theory Dining_Cryptographers
imports "HOL-Probability.Information"
begin
lemma image_ex1_eq: "inj_on f A \<Longrightarrow> (b \<in> f ` A) \<longleftrightarrow> (\<exists>!x \<in> A. b = f x)"
by (unfold inj_on_def) blast
lemma Ex1_eq: "\<exists>!x. P x \<Longrightarrow> P x \<Longrightarrow> P y \<Longrightarrow> x = y"
by auto
subsection \<open>Define the state space\<close>
text \<open>
We introduce the state space on which the algorithm operates.
This contains:
\begin{description}
\item[n]
The number of cryptographers on the table.
\item[payer]
Either one of the cryptographers or the NSA.
\item[coin]
The result of the coin flipping for each cryptographer.
\item[inversion]
The public result for each cryptographer, e.g. the sum of the coin flipping
for the cryptographer, its right neighbour and the information if he paid or
not.
\end{description}
The observables are the \emph{inversions}
\<close>
locale dining_cryptographers_space =
fixes n :: nat
assumes n_gt_3: "n \<ge> 3"
begin
definition "dining_cryptographers =
({None} \<union> Some ` {0..<n}) \<times> {xs :: bool list. length xs = n}"
definition "payer dc = fst dc"
definition coin :: "(nat option \<times> bool list) \<Rightarrow> nat \<Rightarrow> bool" where
"coin dc c = snd dc ! (c mod n)"
definition "inversion dc =
map (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) [0..<n]"
definition "result dc = foldl (\<lambda> a b. a \<noteq> b) False (inversion dc)"
lemma coin_n[simp]: "coin dc n = coin dc 0"
unfolding coin_def by simp
theorem correctness:
assumes "dc \<in> dining_cryptographers"
shows "result dc \<longleftrightarrow> (payer dc \<noteq> None)"
proof -
let ?XOR = "\<lambda>f l. foldl (\<noteq>) False (map f [0..<l])"
have foldl_coin:
"\<not> ?XOR (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) n"
proof -
define n' where "n' = n" \<comment> \<open>Need to hide n, as it is hidden in coin\<close>
have "?XOR (\<lambda>c. coin dc c \<noteq> coin dc (c + 1)) n'
= (coin dc 0 \<noteq> coin dc n')"
by (induct n') auto
thus ?thesis using \<open>n' \<equiv> n\<close> by simp
qed
from assms have "payer dc = None \<or> (\<exists>k<n. payer dc = Some k)"
unfolding dining_cryptographers_def payer_def by auto
thus ?thesis
proof (rule disjE)
assume "payer dc = None"
thus ?thesis unfolding result_def inversion_def
using foldl_coin by simp
next
assume "\<exists>k<n. payer dc = Some k"
then obtain k where "k < n" and "payer dc = Some k" by auto
define l where "l = n" \<comment> \<open>Need to hide n, as it is hidden in coin, payer etc.\<close>
have "?XOR (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) l =
((k < l) \<noteq> ?XOR (\<lambda>c. (coin dc c \<noteq> coin dc (c + 1))) l)"
using \<open>payer dc = Some k\<close> by (induct l) auto
thus ?thesis
unfolding result_def inversion_def l_def
using \<open>payer dc = Some k\<close> foldl_coin \<open>k < n\<close> by simp
qed
qed
text \<open>
We now restrict the state space for the dining cryptographers to the cases when
one of the cryptographer pays.
\<close>
definition
"dc_crypto = dining_cryptographers - {None}\<times>UNIV"
lemma dc_crypto: "dc_crypto = Some ` {0..<n} \<times> {xs :: bool list. length xs = n}"
unfolding dc_crypto_def dining_cryptographers_def by auto
lemma image_payer_dc_crypto: "payer ` dc_crypto = Some ` {0..<n}"
proof -
have *: "{xs. length xs = n} \<noteq> {}"
by (auto intro!: exI[of _ "replicate n undefined"])
show ?thesis
unfolding payer_def [abs_def] dc_crypto fst_image_times if_not_P[OF *] ..
qed
lemma card_payer_and_inversion:
assumes "xs \<in> inversion ` dc_crypto" and "i < n"
shows "card {dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} = 2"
(is "card ?S = 2")
proof -
obtain ys j where xs_inv: "inversion (Some j, ys) = xs" and
"j < n" and "(Some j, ys) \<in> dc_crypto"
using assms(1) by (auto simp: dc_crypto)
hence "length ys = n" by (simp add: dc_crypto)
have [simp]: "length xs = n" using xs_inv[symmetric] by (simp add: inversion_def)
have inj_inv: "inj_on (\<lambda>x. inversion (Some i, x)) {ys. ys ! 0 = b \<and> length ys = length xs}"
for b
proof (rule inj_onI)
fix x y
assume "x \<in> {ys. ys ! 0 = b \<and> length ys = length xs}"
and "y \<in> {ys. ys ! 0 = b \<and> length ys = length xs}"
and inv: "inversion (Some i, x) = inversion (Some i, y)"
hence [simp]: "x ! 0 = y ! 0" "length y = n" "length x = n"
using \<open>length xs = n\<close> by simp_all
have *: "\<And>j. j < n \<Longrightarrow>
(x ! j = x ! (Suc j mod n)) = (y ! j = y ! (Suc j mod n))"
using inv unfolding inversion_def map_eq_conv payer_def coin_def
by fastforce
show "x = y"
proof (rule nth_equalityI, simp)
fix j assume "j < length x" hence "j < n" using \<open>length xs = n\<close> by simp
thus "x ! j = y ! j"
proof (induct j)
case (Suc j)
hence "j < n" by simp
with Suc show ?case using *[OF \<open>j < n\<close>]
by (cases "y ! j") simp_all
qed simp
qed
qed
txt \<open>
We now construct the possible inversions for \<^term>\<open>xs\<close> when the payer is
\<^term>\<open>i\<close>.
\<close>
define zs where "zs = map (\<lambda>p. if p \<in> {min i j<..max i j} then \<not> ys ! p else ys ! p) [0..<n]"
hence [simp]: "length zs = n" by simp
hence [simp]: "0 < length zs" using n_gt_3 by simp
have "\<And>l. l < max i j \<Longrightarrow> Suc l mod n = Suc l"
using \<open>i < n\<close> \<open>j < n\<close> by auto
have "((i = l) = (zs ! l = zs ! (Suc l mod n))) = ((j = l) = (ys ! l = ys ! (Suc l mod n)))"
if "l < n" for l
proof -
from that consider "l < min i j" | "l = min i j" | "min i j < l" "l < max i j"
| "l = max i j" | "max i j < l"
by linarith
thus ?thesis
proof cases
case 1
hence "l \<noteq> i" and "l \<noteq> j" and "zs ! l = ys ! l" and
"zs ! (Suc l mod n) = ys ! (Suc l mod n)" using \<open>i < n\<close> \<open>j < n\<close> unfolding zs_def by auto
thus ?thesis by simp
next
case 2
show ?thesis
proof (cases rule: linorder_cases)
assume "i < j"
hence "l = i" and "Suc l < n" and "i \<noteq> j" and "Suc l \<le> max i j" using \<open>l = min i j\<close> using \<open>j < n\<close> by auto
hence "zs ! l = ys ! l" and "zs ! (Suc l mod n) = (\<not> ys ! (Suc l mod n))"
using \<open>l = min i j\<close>[symmetric] by (simp_all add: zs_def)
thus ?thesis using \<open>l = i\<close> \<open>i \<noteq> j\<close> by simp
next
assume "j < i"
hence "l = j" and "Suc l < n" and "i \<noteq> j" and "Suc l \<le> max i j" using \<open>l = min i j\<close> using \<open>i < n\<close> by auto
hence "zs ! l = ys ! l" and "zs ! (Suc l mod n) = (\<not> ys ! (Suc l mod n))"
using \<open>l = min i j\<close>[symmetric] by (simp_all add: zs_def)
thus ?thesis using \<open>l = j\<close> \<open>i \<noteq> j\<close> by simp
next
assume "i = j"
hence "i = j" and "max i j = l" and "min i j = l" and "zs = ys"
using \<open>l = min i j\<close> by (simp_all add: zs_def \<open>length ys = n\<close>[symmetric] map_nth)
thus ?thesis by simp
qed
next
case 3
hence "i \<noteq> l" and "j \<noteq> l" and "zs ! l = (\<not> ys ! l)"
"zs ! (Suc l mod n) = (\<not> ys ! (Suc l mod n))"
using \<open>i < n\<close> \<open>j < n\<close> by (auto simp: zs_def)
thus ?thesis by simp
next
case 4
show ?thesis
proof (cases rule: linorder_cases)
assume "i < j"
hence "l = j" and "i \<noteq> j" using \<open>l = max i j\<close> using \<open>j < n\<close> by auto
have "zs ! (Suc l mod n) = ys ! (Suc l mod n)"
using \<open>j < n\<close> \<open>i < j\<close> \<open>l = j\<close> by (cases "Suc l = n") (auto simp add: zs_def)
moreover have "zs ! l = (\<not> ys ! l)"
using \<open>j < n\<close> \<open>i < j\<close> by (auto simp add: \<open>l = j\<close> zs_def)
ultimately show ?thesis using \<open>l = j\<close> \<open>i \<noteq> j\<close> by simp
next
assume "j < i"
hence "l = i" and "i \<noteq> j" using \<open>l = max i j\<close> by auto
have "zs ! (Suc l mod n) = ys ! (Suc l mod n)"
using \<open>i < n\<close> \<open>j < i\<close> \<open>l = i\<close> by (cases "Suc l = n") (auto simp add: zs_def)
moreover have "zs ! l = (\<not> ys ! l)"
using \<open>i < n\<close> \<open>j < i\<close> by (auto simp add: \<open>l = i\<close> zs_def)
ultimately show ?thesis using \<open>l = i\<close> \<open>i \<noteq> j\<close> by auto
next
assume "i = j"
hence "i = j" and "max i j = l" and "min i j = l" and "zs = ys"
using \<open>l = max i j\<close> by (simp_all add: zs_def \<open>length ys = n\<close>[symmetric] map_nth)
thus ?thesis by simp
qed
next
case 5
hence "j \<noteq> l" and "i \<noteq> l" by simp_all
have "zs ! (Suc l mod n) = ys ! (Suc l mod n)"
using \<open>l < n\<close> \<open>max i j < l\<close> by (cases "Suc l = n") (auto simp add: zs_def)
moreover have "zs ! l = ys ! l"
using \<open>l < n\<close> \<open>max i j < l\<close> by (auto simp add: zs_def)
ultimately show ?thesis using \<open>j \<noteq> l\<close> \<open>i \<noteq> l\<close> by auto
qed
qed
hence zs: "inversion (Some i, zs) = xs"
by (simp add: xs_inv[symmetric] inversion_def coin_def payer_def)
moreover
from zs have Not_zs: "inversion (Some i, (map Not zs)) = xs"
by (simp add: xs_inv[symmetric] inversion_def coin_def payer_def)
ultimately
have "{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} =
{(Some i, zs), (Some i, map Not zs)}"
using \<open>i < n\<close> [[ hypsubst_thin = true ]]
proof (safe, simp_all add:dc_crypto payer_def)
fix b assume [simp]: "length b = n"
and *: "inversion (Some i, b) = xs" and "b \<noteq> zs"
show "b = map Not zs"
proof (cases "b ! 0 = zs ! 0")
case True
hence zs: "zs \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, zs)"
using zs by simp
have b: "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, b)"
using * by simp
hence "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}" ..
with *[symmetric] have "xs \<in> (\<lambda>x. inversion (Some i, x)) ` {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}"
by (rule image_eqI)
from this[unfolded image_ex1_eq[OF inj_inv]] b zs
have "b = zs" by (rule Ex1_eq)
thus ?thesis using \<open>b \<noteq> zs\<close> by simp
next
case False
hence zs: "map Not zs \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, map Not zs)"
using Not_zs by (simp add: nth_map[OF \<open>0 < length zs\<close>])
have b: "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs} \<and> xs = inversion (Some i, b)"
using * by simp
hence "b \<in> {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}" ..
with *[symmetric] have "xs \<in> (\<lambda>x. inversion (Some i, x)) ` {ys. ys ! 0 = b ! 0 \<and> length ys = length xs}"
by (rule image_eqI)
from this[unfolded image_ex1_eq[OF inj_inv]] b zs
show "b = map Not zs" by (rule Ex1_eq)
qed
qed
moreover
have "zs \<noteq> map Not zs"
using \<open>0 < length zs\<close> by (cases zs) simp_all
ultimately show ?thesis by simp
qed
lemma finite_dc_crypto: "finite dc_crypto"
using finite_lists_length_eq[where A="UNIV :: bool set"]
unfolding dc_crypto by simp
lemma card_inversion:
assumes "xs \<in> inversion ` dc_crypto"
shows "card {dc \<in> dc_crypto. inversion dc = xs} = 2 * n"
proof -
let ?set = "\<lambda>i. {dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs}"
let ?sets = "{?set i | i. i < n}"
have [simp]: "length xs = n" using assms
by (auto simp: dc_crypto inversion_def [abs_def])
have "{dc \<in> dc_crypto. inversion dc = xs} = (\<Union>i < n. ?set i)"
unfolding dc_crypto payer_def by auto
also have "\<dots> = (\<Union>?sets)" by auto
finally have eq_Union: "{dc \<in> dc_crypto. inversion dc = xs} = (\<Union>?sets)" by simp
have card_double: "2 * card ?sets = card (\<Union>?sets)"
proof (rule card_partition)
show "finite ?sets" by simp
have "finite (?set i)" for i
using finite_dc_crypto by auto
thus "finite (\<Union>?sets)" by auto
next
fix c assume "c \<in> ?sets"
thus "card c = 2" using card_payer_and_inversion[OF assms] by auto
next
fix x y assume "x \<in> ?sets" and "y \<in> ?sets" "x \<noteq> y"
then obtain i j where xy: "x = ?set i" "y = ?set j" by auto
hence "i \<noteq> j" using \<open>x \<noteq> y\<close> by auto
thus "x \<inter> y = {}" using xy by auto
qed
have sets: "?sets = ?set ` {..< n}"
unfolding image_def by auto
have "?set i \<noteq> ?set j" if asm: "i \<noteq> j" "i < n" "j < n" for i j
proof -
have False if iasm: "?set i = {}"
proof -
have "card (?set i) = 2"
using card_payer_and_inversion[OF assms \<open>i < n\<close>] by auto
thus ?thesis using iasm by auto
qed
then obtain c where ci: "c \<in> ?set i" by blast
hence cj: "c \<notin> ?set j" using asm by auto
have False if "?set i = ?set j"
using that ci cj by auto
thus ?thesis by auto
qed
hence "inj_on ?set {..< n}" unfolding inj_on_def by auto
from card_image[OF this]
have "card (?set ` {..< n}) = n" by auto
hence "card ?sets = n" using sets by auto
thus ?thesis using eq_Union card_double by auto
qed
lemma card_dc_crypto:
"card dc_crypto = n * 2^n"
unfolding dc_crypto
using card_lists_length_eq[of "UNIV :: bool set"]
by (simp add: card_cartesian_product card_image)
lemma card_image_inversion:
"card (inversion ` dc_crypto) = 2^(n - 1)"
proof -
let ?P = "{inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto}"
have "\<Union>?P = dc_crypto" by auto
have inv_SOME: "inversion (SOME x. inversion x = inversion (a, b) \<and> x \<in> dc_crypto) = inversion (a, b)"
if "(a, b) \<in> dc_crypto" for a b
by (rule someI2) (auto simp: that)
have SOME_inv_dc: "(SOME x. inversion x = inversion (a, b) \<and> x \<in> dc_crypto) \<in> dc_crypto"
if "(a, b) \<in> dc_crypto" for a b
by (rule someI2) (auto simp: that)
have "bij_betw (\<lambda>s. inversion (SOME x. x \<in> s \<and> x \<in> dc_crypto))
{inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto}
(inversion ` dc_crypto)"
unfolding bij_betw_def
by (auto intro!: inj_onI image_eqI simp: inv_SOME SOME_inv_dc)
hence card_eq: "card {inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto} = card (inversion ` dc_crypto)"
by (rule bij_betw_same_card)
have "(2*n) * card (inversion ` dc_crypto) = card (\<Union>?P)"
unfolding card_eq[symmetric]
proof (rule card_partition)
have "\<Union>?P \<subseteq> dc_crypto" by auto
thus "finite (\<Union>?P)" using finite_dc_crypto by (auto intro: finite_subset)
have "?P = (\<lambda>x. inversion -` {x} \<inter> dc_crypto) ` (inversion ` dc_crypto)"
by auto
thus "finite ?P" using finite_dc_crypto by auto
next
fix c assume "c \<in> {inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto}"
then obtain x where "c = inversion -` {x} \<inter> dc_crypto" and x: "x \<in> inversion ` dc_crypto" by auto
hence "c = {dc \<in> dc_crypto. inversion dc = x}" by auto
thus "card c = 2 * n" using card_inversion[OF x] by simp
next
fix x y assume "x \<in> ?P" "y \<in> ?P" and "x \<noteq> y"
then obtain i j where
x: "x = inversion -` {i} \<inter> dc_crypto" and i: "i \<in> inversion ` dc_crypto" and
y: "y = inversion -` {j} \<inter> dc_crypto" and j: "j \<in> inversion ` dc_crypto" by auto
show "x \<inter> y = {}" using x y \<open>x \<noteq> y\<close> by auto
qed
hence "2 * card (inversion ` dc_crypto) = 2 ^ n" unfolding \<open>\<Union>?P = dc_crypto\<close> card_dc_crypto
using n_gt_3 by auto
thus ?thesis by (cases n) auto
qed
end
sublocale dining_cryptographers_space \<subseteq> prob_space "uniform_count_measure dc_crypto"
by (rule prob_space_uniform_count_measure[OF finite_dc_crypto])
(insert n_gt_3, auto simp: dc_crypto intro: exI[of _ "replicate n True"])
sublocale dining_cryptographers_space \<subseteq> information_space "uniform_count_measure dc_crypto" 2
by standard auto
notation (in dining_cryptographers_space)
mutual_information_Pow (\<open>\<I>'( _ ; _ ')\<close>)
notation (in dining_cryptographers_space)
entropy_Pow (\<open>\<H>'( _ ')\<close>)
notation (in dining_cryptographers_space)
conditional_entropy_Pow (\<open>\<H>'( _ | _ ')\<close>)
theorem (in dining_cryptographers_space)
"\<I>( inversion ; payer ) = 0"
proof (rule mutual_information_eq_0_simple)
have n: "0 < n" using n_gt_3 by auto
have card_image_inversion:
"real (card (inversion ` dc_crypto)) = 2^n / 2"
unfolding card_image_inversion using \<open>0 < n\<close> by (cases n) auto
show inversion: "simple_distributed (uniform_count_measure dc_crypto) inversion (\<lambda>x. 2 / 2^n)"
proof (rule simple_distributedI)
show "simple_function (uniform_count_measure dc_crypto) inversion"
using finite_dc_crypto
by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
fix x assume "x \<in> inversion ` space (uniform_count_measure dc_crypto)"
moreover have "inversion -` {x} \<inter> dc_crypto = {dc \<in> dc_crypto. inversion dc = x}" by auto
ultimately show "2 / 2^n = prob (inversion -` {x} \<inter> space (uniform_count_measure dc_crypto))"
using \<open>0 < n\<close>
by (simp add: card_inversion card_dc_crypto finite_dc_crypto
subset_eq space_uniform_count_measure measure_uniform_count_measure)
qed simp
show "simple_distributed (uniform_count_measure dc_crypto) payer (\<lambda>x. 1 / real n)"
proof (rule simple_distributedI)
show "simple_function (uniform_count_measure dc_crypto) payer"
using finite_dc_crypto
by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
fix z assume "z \<in> payer ` space (uniform_count_measure dc_crypto)"
then have "payer -` {z} \<inter> dc_crypto = {z} \<times> {xs. length xs = n}"
by (auto simp: dc_crypto payer_def space_uniform_count_measure cong del: image_cong_simp)
hence "card (payer -` {z} \<inter> dc_crypto) = 2^n"
using card_lists_length_eq[where A="UNIV::bool set"]
by (simp add: card_cartesian_product_singleton)
then show "1 / real n = prob (payer -` {z} \<inter> space (uniform_count_measure dc_crypto))"
using finite_dc_crypto
by (subst measure_uniform_count_measure) (auto simp add: card_dc_crypto space_uniform_count_measure)
qed simp
show "simple_distributed (uniform_count_measure dc_crypto) (\<lambda>x. (inversion x, payer x)) (\<lambda>x. 2 / (real n *2^n))"
proof (rule simple_distributedI)
show "simple_function (uniform_count_measure dc_crypto) (\<lambda>x. (inversion x, payer x))"
using finite_dc_crypto
by (auto simp: simple_function_def space_uniform_count_measure sets_uniform_count_measure)
fix x assume "x \<in> (\<lambda>x. (inversion x, payer x)) ` space (uniform_count_measure dc_crypto)"
then obtain i xs where x: "x = (inversion (Some i, xs), payer (Some i, xs))"
and "i < n" "length xs = n"
by (simp add: image_iff space_uniform_count_measure dc_crypto Bex_def) blast
then have xs: "inversion (Some i, xs) \<in> inversion`dc_crypto" and i: "Some i \<in> Some ` {0..<n}"
and x: "x = (inversion (Some i, xs), Some i)" by (simp_all add: payer_def dc_crypto)
moreover define ys where "ys = inversion (Some i, xs)"
ultimately have ys: "ys \<in> inversion`dc_crypto"
and "Some i \<in> Some ` {0..<n}" "x = (ys, Some i)" by simp_all
then have "(\<lambda>x. (inversion x, payer x)) -` {x} \<inter> space (uniform_count_measure dc_crypto) =
{dc \<in> dc_crypto. payer dc = Some (the (Some i)) \<and> inversion dc = ys}"
by (auto simp add: payer_def space_uniform_count_measure)
then show "2 / (real n * 2 ^ n) = prob ((\<lambda>x. (inversion x, payer x)) -` {x} \<inter> space (uniform_count_measure dc_crypto))"
using \<open>i < n\<close> ys
by (simp add: measure_uniform_count_measure card_payer_and_inversion finite_dc_crypto subset_eq card_dc_crypto)
qed simp
show "\<forall>x\<in>space (uniform_count_measure dc_crypto). 2 / (real n * 2 ^ n) = 2 / 2 ^ n * (1 / real n) "
by simp
qed
end