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dhkem.auth.outsider-auth-lr.proof
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dhkem.auth.outsider-auth-lr.proof
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Initial state
Game 1 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
return((skS: Z_t <- sk; {23}let (skE: Z_t, pkE_2: G_t) = (z <-R Z_t; (z, exp(g, z))) in dh_2: GG_t <- concatDH(exp(pk_R, skE), exp(pk_R, skS)); enc_1: bitstring <- Serialize(pkE_2); pkS: G_t <- exp(g, skS); kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS); zz_2: eae_output_t <- (key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2); info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2); ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info))); AuthEncap_tuple(zz_2, enc_1) else AuthEncap_None))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
return((skR: Z_t <- sk; let Serialize(pkE_3: G_t) = enc_2 in dh_3: GG_t <- concatDH(exp(pkE_3, skR), exp(pk_S, skR)); pkR_1: G_t <- exp(g, skR); kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S); zz_3: eae_output_t <- (key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3); info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3); ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1))); AuthDecap_Some(zz_3) else AuthDecap_None))
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying expand
- Expand if/find/let
- Simplify pattern (skE: Z_t, pkE_2: G_t) (tuple expanded) at 23
- Remove else branch of let at 23
yields
Game 2 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
skS: Z_t <- sk;
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
skE: Z_t <- z;
dh_2: GG_t <- concatDH(exp(pk_R, skE), exp(pk_R, skS));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, skS);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
skR: Z_t <- sk;
let Serialize(pkE_3: G_t) = enc_2 in
dh_3: GG_t <- concatDH(exp(pkE_3, skR), exp(pk_S, skR));
pkR_1: G_t <- exp(g, skR);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying remove assignments of findcond
- Remove assignments on skR (definition removed, all usages removed)
- Remove assignments on skS (definition removed, all usages removed)
- Remove assignments on skE (definition removed, all usages removed)
yields
Game 3 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
{116} dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying insert instruction find ie1_1 = ie1 <= Qeperuser, i1_3 = i1_2 <= N suchthat
defined(zz_2[ie1, i1_2], enc_1[ie1, i1_2], sk[i1_2], pk_R[ie1, i1_2])
&& (exp(g, sk) = pk_R[ie1, i1_2])
&& (exp(g, sk[i1_2]) = pk_S)
&& (enc_1[ie1, i1_2] = enc_2) then
orfind id1_1 = id1 <= Qdperuser suchthat
defined(random_br[id1], zz_3[id1], enc_2[id1], pk_S[id1])
&& (pk_S[id1] = pk_S)
&& (enc_2[id1] = enc_2) then
else
find i1_1 = i1 <= N suchthat
defined(sk[i1])
&& (pk_S = exp(g, sk[i1])) then
random_br <- true at occurrence 116
yields
Game 4 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
{116} find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], enc_1[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && {124}((exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (enc_1[ie1, i1_5] = enc_2)) then
dh_3: GG_t <- {150}concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
orfind id1_1 = id1 <= Qdperuser suchthat defined(random_br[id1], zz_3[id1], enc_2[id1], pk_S[id1]) && {229}((pk_S[id1] = pk_S) && (enc_2[id1] = enc_2)) then
dh_3: GG_t <- {245}concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
random_br: bool <- true;
dh_3: GG_t <- {329}concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying simplify
- Simplification pass
- Replaced concatDH(exp(pkE_3, sk), exp(pk_S, sk)) with concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk))) at 329
- Replaced ((pk_S[id1] = pk_S) && (enc_2[id1] = enc_2)) with ((pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3)) at 229
- Replaced concatDH(exp(pkE_3, sk), exp(pk_S, sk)) with concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk))) at 245
- Replaced defined condition random_br[id1], zz_3[id1], enc_2[id1], pk_S[id1] with random_br[id1], zz_3[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1] in find at 116
- Replaced ((exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (enc_1[ie1, i1_5] = enc_2)) with ((exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3)) at 124
- Replaced concatDH(exp(pkE_3, sk), exp(pk_S, sk)) with concatDH(exp(g, mult(z[ie1_1, i1_4], sk)), exp(g, mult(sk[i1_4], sk))) at 150
- Replaced defined condition zz_2[ie1, i1_5], enc_1[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5] with zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5] in find at 116
yields
Game 5 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
dh_3: GG_t <- concatDH(exp(g, mult(z[ie1_1, i1_4], sk)), exp(g, mult(sk[i1_4], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some({222}zz_3))
orfind id1_1 = id1 <= Qdperuser suchthat defined(random_br[id1], zz_3[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
random_br: bool <- true;
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying replace term at occurrence 222 with zz_2[ie1_1, i1_4]
yields
Game 6 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
dh_3: GG_t <- concatDH(exp(g, mult(z[ie1_1, i1_4], sk)), exp(g, mult(sk[i1_4], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(random_br[id1], zz_3[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
random_br: bool <- true;
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying remove assignments of useless
- Remove assignments on random_br (definition point kept, all usages removed)
- Remove assignments on zz_3 (definition point kept, all usages removed)
- Remove assignments on key_1 (definition removed, all usages removed)
- Remove assignments on info_1 (definition removed, all usages removed)
- Remove assignments on dh_3 (definition removed, all usages removed)
- Remove assignments on kemContext_3 (definition removed, all usages removed)
- Remove assignments on pkR_1 (definition removed, all usages removed)
yields
Game 7 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
zz_3: eae_output_t <- cst_eae_output_t;
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(random_br[id1], zz_3[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some({258}zz_3))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
random_br: bool <- cst_bool;
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying replace term at occurrence 258 with zz_3[id1_1]
yields
Game 8 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
zz_3: eae_output_t <- cst_eae_output_t;
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_3[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
random_br: bool <- cst_bool;
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_3: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_3))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying SA rename zz_3
- Rename variable zz_3 into zz_13, zz_12, zz_11, zz_10
yields
Game 9 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
zz_10: eae_output_t <- cst_eae_output_t;
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_10[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_10[id1_1]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_11[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_11[id1_1]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_12[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12[id1_1]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_13[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
random_br: bool <- cst_bool;
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_12: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_13: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying remove assignments of findcond
- Remove assignments on random_br (definition removed, all usages removed)
yields
Game 10 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
{116} find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
zz_10: eae_output_t <- cst_eae_output_t;
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_10[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_10[id1_1]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_11[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_11[id1_1]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_12[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12[id1_1]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_13[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_12: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_13: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying simplify
- Simplification pass
- Remove branch 5 in find at 116
- Remove branch 3 in find at 116
- Remove branch 2 in find at 116
- Replaced defined condition zz_2[ie1, i1_5], z[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5] with zz_2[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5] in find at 116
yields
Game 11 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
zz_10: eae_output_t <- cst_eae_output_t;
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_12[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_12: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12))
else
dh_3: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_3);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_13: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying SA rename dh_3
- Rename variable dh_3 into dh_12, dh_11, dh_10
yields
Game 12 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
zz_10: eae_output_t <- cst_eae_output_t;
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_12[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
dh_10: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2[id1_1]], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_10);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_11: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
dh_11: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_11);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_12: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12))
else
dh_12: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_12);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_13: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying remove assignments of findcond
- Remove assignments on zz_10 (definition removed, all usages removed)
- Remove assignments on zz_11 (definition removed, all usages removed)
- Remove assignments on key_1 (definition removed, all usages removed)
- Remove assignments on info_1 (definition removed, all usages removed)
- Remove assignments on dh_10 (definition removed, all usages removed)
- Remove assignments on kemContext_3 (definition removed, all usages removed)
- Remove assignments on pkR_1 (definition removed, all usages removed)
yields
Game 13 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
{116} find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_12[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
return(AuthDecap_Some(zz_12[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
dh_11: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_11);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_12: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12))
else
dh_12: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_12);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_13: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (
foreach iH <= Qh do
OH(x1: eae_input_t) :=
return(ExtractAndExpand_inner(key_extr_3, x1))
))
Applying simplify
- Simplification pass
- Replaced defined condition zz_12[id1], sk[i1_2[id1]], pkE_3[id1], pk_S[id1] with zz_12[id1], pkE_3[id1], pk_S[id1] in find at 116
yields
Game 14 is
Ostart() :=
key_extr_3 <-R hash_key_t;
return();
((
foreach i <= N do
Osetup() :=
sk <-R Z_t;
return();
((
foreach ie <= Qeperuser do
OAEncap(pk_R: G_t) :=
z <-R Z_t;
pkE_2: G_t <- exp(g, z);
dh_2: GG_t <- concatDH(exp(pk_R, z), exp(pk_R, sk));
enc_1: bitstring <- Serialize(pkE_2);
pkS: G_t <- exp(g, sk);
kemContext_2: GGG_t <- concatContext(pkE_2, pk_R, pkS);
key: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_2);
info: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_2);
zz_2: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key, info));
return(AuthEncap_tuple(zz_2, enc_1))
) | (
foreach id <= Qdperuser do
OADecap(pk_S: G_t, enc_2: bitstring) :=
let Serialize(pkE_3: G_t) = enc_2 in
find ie1_1 = ie1 <= Qeperuser, i1_4 = i1_5 <= N suchthat defined(zz_2[ie1, i1_5], pkE_2[ie1, i1_5], sk[i1_5], pk_R[ie1, i1_5]) && (exp(g, sk) = pk_R[ie1, i1_5]) && (exp(g, sk[i1_5]) = pk_S) && (pkE_2[ie1, i1_5] = pkE_3) then
return(AuthDecap_Some(zz_2[ie1_1, i1_4]))
orfind id1_1 = id1 <= Qdperuser suchthat defined(zz_12[id1], pkE_3[id1], pk_S[id1]) && (pk_S[id1] = pk_S) && (pkE_3[id1] = pkE_3) then
return(AuthDecap_Some(zz_12[id1_1]))
else
find i1_2 = i1_3 <= N suchthat defined(sk[i1_3]) && (pk_S = exp(g, sk[i1_3])) then
dh_11: GG_t <- concatDH(exp(pkE_3, sk), exp(g, mult(sk[i1_2], sk)));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_11);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_12: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_12))
else
dh_12: GG_t <- concatDH(exp(pkE_3, sk), exp(pk_S, sk));
pkR_1: G_t <- exp(g, sk);
kemContext_3: GGG_t <- concatContext(pkE_3, pkR_1, pk_S);
key_1: extract_key_t <- concatExtract(RFCXXXX, suite_id, label_eae_prk, dh_12);
info_1: expand_info_t <- concatExpand(I2OSP2(Nsecret), RFCXXXX, suite_id, label_shared_secret, kemContext_3);
zz_13: eae_output_t <- ExtractAndExpand_inner(key_extr_3, eae_input(lbytes_empty, key_1, info_1));
return(AuthDecap_Some(zz_13))
else
return(AuthDecap_None)
) | (
Opk() :=
return(exp(g, sk))
))
) | (