swh:1:snp:44a5c6c33fe8f30e42c028fce02d1c96d6b89575
Tip revision: 03fd7f2c77df23d8f806e8b05d08b20b36f5d9d6 authored by Pierre-Yves Strub on 10 October 2017, 09:04:16 UTC
compile with up-to-date toolchain
compile with up-to-date toolchain
Tip revision: 03fd7f2
hashed_elgamal_generic.ec
require import Pred.
require import Distr.
require import Int.
require import Real.
require import FMap.
require import FSet.
require (*--*) CDH.
require (*--*) AWord.
require (*--*) ROM.
require (*--*) PKE.
(* The type of plaintexts: bitstrings of length k *)
type bits.
op k: int.
axiom k_pos: 0 < k.
op uniform: bits distr.
axiom uniformU: isuniform uniform.
axiom uniformX x: mu uniform ((=) x) = 1%r/(2^k)%r.
axiom uniformL: mu uniform True = 1%r.
op (^^): bits -> bits -> bits.
clone import AWord as Bitstring with
type word <- bits,
op length <- k,
op (^) <- (^^),
op Dword.dword <- uniform
proof
Dword.* by smt,
leq0_length by smt.
(* Upper bound on the number of calls to H *)
op qH: int.
axiom qH_pos: 0 < qH.
(* Assumption *)
clone import CDH.Set_CDH as SCDH with
op n <- qH.
import Group.
type pkey = group.
type skey = int.
type plaintext = bits.
type ciphertext = group * bits.
clone import PKE as PKE_ with
type pkey <- pkey,
type skey <- skey,
type plaintext <- plaintext,
type ciphertext <- ciphertext.
module type Hash = {
proc init(): unit
proc hash(x:group): bits
}.
module Hashed_ElGamal (H:Hash): Scheme = {
proc kg(): pkey * skey = {
var sk;
H.init();
sk = $[0..q-1];
return (g ^ sk, sk);
}
proc enc(pk:pkey, m:plaintext): ciphertext = {
var y, h;
y = $[0..q-1];
h = H.hash(pk ^ y);
return (g ^ y, h ^^ m);
}
proc dec(sk:skey, c:ciphertext): plaintext option = {
var gy, h, hm;
(gy, hm) = c;
h = H.hash(gy ^ sk);
return Option.Some (h ^^ hm);
}
}.
clone import ROM.ROM_BadCall as ROC with
type from <- group,
type to <- bits,
op dsample <- fun (x:group), uniform,
op qH <- qH
proof * by smt.
(* Adversary Definitions *)
module type Adversary (O:ARO) = {
proc choose(pk:pkey) : plaintext * plaintext
proc guess(c:ciphertext): bool
}.
(* Specializing and merging the hash function *)
module H:Hash = {
proc init(): unit = { RO.init(); Log.qs = FSet.empty; }
proc hash(x:group): bits = { var y; y = RO.o(x); return y; }
}.
(* The initial scheme *)
module S = Hashed_ElGamal(H).
(* Correctness result *)
hoare Correctness: Correctness(S).main: true ==> res.
proof. by proc; inline *; auto; progress; smt. qed.
(* The reduction *)
module SCDH_from_CPA(A:Adversary,O:Oracle): Top.SCDH.Adversary = {
module BA = A(Bound(O))
proc solve(gx:group, gy:group): group set = {
var m0, m1, h, b';
H.init();
(m0,m1) = BA.choose(gx);
h = $uniform;
b' = BA.guess(gy, h);
return Log.qs;
}
}.
(* We want to bound the probability of A winning CPA(Bounder(A,RO),S) in terms of
the probability of B = CDH_from_CPA(SCDH_from_CPA(A,RO)) winning CDH(B) *)
section.
declare module A: Adversary {RO, Log, OnBound.G1, OnBound.G2}.
axiom chooseL (O <: ARO {A}): islossless O.o => islossless A(O).choose.
axiom guessL (O <: ARO {A}) : islossless O.o => islossless A(O).guess.
local module BA = A(Bound(RO)).
local module G0 = {
var gxy:group
proc main(): bool = {
var m0, m1, c, b, b';
var x, y, h, gx;
H.init();
x = $[0..q-1];
y = $[0..q-1];
gx = g ^ x;
gxy = gx ^ y;
(m0,m1) = BA.choose(gx);
b = ${0,1};
h = H.hash(gxy);
c = (g ^ y, h ^^ (b ? m1 : m0));
b' = BA.guess(c);
return (b' = b);
}
}.
local equiv CPA_G0: CPA(S,BA).main ~ G0.main: ={glob A} ==> ={res}.
proof.
proc.
inline Hashed_ElGamal(H).kg Hashed_ElGamal(H).enc.
swap{1} 8 -5.
call (_: ={glob H, Log.qs}); first by sim.
wp; call (_: ={glob H}); first by sim.
wp; rnd.
call (_: ={glob H, Log.qs}); first by sim.
wp; do !rnd.
by call (_: true ==> ={glob H}); first by sim.
qed.
local lemma Pr_CPA_G0 &m:
Pr[CPA(S,BA).main() @ &m: res] = Pr[G0.main() @ &m: res]
by byequiv CPA_G0.
local module G1 = {
proc main() : bool = {
var m0, m1, c, b, b';
var x, y, h, gx, gxy;
H.init();
x = $[0..q-1];
y = $[0..q-1];
gx = g ^ x;
gxy = gx ^ y;
(m0,m1) = BA.choose(gx);
b = ${0,1};
h = $uniform;
c = (g ^ y, h ^^ (b ? m1 : m0));
b' = BA.guess(c);
return (b' = b);
}
}.
local module G2 = {
var gxy : group
proc main() : bool = {
var m0, m1, c, b, b';
var x, y, h, gx;
H.init();
x = $[0..q-1];
y = $[0..q-1];
gx = g ^ x;
gxy = gx ^ y;
(m0,m1) = BA.choose(gx);
b = ${0,1};
h = $uniform;
c = (g ^ y, h ^^ (b ? m1 : m0));
b' = BA.guess(c);
return mem G2.gxy Log.qs;
}
}.
local module D(H:ARO): ROC.Dist(H) = {
module A = A(H)
var y:int
var b:bool
var m0, m1:plaintext
proc a1(): group = {
var x, gxy, gx;
x = $[0..q-1];
y = $[0..q-1];
gx = g ^ x;
gxy = gx ^ y;
(m0,m1) = A.choose(gx);
b = ${0,1};
return gxy;
}
proc a2(x:bits): bool = {
var c, b';
c = (g ^ y, x ^^ (b ? m1 : m0));
b' = A.guess(c);
return (b' = b);
}
}.
local lemma G0_D &m: Pr[G0.main() @ &m: res] = Pr[OnBound.G0(D,RO).main() @ &m: res].
proof.
byequiv (_: ={glob A} ==> ={res})=> //.
proc.
inline OnBound.G0(D,RO).D.a1 OnBound.G0(D,RO).D.a2; wp.
conseq (_: _ ==> ={b'} /\ b{1} = D.b{2})=> //.
by inline H.hash; sim.
qed.
local lemma G1_D &m: Pr[G1.main() @ &m: res] = Pr[OnBound.G1(D,RO).main() @ &m: res].
proof.
byequiv (_: ={glob A} ==> ={res})=> //.
proc.
inline OnBound.G1(D,RO).D.a1 OnBound.G1(D,RO).D.a2; wp.
conseq (_: _ ==> ={b'} /\ b{1} = D.b{2})=> //.
by inline H.hash; sim.
qed.
local lemma G2_D &m: Pr[G2.main() @ &m: res] = Pr[OnBound.G2(D,RO).main() @ &m: res].
proof.
byequiv (_: ={glob A} ==> ={res})=> //.
proc.
inline OnBound.G2(D,RO).D.a1 OnBound.G2(D,RO).D.a2; wp.
conseq (_: _ ==> ={glob Log, b'} /\ b{1} = D.b{2} /\ G2.gxy{1} = x{2})=> //.
by inline H.hash; sim.
qed.
local lemma G0_G1_G2 &m:
Pr[G0.main() @ &m: res] <= Pr[G1.main() @ &m: res] + Pr[G2.main() @ &m: res].
proof.
rewrite (G0_D &m) (G1_D &m) (G2_D &m).
apply (OnBound.ROM_BadCall D _ _ &m).
by progress; proc; auto; call (chooseL H _)=> //; auto; progress; smt.
by progress; proc; call (guessL H _)=> //; auto.
qed.
local module G1' = {
proc main() : bool = {
var m0, m1, c, b, b';
var x, y, h, gx, gxy;
H.init();
x = $[0..q-1];
y = $[0..q-1];
gx = g ^ x;
gxy = gx ^ y;
(m0,m1) = BA.choose(gx);
b = ${0,1};
h = $uniform;
c = (g ^ y, h);
b' = BA.guess(c);
return (b' = b);
}
}.
local lemma G1_G1' &m: Pr[G1.main() @ &m: res] = Pr[G1'.main() @ &m: res].
proof.
byequiv (_: ={glob A} ==> ={res})=> //.
proc.
call (_: ={glob RO, glob Log}); first by sim.
wp; rnd (fun h, h ^^ if b then m1 else m0){1}; rnd.
call (_: ={glob RO, glob Log}); first by sim.
by inline H.init RO.init; auto; progress; smt.
qed.
local lemma Pr_G1' &m: Pr[G1'.main() @ &m: res] = 1%r/2%r.
proof.
cut RO_o_ll:= RO_o_ll _; first smt.
byphoare (_: true ==> res)=> //.
proc.
swap 7 3.
rnd ((=) b').
call (_: true);
first by progress; apply (guessL O).
by proc; sp; if=> //; wp; call (Log_o_ll RO _).
auto.
call (_: true);
first by progress; apply (chooseL O).
by proc; sp; if=> //; wp; call (Log_o_ll RO _).
by inline H.init RO.init; auto; progress; expect 3 smt.
qed.
local module G2' = {
var gxy : group
proc main() : bool = {
var m0, m1, c, b, b';
var x, y, h, gx;
H.init();
x = $[0..q-1];
y = $[0..q-1];
gx = g ^ x;
gxy = gx ^ y;
(m0,m1) = BA.choose(gx);
b = ${0,1};
h = $uniform;
c = (g ^ y, h);
b' = BA.guess(c);
return mem gxy Log.qs;
}
}.
local lemma G2_G2' &m: Pr[G2.main() @ &m: res] = Pr[G2'.main() @ &m: res].
proof.
byequiv (_: ={glob A} ==> ={res})=> //.
proc.
call (_: ={glob RO, glob Log}); first by sim.
wp; rnd (fun h, h ^^ if b then m1 else m0){1}; rnd.
call (_: ={glob RO, glob Log}); first by sim.
by inline H.init RO.init; auto; progress; smt.
qed.
local equiv G2'_SCDH: G2'.main ~ SCDH(SCDH_from_CPA(A,RO)).main:
={glob A} ==> res{1} = res{2} /\ card Log.qs{1} <= qH.
proof.
proc.
inline SCDH_from_CPA(A,RO).solve.
swap{2} 5 -4; swap{1} 7 3.
rnd{1}; wp.
seq 8 7: (={glob BA} /\
c{1} = (gy, h){2} /\
G2'.gxy{1} = g ^ (x * y){2} /\
card Log.qs{1} <= qH).
wp; rnd; call (_: ={glob H} /\ card Log.qs{1} <= qH).
by proc; sp; if=> //; inline Bound(RO).LO.o RO.o; auto; smt.
by inline H.init RO.init; auto; progress; smt.
call (_: ={glob H} /\ card Log.qs{1} <= qH).
by proc; sp; if=> //; inline Bound(RO).LO.o RO.o; auto; smt.
by skip; smt.
qed.
local lemma Pr_G2'_SCDH &m :
Pr[G2'.main() @ &m: res]
= Pr[SCDH(SCDH_from_CPA(A,RO)).main() @ &m : res]
by byequiv G2'_SCDH.
local lemma Reduction &m :
Pr[CPA(S,BA).main() @ &m : res] <=
1%r / 2%r + Pr[SCDH(SCDH_from_CPA(A,RO)).main() @ &m : res].
proof.
rewrite (Pr_CPA_G0 &m).
rewrite -(Pr_G1' &m) -(G1_G1' &m).
rewrite -(Pr_G2'_SCDH &m) -(G2_G2' &m).
by apply (G0_G1_G2 &m).
qed.
lemma mult_inv_le_r (x y z:real) :
0%r < x => (1%r / x) * y <= z => y <= x * z.
proof.
move=> lt0x ledivxyz.
cut:= mulrMle (1%r / x * y) z x _ _; [by smt | done |].
by rewrite -Real.Comm.Comm -Real.Assoc.Assoc -div_def 2:mul_div // smt.
qed.
(** Composing reduction from CPA to SCDH with reduction from SCDH to CDH *)
lemma Security &m:
Pr[CPA(S,A(Bound(RO))).main() @ &m: res] - 1%r / 2%r <=
qH%r * Pr[CDH.CDH(CDH_from_SCDH(SCDH_from_CPA(A,RO))).main() @ &m: res].
proof.
apply (Trans _ (Pr[SCDH(SCDH_from_CPA(A,RO)).main() @ &m: res]));
first smt.
apply mult_inv_le_r; first smt.
by apply (Top.SCDH.Reduction (SCDH_from_CPA(A,RO)) &m); apply qH_pos.
qed.
end section.
print axiom Security.