain * chain"
where
"step_both X cc j = (case cc of (c, c') \<Rightarrow>
(step_l X c j, step_r X c' j))"
definition steps::"[ block, chain * chain, nat] \<Rightarrow> chain * chain"
where "steps X cc i = foldl (step_both X) cc [0..<i]"
definition round::"[ block, chain ] \<Rightarrow> chain"
where "round X h =
(let (h0, h1, h2, h3, h4) = h in
let ((A, B, C, D, E), (A', B', C', D', E')) = steps X (h, h) 80 in
((* h0 *) h1 + C + D',
(* h1 *) h2 + D + E',
(* h2 *) h3 + E + A',
(* h3 *) h4 + A + B',
(* h4 *) h0 + B + C'))"
definition rmd_body::"[ message, chain, nat ] => chain"
where
"rmd_body X h i = round (X i) h"
definition rounds::"message \<Rightarrow> chain \<Rightarrow> nat \<Rightarrow> chain"
where
"rounds X h i = foldl (rmd_body X) h_0 [0..<i]"
definition rmd :: "message \<Rightarrow> nat \<Rightarrow> chain"
where
"rmd X len = rounds X h_0 len"
end