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hpke.auth.outsider-cca.proof
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hpke.auth.outsider-cca.proof
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Initial state
Game 1 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
let (the_sk: skey_t, the_pk: pkey_t) = (s <-R keypairseed_t; (skgen(s), pkgen(s))) in
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
return((let SetupAuthS_Some(enc_6: kemciph_t, ctx_7: context_t) = (let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = (k_2 <-R kemseed_t; AuthEncap_r(k_2, pk_3, the_sk)) in let KeySchedule_Some(ctx_6: context_t) = (let concat(key_6: key_t, nonce_6: nonce_t) = (r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6); concat(get1(r_3), get2(r_3))) in KeySchedule_Some(Context(key_6, nonce_6, nonce_zero)) else KeySchedule_None) in SetupAuthS_Some(enc_5, ctx_6) else SetupAuthS_None else SetupAuthS_None) in let Context_Seal_Some(ct_8: bitstring) = (let Context(key_7: key_t, nonce_7: nonce_t, seq_3: nonce_t) = ctx_7 in ct_7: bitstring <- (nonce_8: nonce_t <- xor(nonce_7, seq_3); Seal_inner(m, aad_9, key_7, nonce_8)); Context_Seal_Some(ct_7) else Context_Seal_None) in SealAuth_Some(enc_6, ct_8) else SealAuth_None else SealAuth_None))
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
get E(=pk_4, =the_pk, =enc_7, =c, =aad_10, =info_7) in
return(OpenAuth_None)
else
return((info_hash: bitstring <- info_7; let SetupAuthR_Some(ctx_9: context_t) = (let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in let KeySchedule_Some(ctx_8: context_t) = (let concat(key_8: key_t, nonce_9: nonce_t) = (r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_hash); concat(get1(r_4), get2(r_4))) in KeySchedule_Some(Context(key_8, nonce_9, nonce_zero)) else KeySchedule_None) in SetupAuthR_Some(ctx_8) else SetupAuthR_None else SetupAuthR_None) in OpenAuth_Some((let Context(key_9: key_t, nonce_10: nonce_t, seq_4: nonce_t) = ctx_9 in let injbot(pt_7: bitstring) = (nonce_11: nonce_t <- xor(nonce_10, seq_4); Open_inner(c, aad_10, key_9, nonce_11)) in Context_Open_Some(pt_7) else Context_Open_None else Context_Open_None)) else OpenAuth_None))
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
let SealAuth_Some(enc_star: kemciph_t, c_star: bitstring) = (pt_8: bitstring <- test(b, m0, m1); let SetupAuthS_Some(enc_9: kemciph_t, ctx_11: context_t) = (let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = (k_3 <-R kemseed_t; AuthEncap_r(k_3, pk_5, the_sk)) in let KeySchedule_Some(ctx_10: context_t) = (let concat(key_10: key_t, nonce_12: nonce_t) = (r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8); concat(get1(r_5), get2(r_5))) in KeySchedule_Some(Context(key_10, nonce_12, nonce_zero)) else KeySchedule_None) in SetupAuthS_Some(enc_8, ctx_10) else SetupAuthS_None else SetupAuthS_None) in let Context_Seal_Some(ct_10: bitstring) = (let Context(key_11: key_t, nonce_13: nonce_t, seq_5: nonce_t) = ctx_11 in ct_9: bitstring <- (nonce_14: nonce_t <- xor(nonce_13, seq_5); Seal_inner(pt_8, aad_11, key_11, nonce_14)); Context_Seal_Some(ct_9) else Context_Seal_None) in SealAuth_Some(enc_9, ct_10) else SealAuth_None else SealAuth_None) in
insert E(the_pk, pk_5, enc_star, c_star, aad_11, info_8);
return((enc_star, c_star))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying expand get, insert and prove unique annotations
- Expand get/insert for table E
yields
Game 2 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
{8}let (the_sk: skey_t, the_pk: pkey_t) = (s <-R keypairseed_t; (skgen(s), pkgen(s))) in
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
return({29}(let SetupAuthS_Some(enc_6: kemciph_t, ctx_7: context_t) = (let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = (k_2 <-R kemseed_t; AuthEncap_r(k_2, pk_3, the_sk)) in {41}let KeySchedule_Some(ctx_6: context_t) = {42}(let concat(key_6: key_t, nonce_6: nonce_t) = (r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6); concat(get1(r_3), get2(r_3))) in KeySchedule_Some(Context(key_6, nonce_6, nonce_zero)) else KeySchedule_None) in SetupAuthS_Some(enc_5, ctx_6) else SetupAuthS_None else SetupAuthS_None) in {79}let Context_Seal_Some(ct_8: bitstring) = {80}(let Context(key_7: key_t, nonce_7: nonce_t, seq_3: nonce_t) = ctx_7 in ct_7: bitstring <- (nonce_8: nonce_t <- xor(nonce_7, seq_3); Seal_inner(m, aad_9, key_7, nonce_8)); Context_Seal_Some(ct_7) else Context_Seal_None) in SealAuth_Some(enc_6, ct_8) else SealAuth_None else SealAuth_None))
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(E_2[u_2, u_3], E_3[u_2, u_3], E_4[u_2, u_3], E_5[u_2, u_3], E_6[u_2, u_3], E_7[u_2, u_3]) && (E_2[u_2, u_3] = pk_4) && (E_3[u_2, u_3] = the_pk) && (E_4[u_2, u_3] = enc_7) && (E_5[u_2, u_3] = c) && (E_6[u_2, u_3] = aad_10) && (E_7[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
return((info_hash: bitstring <- info_7; {194}let SetupAuthR_Some(ctx_9: context_t) = (let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in {205}let KeySchedule_Some(ctx_8: context_t) = {206}(let concat(key_8: key_t, nonce_9: nonce_t) = (r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_hash); concat(get1(r_4), get2(r_4))) in KeySchedule_Some(Context(key_8, nonce_9, nonce_zero)) else KeySchedule_None) in SetupAuthR_Some(ctx_8) else SetupAuthR_None else SetupAuthR_None) in OpenAuth_Some({241}(let Context(key_9: key_t, nonce_10: nonce_t, seq_4: nonce_t) = ctx_9 in let injbot(pt_7: bitstring) = (nonce_11: nonce_t <- xor(nonce_10, seq_4); Open_inner(c, aad_10, key_9, nonce_11)) in Context_Open_Some(pt_7) else Context_Open_None else Context_Open_None)) else OpenAuth_None))
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
{297} let SealAuth_Some(enc_star: kemciph_t, c_star: bitstring) = (pt_8: bitstring <- test(b, m0, m1); {307}let SetupAuthS_Some(enc_9: kemciph_t, ctx_11: context_t) = (let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = (k_3 <-R kemseed_t; AuthEncap_r(k_3, pk_5, the_sk)) in {319}let KeySchedule_Some(ctx_10: context_t) = {320}(let concat(key_10: key_t, nonce_12: nonce_t) = (r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8); concat(get1(r_5), get2(r_5))) in KeySchedule_Some(Context(key_10, nonce_12, nonce_zero)) else KeySchedule_None) in SetupAuthS_Some(enc_8, ctx_10) else SetupAuthS_None else SetupAuthS_None) in {357}let Context_Seal_Some(ct_10: bitstring) = {358}(let Context(key_11: key_t, nonce_13: nonce_t, seq_5: nonce_t) = ctx_11 in ct_9: bitstring <- (nonce_14: nonce_t <- xor(nonce_13, seq_5); Seal_inner(pt_8, aad_11, key_11, nonce_14)); Context_Seal_Some(ct_9) else Context_Seal_None) in SealAuth_Some(enc_9, ct_10) else SealAuth_None else SealAuth_None) in
E_2: pkey_t <- the_pk;
E_3: pkey_t <- pk_5;
E_4: kemciph_t <- enc_star;
E_5: bitstring <- c_star;
E_6: bitstring <- aad_11;
E_7: bitstring <- info_8;
return((enc_star, c_star))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying expand
- Expand if/find/let
- Simplify pattern (the_sk: skey_t, the_pk: pkey_t) (tuple expanded) at 8
- Remove let at 297
- Remove let at 307
- Remove let at 297
- Simplify pattern concat(key_10: key_t, nonce_12: nonce_t) (tuple expanded) at 320
- Remove else branch of let at 320
- Simplify pattern KeySchedule_Some(ctx_10: context_t) (tuple expanded) at 319
- Remove else branch of let at 319
- Simplify pattern SetupAuthS_Some(enc_9: kemciph_t, ctx_11: context_t) (tuple expanded) at 307
- Remove else branch of let at 307
- Simplify pattern Context(key_11: key_t, nonce_13: nonce_t, seq_5: nonce_t) (tuple expanded) at 358
- Remove else branch of let at 358
- Simplify pattern Context_Seal_Some(ct_10: bitstring) (tuple expanded) at 357
- Remove else branch of let at 357
- Simplify pattern SealAuth_Some(enc_star: kemciph_t, c_star: bitstring) (tuple expanded) at 297
- Remove else branch of let at 297
- Remove let at 194
- Simplify pattern concat(key_8: key_t, nonce_9: nonce_t) (tuple expanded) at 206
- Remove else branch of let at 206
- Simplify pattern KeySchedule_Some(ctx_8: context_t) (tuple expanded) at 205
- Remove else branch of let at 205
- Simplify pattern SetupAuthR_Some(ctx_9: context_t) (tuple expanded) at 194
- Remove else branch of let at 194
- Simplify pattern Context(key_9: key_t, nonce_10: nonce_t, seq_4: nonce_t) (tuple expanded) at 241
- Remove else branch of let at 241
- Remove let at 29
- Simplify pattern concat(key_6: key_t, nonce_6: nonce_t) (tuple expanded) at 42
- Remove else branch of let at 42
- Simplify pattern KeySchedule_Some(ctx_6: context_t) (tuple expanded) at 41
- Remove else branch of let at 41
- Simplify pattern SetupAuthS_Some(enc_6: kemciph_t, ctx_7: context_t) (tuple expanded) at 29
- Remove else branch of let at 29
- Simplify pattern Context(key_7: key_t, nonce_7: nonce_t, seq_3: nonce_t) (tuple expanded) at 80
- Remove else branch of let at 80
- Simplify pattern Context_Seal_Some(ct_8: bitstring) (tuple expanded) at 79
- Remove else branch of let at 79
yields
Game 3 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
the_pk: pkey_t <- pkgen(s);
the_sk: skey_t <- skgen(s);
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, the_sk) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ctx_6: context_t <- Context(key_6, nonce_6, nonce_zero);
ctx_7: context_t <- ctx_6;
enc_6: kemciph_t <- enc_5;
seq_3: nonce_t <- nonce_zero;
nonce_7: nonce_t <- nonce_6;
key_7: key_t <- key_6;
nonce_8: nonce_t <- xor(nonce_7, seq_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_7, nonce_8);
ct_8: bitstring <- ct_7;
return(SealAuth_Some(enc_6, ct_8))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(E_2[u_2, u_3], E_3[u_2, u_3], E_4[u_2, u_3], E_5[u_2, u_3], E_6[u_2, u_3], E_7[u_2, u_3]) && (E_2[u_2, u_3] = pk_4) && (E_3[u_2, u_3] = the_pk) && (E_4[u_2, u_3] = enc_7) && (E_5[u_2, u_3] = c) && (E_6[u_2, u_3] = aad_10) && (E_7[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
info_hash: bitstring <- info_7;
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_hash);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
ctx_8: context_t <- Context(key_8, nonce_9, nonce_zero);
ctx_9: context_t <- ctx_8;
seq_4: nonce_t <- nonce_zero;
nonce_10: nonce_t <- nonce_9;
key_9: key_t <- key_8;
nonce_11: nonce_t <- xor(nonce_10, seq_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_9, nonce_11) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
pt_8: bitstring <- test(b, m0, m1);
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, the_sk) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
ctx_10: context_t <- Context(key_10, nonce_12, nonce_zero);
ctx_11: context_t <- ctx_10;
enc_9: kemciph_t <- enc_8;
seq_5: nonce_t <- nonce_zero;
nonce_13: nonce_t <- nonce_12;
key_11: key_t <- key_10;
nonce_14: nonce_t <- xor(nonce_13, seq_5);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_11, nonce_14);
ct_10: bitstring <- ct_9;
c_star: bitstring <- ct_10;
enc_star: kemciph_t <- enc_9;
E_2: pkey_t <- the_pk;
E_3: pkey_t <- pk_5;
E_4: kemciph_t <- enc_star;
E_5: bitstring <- c_star;
E_6: bitstring <- aad_11;
E_7: bitstring <- info_8;
return((enc_star, c_star))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying remove assignments of findcond
- Remove assignments on ctx_11 (definition removed, all usages removed)
- Remove assignments on enc_9 (definition removed, all usages removed)
- Remove assignments on nonce_13 (definition removed, all usages removed)
- Remove assignments on key_11 (definition removed, all usages removed)
- Remove assignments on ct_10 (definition removed, all usages removed)
- Remove assignments on c_star (definition removed, all usages removed)
- Remove assignments on enc_star (definition removed, all usages removed)
- Remove assignments on E_2 (definition removed, all usages removed)
- Remove assignments on E_3 (definition removed, all usages removed)
- Remove assignments on E_4 (definition removed, all usages removed)
- Remove assignments on E_5 (definition removed, all usages removed)
- Remove assignments on E_6 (definition removed, all usages removed)
- Remove assignments on E_7 (definition removed, all usages removed)
- Remove assignments on info_hash (definition removed, all usages removed)
- Remove assignments on ctx_9 (definition removed, all usages removed)
- Remove assignments on nonce_10 (definition removed, all usages removed)
- Remove assignments on key_9 (definition removed, all usages removed)
- Remove assignments on ctx_7 (definition removed, all usages removed)
- Remove assignments on enc_6 (definition removed, all usages removed)
- Remove assignments on nonce_7 (definition removed, all usages removed)
- Remove assignments on key_7 (definition removed, all usages removed)
- Remove assignments on ct_8 (definition removed, all usages removed)
- Remove assignments on ctx_10 (definition removed, all usages removed)
- Remove assignments on ctx_8 (definition removed, all usages removed)
- Remove assignments on ctx_6 (definition removed, all usages removed)
yields
Game 4 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
the_pk: pkey_t <- pkgen(s);
the_sk: skey_t <- skgen(s);
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, the_sk) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
seq_3: nonce_t <- nonce_zero;
nonce_8: nonce_t <- {59}xor(nonce_6, seq_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_8);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
{104} find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(the_pk[u_3], pk_5[u_2, u_3], enc_8[u_2, u_3], aad_11[u_2, u_3], info_8[u_2, u_3], ct_9[u_2, u_3]) && {116}((the_pk[u_3] = pk_4) && (pk_5[u_2, u_3] = the_pk) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7)) then
return(OpenAuth_None)
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
seq_4: nonce_t <- nonce_zero;
nonce_11: nonce_t <- {196}xor(nonce_9, seq_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_11) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
pt_8: bitstring <- test(b, m0, m1);
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, the_sk) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
seq_5: nonce_t <- nonce_zero;
nonce_14: nonce_t <- {302}xor(nonce_12, seq_5);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_14);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying simplify [probability N^2 * P_pk_coll]
- Simplification pass
- Replaced xor(nonce_12, seq_5) with nonce_12 at 302
- Replaced xor(nonce_9, seq_4) with nonce_9 at 196
- Replaced ((the_pk[u_3] = pk_4) && (pk_5[u_2, u_3] = the_pk) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7)) with ((the_pk[u_3] = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7)) at 116
- Replaced defined condition the_pk[u_3], pk_5[u_2, u_3], enc_8[u_2, u_3], aad_11[u_2, u_3], info_8[u_2, u_3], ct_9[u_2, u_3] with info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3], the_pk[u_3] in find at 104
- Replaced xor(nonce_6, seq_3) with nonce_6 at 59
yields
Game 5 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
the_pk: pkey_t <- pkgen(s);
the_sk: skey_t <- skgen(s);
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, the_sk) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
seq_3: nonce_t <- nonce_zero;
nonce_8: nonce_t <- nonce_6;
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_8);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3], the_pk[u_3]) && (the_pk[u_3] = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
seq_4: nonce_t <- nonce_zero;
nonce_11: nonce_t <- nonce_9;
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_11) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
pt_8: bitstring <- test(b, m0, m1);
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, the_sk) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
seq_5: nonce_t <- nonce_zero;
nonce_14: nonce_t <- nonce_12;
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_14);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying move all binders
- Move assignment to pt_8
- Move assignment to seq_4
yields
Game 6 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
the_pk: pkey_t <- pkgen(s);
the_sk: skey_t <- skgen(s);
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, the_sk) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
seq_3: nonce_t <- nonce_zero;
nonce_8: nonce_t <- nonce_6;
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_8);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3], the_pk[u_3]) && (the_pk[u_3] = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
nonce_11: nonce_t <- nonce_9;
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_11) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, the_sk) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
seq_5: nonce_t <- nonce_zero;
nonce_14: nonce_t <- nonce_12;
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_14);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying remove assignments of findcond
- Remove assignments on seq_5 (definition removed, all usages removed)
- Remove assignments on nonce_14 (definition removed, all usages removed)
- Remove assignments on nonce_11 (definition removed, all usages removed)
- Remove assignments on seq_3 (definition removed, all usages removed)
- Remove assignments on nonce_8 (definition removed, all usages removed)
yields
Game 7 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
the_pk: pkey_t <- pkgen(s);
the_sk: skey_t <- skgen(s);
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, the_sk) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3], the_pk[u_3]) && (the_pk[u_3] = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, the_sk, pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, the_sk) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying remove assignments of binder the_sk
- Remove assignments on the_sk (definition removed, all usages removed)
yields
Game 8 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
the_pk: pkey_t <- pkgen(s);
return(the_pk);
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, skgen(s)) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3], the_pk[u_3]) && (the_pk[u_3] = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, skgen(s), pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(the_pk[i']) && (the_pk[i'] = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, skgen(s)) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying remove assignments of binder the_pk
- Remove assignments on the_pk (definition removed, all usages removed)
yields
Game 9 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
return(pkgen(s));
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, skgen(s)) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(s[u_3], info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3]) && (pkgen(s[u_3]) = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, skgen(s), pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(s[i']) && (pkgen(s[i']) = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_r(k_3, pk_5, skgen(s)) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying equivalence outsider_cca(AuthEncap) with variables: s -> s_1 [probability Adv_Outsider_CCA(time_1, N, #Ochall + #Oaenc, #Oadec)]
- Equivalence outsider_cca(AuthEncap) with variables: k_3 -> ks, k_2 -> ks, s -> s_1
yields
Game 10 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
return(pkgen(s));
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
k'_1 <-R kemkey_t;
{22} let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = (pk_R: pkey_t <- pk_3; find u_9 = ri_5 <= N suchthat defined(s[ri_5]) && (pk_R = pkgen(s[ri_5])) then let AuthEncap_tuple(k_4: kemkey_t, ce: kemciph_t) = AuthEncap_r(k_2, pk_R, skgen(s)) in k': kemkey_t <- cst_kemkey_t; E_8: pkey_t <- pkgen(s); AuthEncap_tuple(k'_1, ce) else AuthEncap_None else AuthEncap_r(k_2, pk_R, skgen(s))) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(s[u_3], info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3]) && (pkgen(s[u_3]) = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
{186} let AuthDecap_Some(shared_secret_4: kemkey_t) = (pk_S: pkey_t <- pk_4; cd: kemciph_t <- enc_7; find u_7 = ri_3 <= Qeperuser, u_8 = ri_4 <= N suchthat defined(k'_1[ri_3, ri_4], ce[ri_3, ri_4], pk_R[ri_3, ri_4], E_8[ri_3, ri_4]) && (E_8[ri_3, ri_4] = pk_S) && (pk_R[ri_3, ri_4] = pkgen(s)) && (ce[ri_3, ri_4] = cd) then AuthDecap_Some(k'_1[u_7, u_8]) orfind u_5 = ri_1 <= Qcperuser, u_6 = ri_2 <= N suchthat defined(k'_3[ri_1, ri_2], ce_1[ri_1, ri_2], pk_R_1[ri_1, ri_2], E_9[ri_1, ri_2]) && (E_9[ri_1, ri_2] = pk_S) && (pk_R_1[ri_1, ri_2] = pkgen(s)) && (ce_1[ri_1, ri_2] = cd) then AuthDecap_Some(k'_3[u_5, u_6]) else AuthDecap(cd, skgen(s), pk_S)) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(s[i']) && (pkgen(s[i']) = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
k'_3 <-R kemkey_t;
{361} let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = (pk_R_1: pkey_t <- pk_5; {366}find u_4 = ri <= N suchthat defined(s[ri]) && (pk_R_1 = pkgen(s[ri])) then let AuthEncap_tuple(k_5: kemkey_t, ce_1: kemciph_t) = AuthEncap_r(k_3, pk_R_1, skgen(s)) in k'_2: kemkey_t <- cst_kemkey_t; E_9: pkey_t <- pkgen(s); AuthEncap_tuple(k'_3, ce_1) else AuthEncap_None else AuthEncap_r(k_3, pk_R_1, skgen(s))) in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying expand [probability N^2 * P_pk_coll]
- Expand if/find/let
- Remove else branch of find at 366
- Simplify pattern AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) (tuple expanded) at 361
- Remove else branch of let at 361
- A single branch always succeeds in find at 366
- Simplify pattern AuthDecap_Some(shared_secret_4: kemkey_t) (tuple expanded) at 186
- Remove else branch of let at 186
- Simplify pattern AuthDecap_Some(shared_secret_4: kemkey_t) (tuple expanded) at 186
- Remove else branch of let at 186
- Simplify pattern AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) (tuple expanded) at 22
- Remove else branch of let at 22
yields
Game 11 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
return(pkgen(s));
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
k'_1 <-R kemkey_t;
pk_R: pkey_t <- pk_3;
find u_9 = ri_5 <= N suchthat defined(s[ri_5]) && (pk_R = pkgen(s[ri_5])) then
let AuthEncap_tuple(k_4: kemkey_t, ce: kemciph_t) = AuthEncap_r(k_2, pk_R, skgen(s)) in
k': kemkey_t <- cst_kemkey_t;
E_8: pkey_t <- pkgen(s);
enc_5: kemciph_t <- ce;
shared_secret_3: kemkey_t <- k'_1;
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_None in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
else
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_R, skgen(s)) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(s[u_3], info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3]) && (pkgen(s[u_3]) = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
pk_S: pkey_t <- pk_4;
cd: kemciph_t <- enc_7;
find u_7 = ri_3 <= Qeperuser, u_8 = ri_4 <= N suchthat defined(k'_1[ri_3, ri_4], ce[ri_3, ri_4], pk_R[ri_3, ri_4], E_8[ri_3, ri_4]) && (E_8[ri_3, ri_4] = pk_S) && (pk_R[ri_3, ri_4] = pkgen(s)) && (ce[ri_3, ri_4] = cd) then
shared_secret_4: kemkey_t <- k'_1[u_7, u_8];
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
orfind u_5 = ri_1 <= Qcperuser, u_6 = ri_2 <= N suchthat defined(k'_3[ri_1, ri_2], ce_1[ri_1, ri_2], pk_R_1[ri_1, ri_2], E_9[ri_1, ri_2]) && (E_9[ri_1, ri_2] = pk_S) && (pk_R_1[ri_1, ri_2] = pkgen(s)) && (ce_1[ri_1, ri_2] = cd) then
shared_secret_4: kemkey_t <- k'_3[u_5, u_6];
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(cd, skgen(s), pk_S) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(s[i']) && (pkgen(s[i']) = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
k'_3 <-R kemkey_t;
pk_R_1: pkey_t <- pk_5;
let AuthEncap_tuple(k_5: kemkey_t, ce_1: kemciph_t) = AuthEncap_r(k_3, pk_R_1, skgen(s)) in
k'_2: kemkey_t <- cst_kemkey_t;
E_9: pkey_t <- pkgen(s);
enc_8: kemciph_t <- ce_1;
shared_secret_5: kemkey_t <- k'_3;
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_None in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying remove assignments of findcond
- Remove assignments on pk_R_1 (definition removed, all usages removed)
- Remove assignments on k'_2 (definition removed, all usages removed)
- Remove assignments on shared_secret_5 (definition removed, all usages removed)
- Remove assignments on enc_8 (definition kept, array references kept)
- Remove assignments on pk_S (definition removed, all usages removed)
- Remove assignments on cd (definition removed, all usages removed)
- Remove assignments on shared_secret_4 (definition removed, all usages removed)
- Remove assignments on shared_secret_4 (definition removed, all usages removed)
- Remove assignments on pk_R (definition removed, all usages removed)
- Remove assignments on k' (definition removed, all usages removed)
- Remove assignments on enc_5 (definition removed, all usages removed)
- Remove assignments on shared_secret_3 (definition removed, all usages removed)
yields
Game 12 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
return(pkgen(s));
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
k'_1 <-R kemkey_t;
find u_9 = ri_5 <= N suchthat defined(s[ri_5]) && (pk_3 = pkgen(s[ri_5])) then
let AuthEncap_tuple(k_4: kemkey_t, ce: kemciph_t) = AuthEncap_r(k_2, pk_3, skgen(s)) in
E_8: pkey_t <- pkgen(s);
r_3: keys_t <- KeySchedule_auth(k'_1, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(ce, ct_7))
else
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_None in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
else
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, skgen(s)) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
) | (
foreach iad <= Qdperuser do
Oadec(pk_4: pkey_t, enc_7: kemciph_t, c: bitstring, aad_10: bitstring, info_7: bitstring) :=
find u = u_2 <= Qcperuser, u_1 = u_3 <= N suchthat defined(s[u_3], info_8[u_2, u_3], aad_11[u_2, u_3], ct_9[u_2, u_3], enc_8[u_2, u_3], i'_1[u_2, u_3]) && (pkgen(s[u_3]) = pk_4) && (i'_1[u_2, u_3] = i) && (enc_8[u_2, u_3] = enc_7) && (ct_9[u_2, u_3] = c) && (aad_11[u_2, u_3] = aad_10) && (info_8[u_2, u_3] = info_7) then
return(OpenAuth_None)
else
{271} find u_7 = ri_3 <= Qeperuser, u_8 = ri_4 <= N suchthat defined(pk_3[ri_3, ri_4], k'_1[ri_3, ri_4], ce[ri_3, ri_4], E_8[ri_3, ri_4]) && {280}((E_8[ri_3, ri_4] = pk_4) && (pk_3[ri_3, ri_4] = pkgen(s)) && (ce[ri_3, ri_4] = enc_7)) then
r_4: keys_t <- KeySchedule_auth(k'_1[u_7, u_8], info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
orfind u_5 = ri_1 <= Qcperuser, u_6 = ri_2 <= N suchthat defined(pk_5[ri_1, ri_2], k'_3[ri_1, ri_2], ce_1[ri_1, ri_2], E_9[ri_1, ri_2]) && {363}((E_9[ri_1, ri_2] = pk_4) && (pk_5[ri_1, ri_2] = pkgen(s)) && (ce_1[ri_1, ri_2] = enc_7)) then
r_4: keys_t <- KeySchedule_auth(k'_3[u_5, u_6], info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
let AuthDecap_Some(shared_secret_4: kemkey_t) = AuthDecap(enc_7, skgen(s), pk_4) in
r_4: keys_t <- KeySchedule_auth(shared_secret_4, info_7);
nonce_9: nonce_t <- get2(r_4);
key_8: key_t <- get1(r_4);
let injbot(pt_7: bitstring) = Open_inner(c, aad_10, key_8, nonce_9) in
return(OpenAuth_Some(Context_Open_Some(pt_7)))
else
return(OpenAuth_Some(Context_Open_None))
else
return(OpenAuth_None)
) | (
foreach ic <= Qcperuser do
Ochall(pk_5: pkey_t, m0: bitstring, m1: bitstring, aad_11: bitstring, info_8: bitstring) :=
find i'_1 = i' <= N suchthat defined(s[i']) && (pkgen(s[i']) = pk_5) then
if Length(m0) = Length(m1) then
k_3 <-R kemseed_t;
k'_3 <-R kemkey_t;
let AuthEncap_tuple(k_5: kemkey_t, ce_1: kemciph_t) = AuthEncap_r(k_3, pk_5, skgen(s)) in
E_9: pkey_t <- pkgen(s);
enc_8: kemciph_t <- ce_1;
r_5: keys_t <- KeySchedule_auth(k'_3, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((ce_1, ct_9))
else
let AuthEncap_tuple(shared_secret_5: kemkey_t, enc_8: kemciph_t) = AuthEncap_None in
r_5: keys_t <- KeySchedule_auth(shared_secret_5, info_8);
nonce_12: nonce_t <- get2(r_5);
key_10: key_t <- get1(r_5);
pt_8: bitstring <- test(b, m0, m1);
ct_9: bitstring <- Seal_inner(pt_8, aad_11, key_10, nonce_12);
return((enc_8, ct_9))
else
return(bottom)
else
return(bottom)
else
return(bottom)
))
Applying simplify [probability N^2 * P_pk_coll]
- Simplification pass
- Replaced ((E_9[ri_1, ri_2] = pk_4) && (pk_5[ri_1, ri_2] = pkgen(s)) && (ce_1[ri_1, ri_2] = enc_7)) with ((E_9[ri_1, ri_2] = pk_4) && (i'_1[ri_1, ri_2] = i) && (ce_1[ri_1, ri_2] = enc_7)) at 363
- Replaced defined condition pk_5[ri_1, ri_2], k'_3[ri_1, ri_2], ce_1[ri_1, ri_2], E_9[ri_1, ri_2] with k'_3[ri_1, ri_2], ce_1[ri_1, ri_2], i'_1[ri_1, ri_2], E_9[ri_1, ri_2] in find at 271
- Replaced ((E_8[ri_3, ri_4] = pk_4) && (pk_3[ri_3, ri_4] = pkgen(s)) && (ce[ri_3, ri_4] = enc_7)) with ((E_8[ri_3, ri_4] = pk_4) && (u_9[ri_3, ri_4] = i) && (ce[ri_3, ri_4] = enc_7)) at 280
- Replaced defined condition pk_3[ri_3, ri_4], k'_1[ri_3, ri_4], ce[ri_3, ri_4], E_8[ri_3, ri_4] with k'_1[ri_3, ri_4], ce[ri_3, ri_4], u_9[ri_3, ri_4], E_8[ri_3, ri_4] in find at 271
yields
Game 13 is
Ostart() :=
b <-R bool;
return();
foreach i <= N do
Osetup() :=
s <-R keypairseed_t;
return(pkgen(s));
((
foreach iae <= Qeperuser do
Oaenc(pk_3: pkey_t, m: bitstring, aad_9: bitstring, info_6: bitstring) :=
k_2 <-R kemseed_t;
k'_1 <-R kemkey_t;
find u_9 = ri_5 <= N suchthat defined(s[ri_5]) && (pk_3 = pkgen(s[ri_5])) then
let AuthEncap_tuple(k_4: kemkey_t, ce: kemciph_t) = AuthEncap_r(k_2, pk_3, skgen(s)) in
E_8: pkey_t <- pkgen(s);
r_3: keys_t <- KeySchedule_auth(k'_1, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(ce, ct_7))
else
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_None in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);
nonce_6: nonce_t <- get2(r_3);
key_6: key_t <- get1(r_3);
ct_7: bitstring <- Seal_inner(m, aad_9, key_6, nonce_6);
return(SealAuth_Some(enc_5, ct_7))
else
return(SealAuth_None)
else
let AuthEncap_tuple(shared_secret_3: kemkey_t, enc_5: kemciph_t) = AuthEncap_r(k_2, pk_3, skgen(s)) in
r_3: keys_t <- KeySchedule_auth(shared_secret_3, info_6);