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hpke.m4.cv
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hpke.m4.cv
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(* Mechanised cryptographic proofs for HPKE using CryptoVerif.
Following as close as possible the notation of the specification at
https://cfrg.github.io/draft-irtf-cfrg-hpke/draft-irtf-cfrg-hpke.html
Generate individual model files with:
./run false
Generate and run (takes a while and writes a lot of files to disk):
./run
Run individual model with (example file):
f=hpke.curveprime.base.oneshot; mkdir -p $f; cryptoverif -oproof $f/$f.proof -o $f $f.cv
2019-2020, Benjamin Lipp, INRIA Paris, Prosecco
*)
(* BEGIN m4 variables are processed that are given as arguments. *)
ifdef(`m4_base',`
(* mode_base is es:
- no skS
- no psk
- corrupting skR leads to trivial break *)
define(`m4_secrecyquery')
')
ifdef(`m4_auth',`
(* mode_auth is es,ss:
- no psk
- corrupting skR leads to trivial break *)
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_dyn_corrupt_skS')
')
ifdef(`m4_psk__skR',`define(`m4_psk')')
ifdef(`m4_psk__psk',`define(`m4_psk')')
ifdef(`m4_psk__skR_oneshot',`define(`m4_psk_oneshot')')
ifdef(`m4_psk__psk_oneshot',`define(`m4_psk_oneshot')')
ifdef(`m4_psk',`
(* mode_psk is es,psk:
- no skS
- guarantees as long as skR or psk are not both compromised *)
')
ifdef(`m4_psk__skR',`
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_dyn_corrupt_skR')
')
ifdef(`m4_psk__psk',`
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_dyn_corrupt_psk')
')
ifdef(`m4_auth_psk__skR', `define(`m4_auth_psk')')
ifdef(`m4_auth_psk__skS_psk', `define(`m4_auth_psk')')
ifdef(`m4_auth_psk__psk', `define(`m4_auth_psk')')
ifdef(`m4_auth_psk__static_skS_skR', `define(`m4_auth_psk')')
ifdef(`m4_auth_psk__static_psk', `define(`m4_auth_psk')')
ifdef(`m4_auth_psk__static_psk_skS', `define(`m4_auth_psk')')
ifdef(`m4_auth_psk__skR_oneshot', `define(`m4_auth_psk_oneshot')')
ifdef(`m4_auth_psk__skS_psk_oneshot',`define(`m4_auth_psk_oneshot')')
ifdef(`m4_auth_psk__psk_oneshot', `define(`m4_auth_psk_oneshot')')
ifdef(`m4_auth_psk__static_skS_skR_oneshot', `define(`m4_auth_psk_oneshot')')
ifdef(`m4_auth_psk__static_psk_oneshot', `define(`m4_auth_psk_oneshot')')
ifdef(`m4_auth_psk__static_psk_skS_oneshot', `define(`m4_auth_psk_oneshot')')
ifdef(`m4_auth_psk',`
(* mode_auth_psk is es,ss,psk:
- secrecy as long as skR and psk are not both compromised
- auth as long as skS and psk are not both compromised *)
')
ifdef(`m4_auth_psk__skR',`
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_dyn_corrupt_skS')
define(`m4_dyn_corrupt_skR')
')
ifdef(`m4_auth_psk__skS_psk',`
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_dyn_corrupt_skS_psk')
')
ifdef(`m4_auth_psk__psk',`
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_dyn_corrupt_psk')
')
ifdef(`m4_auth_psk__static_skS_skR',`
define(`m4_authqueries')
define(`m4_secrecyquery')
define(`m4_static_corrupt_skR')
define(`m4_static_corrupt_skS')
')
ifdef(`m4_auth_psk__static_psk',`
define(`m4_secrecyquery')
define(`m4_authqueries')
define(`m4_static_corrupt_psk')
')
ifdef(`m4_auth_psk__static_psk_skS',`
define(`m4_secrecyquery')
define(`m4_static_corrupt_psk')
define(`m4_static_corrupt_skS')
')
(* TODO Use m4_authqueries to put success only when needed *)
(* END *)
(* BEGIN process m4 variables created from input variables. *)
ifdef(`m4_psk', `define(`m4_pskgen')')
ifdef(`m4_auth_psk',`define(`m4_pskgen')')
ifdef(`m4_base', `define(`m4_Extract')')
ifdef(`m4_psk', `define(`m4_Extract')')
ifdef(`m4_auth', `define(`m4_ExtractAuth')')
ifdef(`m4_auth_psk',`define(`m4_ExtractAuth')')
(* END *)
(* BEGIN other m4 macros. *)
(* The macro m4_out_game outputs the current game to a file.
The file name contains a number that is increased at each output. *)
define(`m4_gamenumber', 0)
define(`m4_out_game',`define(`m4_gamenumber', incr(m4_gamenumber))out_game "m4_name.m4_gamenumber.out.cv" occ')
(* END *)
ifdef(`m4_base',`
proof {
SArename pkE_4; (* the proof is significantly slower without *)
out_game "g01.out.cv";
insert after "Marshal(pkE_5"
"find j <= N_initiators_base suchthat
defined(pkE_9[j]) && pkE_5 = pkE_9[j] then";
out_game "g02.out.cv";
simplify;
out_game "g03.out.cv";
insert after "in(ch1_1\\["
ifdef(`m4_coll',`
"let extract_input(psk: psk_t, G_to_bitstring(zz: G_t)) = x1 in";
',`
"let extract_input(psk: psk_t, G_to_bitstring(zz: G_t)) = x1_1 in";
')
out_game "g03.1.out.cv";
simplify;
out_game "g03.2.out.cv";
crypto rom(Extract_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
out_game "g04.out.cv";
simplify;
out_game "g05.out.cv";
crypto gdh(exp) z_4;
simplify;
out_game "g06.out.cv";
crypto prf(Expand) **;
simplify;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
simplify;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g07.out.cv";
crypto int_ctxt(Seal_inner) **;
simplify;
out_game "g08.out.cv";
crypto ind_cpa(Seal_inner) **;
success
}
')
ifdef(`m4_auth',`
proof {
out_game "g00.out.cv";
SArename pkE_4; (* the proof is significantly slower without *)
out_game "g01.out.cv";
SArename pkE_5;
out_game "g02.out.cv";
insert after "Marshal(pkE_11"
"find j <= N_initiators_auth suchthat
defined(pkE_9[j]) && pkE_11 = pkE_9[j] then";
out_game "g03.out.cv";
simplify;
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g04.out.cv";
insert_event break_auth before_nth 3 "rcvd";
simplify;
focus "query event(break_auth)";
success simplify;
out_game "g05.out.cv";
simplify;
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
out_game "g06.out.cv";
crypto gdh(exp) z_4;
simplify;
out_game "g07.out.cv";
crypto prf(Expand) **;
simplify;
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g08.out.cv";
crypto int_ctxt(Seal_inner) **;
simplify;
success; (* This is for the focus query. *)
(* Now we proved that break_auth has only a negligible probability. *)
out_game "g09.out.cv";
simplify;
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
out_game "g10.out.cv";
crypto gdh(exp) z_4;
simplify;
out_game "g11.out.cv";
crypto prf(Expand) **;
simplify;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
simplify;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g12.out.cv";
crypto int_ctxt(Seal_inner) **;
simplify;
success;
simplify;
out_game "g13.out.cv";
crypto ind_cpa(Seal_inner) **;
success
}
')
ifdef(`m4_psk__psk',`
proof {
out_game "g00.out.cv";
insert after "in(ch1_1\\["
ifdef(`m4_coll',`
"let extract_input(psk: psk_t, G_to_bitstring(zz: G_t)) = x1 in";
',`
"let extract_input(psk: psk_t, G_to_bitstring(zz: G_t)) = x1_1 in";
')
out_game "g01.out.cv";
SArename pkE_8;
out_game "g02.out.cv";
insert after_nth 1 "Marshal(pkE_15"
"find j <= N_initiators_psk suchthat
defined(pkE_7[j]) && pkE_15 = pkE_7[j] then";
out_game "g03.out.cv";
insert_event break_auth before_nth 3 "rcvd";
simplify;
focus "query event(break_auth)";
success simplify;
out_game "g04.out.cv";
set uniqueBranchReorganize = false;
crypto rom(Extract_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
out_game "g06.out.cv";
(* not needed because the proof can rely on the psk only.
Also gdh would actually fail here, because there is no
occurrence of "m = exp(g, mult(b[j], a))" in the game, and
thus CryptoVerif complains that the transformation does no
"useful change".
crypto gdh(exp) z_4 z_5;
simplify;*)
out_game "g07.out.cv";
crypto prf(Expand) **;
out_game "g08.out.cv";
(* simplify;*)
crypto truncate(truncate_to_Nn) **;
out_game "g09.out.cv";
crypto truncate(truncate_to_Nk) **;
(* simplify;*)
out_game "g10.out.cv";
crypto int_ctxt(Seal_inner) **;
simplify;
success; (* This is for the focus query. *)
crypto rom(Extract_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto gdh(exp) z_5;
simplify;
crypto prf(Expand) **;
simplify;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
(*simplify;*)
crypto prf(Expand) **;
(*simplify;*)
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
(*simplify;*)
crypto int_ctxt(Seal_inner) **;
success;
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
success
}
')
ifdef(`m4_psk__skR',`
proof {
out_game "g00.out.cv";
insert after "in(ch1_1\\["
ifdef(`m4_coll',`
"let extract_input(psk: psk_t, G_to_bitstring(zz: G_t)) = x1 in";
',`
"let extract_input(psk: psk_t, G_to_bitstring(zz: G_t)) = x1_1 in";
')
out_game "g01.out.cv";
set uniqueBranchReorganize = false;
crypto rom(Extract_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto prf(Expand) **;
simplify;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
crypto int_ctxt(Seal_inner) **;
success;
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
success
}
')
ifdef(`m4_auth_psk__skR',`
proof {
out_game "g00.out.cv";
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g01.out.cv";
crypto rom(ExtractAuth_inner);
simplify;
out_game "g02.out.cv";
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto prf(Expand) **;
simplify;
out_game "g03.out.cv";
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
out_game "g04.out.cv";
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g05.out.cv";
crypto int_ctxt(Seal_inner) **;
success;
simplify;
out_game "g06.out.cv";
crypto ind_cpa(Seal_inner) **;
simplify;
success
}
')
ifdef(`m4_auth_psk__skS_psk',`
proof {
out_game "g00.out.cv";
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g01.out.cv";
(* SArename pkE_8;*)
out_game "g02.out.cv";
insert after_nth 2 "Marshal(pkE_9"
"find j <= N_initiators_auth_psk suchthat
defined(pkE_8[j]) && pkE_9 = pkE_8[j] then";
out_game "g03.out.cv";
insert_event break_auth before_nth 3 "rcvd";
simplify;
focus "query event(break_auth)";
success simplify;
out_game "g04.out.cv";
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
out_game "g06.out.cv";
crypto prf(Expand) **;
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g07.out.cv";
crypto int_ctxt(Seal_inner) **;
simplify;
success; (* This is for the focus query. *)
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto gdh(exp) z_6;
simplify;
crypto prf(Expand) **;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
crypto int_ctxt(Seal_inner) **;
success;
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
success
}
')
ifdef(`m4_auth_psk__psk',`
proof {
out_game "g01.out.cv";
insert after "Marshal(pkE_9"
"find j <= N_initiators_auth_psk suchthat
defined(pkE_8[j]) && pkE_9 = pkE_8[j] then";
out_game "g02.out.cv";
simplify;
out_game "g03.out.cv";
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g03.1.out.cv";
simplify;
out_game "g03.2.out.cv";
(* wouldn"t finish in reasonable time otherwise *)
set uniqueBranchReorganize = false;
(* To speed up the proof *)
set autoMergeBranches = false;
set elsefindFactsInSimplify = false;
set noAdviceCrypto = true;
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto gdh(exp) z_6;
simplify;
out_game "g04.out.cv";
simplify;
out_game "g05.out.cv";
crypto prf(Expand) **;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g07.out.cv";
crypto int_ctxt(Seal_inner) **;
success; (* This is for the authentication query. *)
simplify;
out_game "g08.out.cv";
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
out_game "g09.out.cv" occ;
success
}
')
ifdef(`m4_auth_psk__static_skS_skR',`
proof {
out_game "g00.out.cv";
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g01.out.cv";
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto prf(Expand) **;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
crypto int_ctxt(Seal_inner) **;
success;
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
success
}
')
ifdef(`m4_auth_psk__static_psk',`
proof {
out_game "g01.out.cv";
insert after "Marshal(pkE_9"
"find j <= N_initiators_auth_psk suchthat
defined(pkE_8[j]) && pkE_9 = pkE_8[j] then";
out_game "g02.out.cv";
simplify;
out_game "g03.out.cv";
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g03.1.out.cv";
simplify;
out_game "g03.2.out.cv";
(* wouldn"t finish in reasonable time otherwise *)
set uniqueBranchReorganize = false;
(* To speed up the proof *)
set autoMergeBranches = false;
set elsefindFactsInSimplify = false;
set noAdviceCrypto = true;
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto gdh(exp) z_6;
simplify;
out_game "g04.out.cv";
simplify;
out_game "g05.out.cv";
crypto prf(Expand) **;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
out_game "g07.out.cv";
crypto int_ctxt(Seal_inner) **;
success; (* This is for the authentication query. *)
simplify;
out_game "g08.out.cv";
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
out_game "g09.out.cv" occ;
success
}
')
ifdef(`m4_auth_psk__static_psk_skS',`
proof {
out_game "g00.out.cv";
insert after "in(ch1_2\\["
ifdef(`m4_coll',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_1 in";
',`
"let extract_auth_input(psk: psk_t, concat2G(dh1: G_t, dh2: G_t)) = x1_2 in";
')
out_game "g01.out.cv";
(* SArename pkE_8;*)
out_game "g02.out.cv";
insert after "Marshal(pkE_9"
"find j <= N_initiators_auth_psk suchthat
defined(pkE_8[j]) && pkE_9 = pkE_8[j] then";
out_game "g03.out.cv";
crypto rom(ExtractAuth_inner);
simplify;
ifdef(`m4_coll',`',`
crypto rom(Hash_inner);
simplify;
')
crypto gdh(exp) z_7;
simplify;
crypto prf(Expand) **;
ifdef(`m4_export',`
crypto truncate(truncate_to_Nh) **;
simplify;
crypto prf(Expand) **;
')
crypto truncate(truncate_to_Nn) **;
crypto truncate(truncate_to_Nk) **;
simplify;
crypto int_ctxt(Seal_inner) **;
simplify;
crypto ind_cpa(Seal_inner) **;
simplify;
success
}
')
(* TODO make sure the channel names are all unique *)
channel c_start, c_setup, c_config_init, c_init, c_resp.
channel c_config_init_sw, c_init_sw.
channel c_init_send, c_init_send_config.
channel c_resp_mult.
channel c_resp_send_config.
channel c_msgs_send_config, c_msgs_send, c_msgs_recv, c_msgs_recv_finish.
channel c_corrupt_skS, c_corrupt_skR, c_corrupt_psk, c_corrupt_skS_psk.
(* This session index is not part of the specification but assumed
by the model. In an implementation, a session index is likely needed
to match subsequent messages to a session and thus the appropriate
decryption context. The session index could be explicit, or implicit,
i.e. a TCP connection. *)
(*type session_index_t [large,fixed].
table sent_seqs(session_index_t, nonce_t).
table rcvd_seqs(session_index_t, nonce_t).*)
type psk_t [large,fixed].
const default_psk: psk_t.
fun psk_to_bitstring(psk_t): bitstring [data].
const default_pskID: bitstring.
const default_pskID_hash: bitstring.
(* We believe an equation like
equation Hash(default_pskID) = default_pskID_hash.
is not necessary, because the default psk is only used in modes where
the psk in not used and thus a formal link between default_psk and
default_pskID is not necessary. *)
const default_pkSm: bitstring.
type expand_t [large,fixed].
(* This covers the maximum length of HKDF-Expand's output. For
RFC 5869, this is 255*Nh. *)
type key_t [large,fixed].
type nonce_t [large,fixed].
(* Application constants *)
const app_info: bitstring.
expand Xor(
nonce_t,
xor,
nonce_zero (* also used for seq that starts at zero *)
).
(* DH-based KEM *)
ifdef(`m4_curveprime',`
(* For P-256 and P-521
- to model Unmarshal(), use a left-hand-side pattern matching of Marshal
- TODO figure out if Marshal and G_to_bitstring are actually the same:
I think they are the same, because looking at the F* specs for
P256 and Curve25519, the scalarmult function is returning an
encoded point.
*)
type G_t [bounded,large].
fun Marshal(G_t): bitstring [data].
fun G_to_bitstring(G_t): bitstring [data].
type Z_t [bounded,large].
expand DH_good_group(
G_t,
Z_t,
g,
exp,
exp_1,
mult
).
proba P_GDH.
expand GDH(
(* types *)
G_t, (* Group elements *)
Z_t, (* Exponents *)
(* variables *)
g, (* a generator of the group *)
exp, (* exponentiation function *)
exp_1, (* a symbol that replaces exp after game transformation *)
mult, (* multiplication function for exponents *)
(* probabilities *)
P_GDH (* probability of breaking the GDH assumption *)
).
')
ifdef(`m4_curve25519',`
(* For Curve25519
- to model Unmarshal(), use a left-hand-side pattern matching of Marshal
*)
type G_t [bounded,large].
type G8_t [bounded,large].
fun Marshal(G_t): bitstring [data].
fun G_to_bitstring(G_t): bitstring [data].
type Z_t [bounded,large,nonuniform].
expand DH_X25519(
G_t,
Z_t,
g,
exp,
mult,
G8_t,
g8,
exp_div8,
exp_div8_1,
pow8,
G8_to_G,
zero,
zero8
).
proba P_GDH.
expand GDH(
G8_t,
Z_t,
g8,
exp_div8,
exp_div8_1,
mult,
P_GDH
).
')
ifdef(`m4_curve448',`
(* For Curve448
- to model Unmarshal(), use a left-hand-side pattern matching of Marshal
- Znw_t are the exponents without weak keys
*)
type G_t [bounded,large].
type G8_t [bounded,large].
fun Marshal(G_t): bitstring [data].
fun G_to_bitstring(G_t): bitstring [data].
type Z_t [bounded,large,nonuniform].
type Znw_t [bounded,large,nonuniform].
expand DH_X448(
G_t,
Z_t,
g,
exp,
mult,
G8_t,
Znw_t,
Znw_to_Z,
g8,
exp_div8,
exp_div8_1,
pow8,
G8_to_G,
zero,
zero8
).
proba P_GDH.
expand GDH(
G8_t,
Znw_t,
g8,
exp_div8,
exp_div8_1,
mult,
P_GDH
).
')
const default_pkS: G_t.
equation Marshal(default_pkS) = default_pkSm.
letfun DH(exponent: Z_t, group_element: G_t) =
exp(group_element, exponent).
letfun pk(exponent: Z_t) =
exp(g, exponent).
letfun GenerateKeyPair() =
new z: Z_t;
(z, exp(g, z)).
(* Key Derivation Function *)
type hash_key_t [fixed].
type hash_output_t [large,fixed].
fun hash_output_to_bitstring(hash_output_t): bitstring [data].
ifdef(`m4_coll',`
proba P_hash. (* probability of breaking collision resistance *)
expand CollisionResistant_hash(
(* types *)
hash_key_t, (* key of the hash function, models the choice of *)
(* the hash function *)
bitstring, (* input that gets hashed *)
hash_output_t, (* output type of the hash function *)
(* functions *)
Hash_inner, (* name of the hash function: *)
(* hash(hashkey_t, bitstring): hash_output_t *)
(* processes *)
Hash_oracle, (* name of the oracle that will make available the *)
(* hash key to the attacker *)
(* parameters *)
P_hash (* probability of breaking collision resistance *)
).
',`
expand ROM_hash_1(
(* types *)
hash_key_t,
bitstring, (* hashinput1: pskID and info are both bitstrings *)
hash_output_t,
(* functions *)
Hash_inner,
(* processes *)
Hash_oracle,
(* parameters *)
N_qHash (* number of queries to the oracle by the adversary *)
).
')
letfun Hash(key_hash: hash_key_t, input: bitstring) =
hash_output_to_bitstring(Hash_inner(key_hash, input)).
(*
Extract(salt, IKM):
Extract a pseudorandom key of fixed length from
input keying material IKM and an optional octet string salt.
Extract(salt, IKM) is HMAC-Hash(salt, IKM)
*)
type extract_input_t.
fun extract_input(psk_t, bitstring): extract_input_t [data].
type extract_auth_input_t.
fun extract_auth_input(psk_t, bitstring): extract_auth_input_t [data].
type extract_output_t [large,fixed]. (* size: Nh bytes *)
expand ROM_hash_1(
(* types *)
hash_key_t,
extract_input_t,
extract_output_t,
(* functions *)
Extract_inner,
(* processes *)
Extract_oracle,
(* parameters *)
N_qExtract (* number of queries to the oracle by the adversary *)
).
expand ROM_hash_1(
(* types *)
hash_key_t,
extract_auth_input_t,
extract_output_t,
(* functions *)
ExtractAuth_inner,
(* processes *)
ExtractAuth_oracle,
(* parameters *)
N_qExtractAuth (* number of queries to the oracle by the adversary *)
).
letfun Extract(key_extract: hash_key_t, psk: psk_t, zz: bitstring) =
Extract_inner(key_extract, extract_input(psk, zz)).
letfun ExtractAuth(key_extract_auth: hash_key_t, psk: psk_t, zz: bitstring) =
ExtractAuth_inner(key_extract_auth, extract_auth_input(psk, zz)).
(*
Expand(PRK, info, L):
Expand a pseudorandom key PRK using optional string info into L bytes
of output keying material.
*)
proba P_PRF.
expand PRF_large(
extract_output_t,
bitstring,
expand_t,
Expand,
P_PRF
).
def truncate(input_t, output_t, truncate_f) {
param N.
fun truncate_f(input_t): output_t.
(* If we truncate a uniformly distributed random value,
we obtain a uniformly distributed random value *)
equiv(truncate(truncate_f))
foreach i<=N do h <-R input_t;
O_trunc() := return(truncate_f(h))
<=(0)=>
foreach i<=N do k <-R output_t;
O_trunc() := return(k).
}
expand truncate(
expand_t,
nonce_t,
truncate_to_Nn
).
expand truncate(
expand_t,
key_t,
truncate_to_Nk
).
(* Nh: The output size of the Hash and Extract functions *)
expand truncate(
expand_t,
extract_output_t,
truncate_to_Nh
).
letfun Expand_Nk(key: extract_output_t, input: bitstring) =