(* Title: HOL/SPARK/Examples/RIPEMD-160/Round.thy
Author: Fabian Immler, TU Muenchen
Verification of the RIPEMD-160 hash function
*)
theory Round
imports RMD_Specification
begin
spark_open "rmd/round"
abbreviation from_chain :: "chain \<Rightarrow> RMD.chain" where
"from_chain c \<equiv> (
word_of_int (h0 c),
word_of_int (h1 c),
word_of_int (h2 c),
word_of_int (h3 c),
word_of_int (h4 c))"
abbreviation from_chain_pair :: "chain_pair \<Rightarrow> RMD.chain \<times> RMD.chain" where
"from_chain_pair cc \<equiv> (
from_chain (left cc),
from_chain (right cc))"
abbreviation to_chain :: "RMD.chain \<Rightarrow> chain" where
"to_chain c \<equiv>
(let (h0, h1, h2, h3, h4) = c in
(|h0 = uint h0,
h1 = uint h1,
h2 = uint h2,
h3 = uint h3,
h4 = uint h4|))"
abbreviation to_chain_pair :: "RMD.chain \<times> RMD.chain \<Rightarrow> chain_pair" where
"to_chain_pair c == (let (c1, c2) = c in
(| left = to_chain c1,
right = to_chain c2 |))"
abbreviation steps' :: "chain_pair \<Rightarrow> int \<Rightarrow> block \<Rightarrow> chain_pair" where
"steps' cc i b == to_chain_pair (steps
(\<lambda>n. word_of_int (b (int n)))
(from_chain_pair cc)
(nat i))"
abbreviation round_spec :: "chain \<Rightarrow> block \<Rightarrow> chain" where
"round_spec c b == to_chain (round (\<lambda>n. word_of_int (b (int n))) (from_chain c))"
spark_proof_functions
steps = steps'
round_spec = round_spec
lemma uint_word_of_int_id:
assumes "0 <= (x::int)"
assumes "x <= 4294967295"
shows"uint(word_of_int x::word32) = x"
unfolding int_word_uint
using assms
by (simp add:int_mod_eq')
lemma steps_step: "steps X cc (Suc i) = step_both X (steps X cc i) i"
unfolding steps_def
by (induct i) simp_all
lemma from_to_id: "from_chain_pair (to_chain_pair CC) = CC"
proof (cases CC)
fix a::RMD.chain
fix b c d e f::word32
assume "CC = (a, b, c, d, e, f)"
thus ?thesis by (cases a) simp
qed
lemma steps_to_steps':
"F A (steps X cc i) B =
F A (from_chain_pair (to_chain_pair (steps X cc i))) B"
unfolding from_to_id ..
lemma steps'_step:
assumes "0 <= i"
shows
"steps' cc (i + 1) X = to_chain_pair (
step_both
(\<lambda>n. word_of_int (X (int n)))
(from_chain_pair (steps' cc i X))
(nat i))"
proof -
have "nat (i + 1) = Suc (nat i)" using assms by simp
show ?thesis
unfolding `nat (i + 1) = Suc (nat i)` steps_step steps_to_steps'
..
qed
lemma step_from_hyp:
assumes
step_hyp:
"\<lparr>left =
\<lparr>h0 = a, h1 = b, h2 = c, h3 = d, h4 = e\<rparr>,
right =
\<lparr>h0 = a', h1 = b', h2 = c', h3 = d', h4 = e'\<rparr>\<rparr> =
steps'
(\<lparr>left =
\<lparr>h0 = a_0, h1 = b_0, h2 = c_0,
h3 = d_0, h4 = e_0\<rparr>,
right =
\<lparr>h0 = a_0, h1 = b_0, h2 = c_0,
h3 = d_0, h4 = e_0\<rparr>\<rparr>)
j x"
assumes "a <= 4294967295" (is "_ <= ?M")
assumes "b <= ?M" and "c <= ?M" and "d <= ?M" and "e <= ?M"
assumes "a' <= ?M" and "b' <= ?M" and "c' <= ?M" and "d' <= ?M" and "e' <= ?M"
assumes "0 <= a " and "0 <= b " and "0 <= c " and "0 <= d " and "0 <= e "
assumes "0 <= a'" and "0 <= b'" and "0 <= c'" and "0 <= d'" and "0 <= e'"
assumes "0 <= x (r_l_spec j)" and "x (r_l_spec j) <= ?M"
assumes "0 <= x (r_r_spec j)" and "x (r_r_spec j) <= ?M"
assumes "0 <= j" and "j <= 79"
shows
"\<lparr>left =
\<lparr>h0 = e,
h1 =
(rotate_left (s_l_spec j)
((((a + f_spec j b c d) mod 4294967296 +
x (r_l_spec j)) mod
4294967296 +
k_l_spec j) mod
4294967296) +
e) mod
4294967296,
h2 = b, h3 = rotate_left 10 c,
h4 = d\<rparr>,
right =
\<lparr>h0 = e',
h1 =
(rotate_left (s_r_spec j)
((((a' + f_spec (79 - j) b' c' d') mod
4294967296 +
x (r_r_spec j)) mod
4294967296 +
k_r_spec j) mod
4294967296) +
e') mod
4294967296,
h2 = b', h3 = rotate_left 10 c',
h4 = d'\<rparr>\<rparr> =
steps'
(\<lparr>left =
\<lparr>h0 = a_0, h1 = b_0, h2 = c_0,
h3 = d_0, h4 = e_0\<rparr>,
right =
\<lparr>h0 = a_0, h1 = b_0, h2 = c_0,
h3 = d_0, h4 = e_0\<rparr>\<rparr>)
(j + 1) x"
using step_hyp
proof -
let ?MM = 4294967296
have AL: "uint(word_of_int e::word32) = e"
by (rule uint_word_of_int_id[OF `0 <= e` `e <= ?M`])
have CL: "uint(word_of_int b::word32) = b"
by (rule uint_word_of_int_id[OF `0 <= b` `b <= ?M`])
have DL: "True" ..
have EL: "uint(word_of_int d::word32) = d"
by (rule uint_word_of_int_id[OF `0 <= d` `d <= ?M`])
have AR: "uint(word_of_int e'::word32) = e'"
by (rule uint_word_of_int_id[OF `0 <= e'` `e' <= ?M`])
have CR: "uint(word_of_int b'::word32) = b'"
by (rule uint_word_of_int_id[OF `0 <= b'` `b' <= ?M`])
have DR: "True" ..
have ER: "uint(word_of_int d'::word32) = d'"
by (rule uint_word_of_int_id[OF `0 <= d'` `d' <= ?M`])
have BL:
"(uint
(word_rotl (s (nat j))
((word_of_int::int\<Rightarrow>word32)
((((a + f_spec j b c d) mod ?MM +
x (r_l_spec j)) mod ?MM +
k_l_spec j) mod ?MM))) +
e) mod ?MM
=
uint
(word_rotl (s (nat j))
(word_of_int a +
f (nat j) (word_of_int b)
(word_of_int c) (word_of_int d) +
word_of_int (x (r_l_spec j)) +
K (nat j)) +
word_of_int e)"
(is "(uint (word_rotl _ (_ ((((_ + ?F) mod _ + ?X) mod _ + _) mod _))) + _) mod _ = _")
proof -
have "a mod ?MM = a" using `0 <= a` `a <= ?M`
by (simp add: int_mod_eq')
have "?X mod ?MM = ?X" using `0 <= ?X` `?X <= ?M`
by (simp add: int_mod_eq')
have "e mod ?MM = e" using `0 <= e` `e <= ?M`
by (simp add: int_mod_eq')
have "(?MM::int) = 2 ^ len_of TYPE(32)" by simp
show ?thesis
unfolding
word_add_def
uint_word_of_int_id[OF `0 <= a` `a <= ?M`]
uint_word_of_int_id[OF `0 <= ?X` `?X <= ?M`]
int_word_uint
unfolding `?MM = 2 ^ len_of TYPE(32)`
unfolding word_uint.Abs_norm
by (simp add:
`a mod ?MM = a`
`e mod ?MM = e`
`?X mod ?MM = ?X`)
qed
have BR:
"(uint
(word_rotl (s' (nat j))
((word_of_int::int\<Rightarrow>word32)
((((a' + f_spec (79 - j) b' c' d') mod ?MM +
x (r_r_spec j)) mod ?MM +
k_r_spec j) mod ?MM))) +
e') mod ?MM
=
uint
(word_rotl (s' (nat j))
(word_of_int a' +
f (79 - nat j) (word_of_int b')
(word_of_int c') (word_of_int d') +
word_of_int (x (r_r_spec j)) +
K' (nat j)) +
word_of_int e')"
(is "(uint (word_rotl _ (_ ((((_ + ?F) mod _ + ?X) mod _ + _) mod _))) + _) mod _ = _")
proof -
have "a' mod ?MM = a'" using `0 <= a'` `a' <= ?M`
by (simp add: int_mod_eq')
have "?X mod ?MM = ?X" using `0 <= ?X` `?X <= ?M`
by (simp add: int_mod_eq')
have "e' mod ?MM = e'" using `0 <= e'` `e' <= ?M`
by (simp add: int_mod_eq')
have "(?MM::int) = 2 ^ len_of TYPE(32)" by simp
have nat_transfer: "79 - nat j = nat (79 - j)"
using nat_diff_distrib `0 <= j` `j <= 79`
by simp
show ?thesis
unfolding
word_add_def
uint_word_of_int_id[OF `0 <= a'` `a' <= ?M`]
uint_word_of_int_id[OF `0 <= ?X` `?X <= ?M`]
int_word_uint
nat_transfer
unfolding `?MM = 2 ^ len_of TYPE(32)`
unfolding word_uint.Abs_norm
by (simp add:
`a' mod ?MM = a'`
`e' mod ?MM = e'`
`?X mod ?MM = ?X`)
qed
show ?thesis
unfolding steps'_step[OF `0 <= j`] step_hyp[symmetric]
step_both_def step_r_def step_l_def
by (simp add: AL BL CL DL EL AR BR CR DR ER)
qed
spark_vc procedure_round_61
proof -
let ?M = "4294967295::int"
have step_hyp:
"\<lparr>left =
\<lparr>h0 = ca, h1 = cb, h2 = cc,
h3 = cd, h4 = ce\<rparr>,
right =
\<lparr>h0 = ca, h1 = cb, h2 = cc,
h3 = cd, h4 = ce\<rparr>\<rparr> =
steps'
(\<lparr>left =
\<lparr>h0 = ca, h1 = cb, h2 = cc,
h3 = cd, h4 = ce\<rparr>,
right =
\<lparr>h0 = ca, h1 = cb, h2 = cc,
h3 = cd, h4 = ce\<rparr>\<rparr>)
0 x"
unfolding steps_def
by (simp add:
uint_word_of_int_id[OF `0 <= ca` `ca <= ?M`]
uint_word_of_int_id[OF `0 <= cb` `cb <= ?M`]
uint_word_of_int_id[OF `0 <= cc` `cc <= ?M`]
uint_word_of_int_id[OF `0 <= cd` `cd <= ?M`]
uint_word_of_int_id[OF `0 <= ce` `ce <= ?M`])
let ?rotate_arg_l =
"((((ca + f 0 cb cc cd) mod 4294967296 +
x (r_l 0)) mod 4294967296 + k_l 0) mod 4294967296)"
let ?rotate_arg_r =
"((((ca + f 79 cb cc cd) mod 4294967296 +
x (r_r 0)) mod 4294967296 + k_r 0) mod 4294967296)"
note returns =
`wordops__rotate (s_l 0) ?rotate_arg_l =
rotate_left (s_l 0) ?rotate_arg_l`
`wordops__rotate (s_r 0) ?rotate_arg_r =
rotate_left (s_r 0) ?rotate_arg_r`
`wordops__rotate 10 cc = rotate_left 10 cc`
`f 0 cb cc cd = f_spec 0 cb cc cd`
`f 79 cb cc cd = f_spec 79 cb cc cd`
`k_l 0 = k_l_spec 0`
`k_r 0 = k_r_spec 0`
`r_l 0 = r_l_spec 0`
`r_r 0 = r_r_spec 0`
`s_l 0 = s_l_spec 0`
`s_r 0 = s_r_spec 0`
note x_borders = `\<forall>i. 0 \<le> i \<and> i \<le> 15 \<longrightarrow> 0 \<le> x i \<and> x i \<le> ?M`
from `0 <= r_l 0` `r_l 0 <= 15` x_borders
have "0 \<le> x (r_l 0)" by blast
hence x_lower: "0 <= x (r_l_spec 0)" unfolding returns .
from `0 <= r_l 0` `r_l 0 <= 15` x_borders
have "x (r_l 0) <= ?M" by blast
hence x_upper: "x (r_l_spec 0) <= ?M" unfolding returns .
from `0 <= r_r 0` `r_r 0 <= 15` x_borders
have "0 \<le> x (r_r 0)" by blast
hence x_lower': "0 <= x (r_r_spec 0)" unfolding returns .
from `0 <= r_r 0` `r_r 0 <= 15` x_borders
have "x (r_r 0) <= ?M" by blast
hence x_upper': "x (r_r_spec 0) <= ?M" unfolding returns .
have "0 <= (0::int)" by simp
have "0 <= (79::int)" by simp
note step_from_hyp [OF
step_hyp
H2 H4 H6 H8 H10 H2 H4 H6 H8 H10 (* upper bounds *)
H1 H3 H5 H7 H9 H1 H3 H5 H7 H9 (* lower bounds *)
]
from this[OF x_lower x_upper x_lower' x_upper' `0 <= 0` `0 <= 79`]
`0 \<le> ca` `0 \<le> ce` x_lower x_lower'
show ?thesis unfolding returns(1) returns(2) unfolding returns
by simp
qed
spark_vc procedure_round_62
proof -
let ?M = "4294967295::int"
let ?rotate_arg_l =
"((((cla + f (loop__1__j + 1) clb clc cld) mod 4294967296 +
x (r_l (loop__1__j + 1))) mod 4294967296 +
k_l (loop__1__j + 1)) mod 4294967296)"
let ?rotate_arg_r =
"((((cra + f (79 - (loop__1__j + 1)) crb crc crd) mod
4294967296 + x (r_r (loop__1__j + 1))) mod 4294967296 +
k_r (loop__1__j + 1)) mod 4294967296)"
have s: "78 - loop__1__j = (79 - (loop__1__j + 1))" by simp
note returns =
`wordops__rotate (s_l (loop__1__j + 1)) ?rotate_arg_l =
rotate_left (s_l (loop__1__j + 1)) ?rotate_arg_l`
`wordops__rotate (s_r (loop__1__j + 1)) ?rotate_arg_r =
rotate_left (s_r (loop__1__j + 1)) ?rotate_arg_r`
`f (loop__1__j + 1) clb clc cld =
f_spec (loop__1__j + 1) clb clc cld`
`f (78 - loop__1__j) crb crc crd =
f_spec (78 - loop__1__j) crb crc crd`[simplified s]
`wordops__rotate 10 clc = rotate_left 10 clc`
`wordops__rotate 10 crc = rotate_left 10 crc`
`k_l (loop__1__j + 1) = k_l_spec (loop__1__j + 1)`
`k_r (loop__1__j + 1) = k_r_spec (loop__1__j + 1)`
`r_l (loop__1__j + 1) = r_l_spec (loop__1__j + 1)`
`r_r (loop__1__j + 1) = r_r_spec (loop__1__j + 1)`
`s_l (loop__1__j + 1) = s_l_spec (loop__1__j + 1)`
`s_r (loop__1__j + 1) = s_r_spec (loop__1__j + 1)`
note x_borders = `\<forall>i. 0 \<le> i \<and> i \<le> 15 \<longrightarrow> 0 \<le> x i \<and> x i \<le> ?M`
from `0 <= r_l (loop__1__j + 1)` `r_l (loop__1__j + 1) <= 15` x_borders
have "0 \<le> x (r_l (loop__1__j + 1))" by blast
hence x_lower: "0 <= x (r_l_spec (loop__1__j + 1))" unfolding returns .
from `0 <= r_l (loop__1__j + 1)` `r_l (loop__1__j + 1) <= 15` x_borders
have "x (r_l (loop__1__j + 1)) <= ?M" by blast
hence x_upper: "x (r_l_spec (loop__1__j + 1)) <= ?M" unfolding returns .
from `0 <= r_r (loop__1__j + 1)` `r_r (loop__1__j + 1) <= 15` x_borders
have "0 \<le> x (r_r (loop__1__j + 1))" by blast
hence x_lower': "0 <= x (r_r_spec (loop__1__j + 1))" unfolding returns .
from `0 <= r_r (loop__1__j + 1)` `r_r (loop__1__j + 1) <= 15` x_borders
have "x (r_r (loop__1__j + 1)) <= ?M" by blast
hence x_upper': "x (r_r_spec (loop__1__j + 1)) <= ?M" unfolding returns .
from `0 <= loop__1__j` have "0 <= loop__1__j + 1" by simp
from `loop__1__j <= 78` have "loop__1__j + 1 <= 79" by simp
have "loop__1__j + 1 + 1 = loop__1__j + 2" by simp
note step_from_hyp[OF H1
`cla <= ?M`
`clb <= ?M`
`clc <= ?M`
`cld <= ?M`
`cle <= ?M`
`cra <= ?M`
`crb <= ?M`
`crc <= ?M`
`crd <= ?M`
`cre <= ?M`
`0 <= cla`
`0 <= clb`
`0 <= clc`
`0 <= cld`
`0 <= cle`
`0 <= cra`
`0 <= crb`
`0 <= crc`
`0 <= crd`
`0 <= cre`]
from this[OF
x_lower x_upper x_lower' x_upper'
`0 <= loop__1__j + 1` `loop__1__j + 1 <= 79`]
`0 \<le> cla` `0 \<le> cle` `0 \<le> cra` `0 \<le> cre` x_lower x_lower'
show ?thesis unfolding `loop__1__j + 1 + 1 = loop__1__j + 2`
unfolding returns(1) returns(2) unfolding returns
by simp
qed
spark_vc procedure_round_76
proof -
let ?M = "4294967295 :: int"
let ?INIT_CHAIN =
"\<lparr>h0 = ca_init, h1 = cb_init,
h2 = cc_init, h3 = cd_init,
h4 = ce_init\<rparr>"
have steps_to_steps':
"steps
(\<lambda>n\<Colon>nat. word_of_int (x (int n)))
(from_chain ?INIT_CHAIN, from_chain ?INIT_CHAIN)
80 =
from_chain_pair (
steps'
(\<lparr>left = ?INIT_CHAIN, right = ?INIT_CHAIN\<rparr>)
80
x)"
unfolding from_to_id by simp
from
`0 \<le> ca_init` `ca_init \<le> ?M`
`0 \<le> cb_init` `cb_init \<le> ?M`
`0 \<le> cc_init` `cc_init \<le> ?M`
`0 \<le> cd_init` `cd_init \<le> ?M`
`0 \<le> ce_init` `ce_init \<le> ?M`
`0 \<le> cla` `cla \<le> ?M`
`0 \<le> clb` `clb \<le> ?M`
`0 \<le> clc` `clc \<le> ?M`
`0 \<le> cld` `cld \<le> ?M`
`0 \<le> cle` `cle \<le> ?M`
`0 \<le> cra` `cra \<le> ?M`
`0 \<le> crb` `crb \<le> ?M`
`0 \<le> crc` `crc \<le> ?M`
`0 \<le> crd` `crd \<le> ?M`
`0 \<le> cre` `cre \<le> ?M`
show ?thesis
unfolding round_def
unfolding steps_to_steps'
unfolding H1[symmetric]
by (simp add: uint_word_ariths(1) rdmods
uint_word_of_int_id)
qed
spark_end
end