This repository contains the Coq formalisation of the paper:
SSProve: A Foundational Framework for Modular Cryptographic Proofs in Coq
- Conference version published at CSF 2021 (distinguished paper award). Carmine Abate, Philipp G. Haselwarter, Exequiel Rivas, Antoine Van Muylder, Théo Winterhalter, Cătălin Hrițcu, Kenji Maillard, and Bas Spitters. (ieee, eprint)
- Extended version under journal review. Carmine Abate, Philipp G. Haselwarter, Exequiel Rivas, Antoine Van Muylder, Théo Winterhalter, Nikolaj Sidorenco, Cătălin Hrițcu, Kenji Maillard, and Bas Spitters. (eprint)
This README serves as a guide to running verification and finding the
correspondence between the claims in the paper and the formal proofs in Coq, as
well as listing the small set of axioms on which the formalisation relies
(either entirely standard ones or transitive ones from mathcomp-analysis).
A documentation is available in DOC.md.
- CSF'21: Video accompanying the publication introducing the general framework (speaker: Philipp Haselwarter)
- TYPES'21: Video focused on semantics and programming logic (speaker: Antoine Van Muylder)
- Coq Workshop '21: Video illustrating the formalisation (speaker: Théo Winterhalter)
- OCaml
>=4.05.0 & <4.13.0 - Coq
8.14.0 - Equations
1.3+8.14 - Mathcomp
1.13.0 - Mathcomp analysis
0.3.13 - Coq Extructures
0.3.1 - Coq Deriving
0.1
You can get them all from the opam package manager for OCaml:
opam repo add coq-released https://coq.inria.fr/opam/released
opam update
opam install ./ssprove.opamTo build the dependency graph, you can optionally install graphviz.
On macOS, gsed is additionally required for this.
Run make from this directory to verify all the Coq files.
This should succeed displaying only the list of axioms used for our listed
results.
Run make graph to build a graph of dependencies between sources.
| Directory | Description |
|---|---|
| theories | Root of all the Coq files |
| theories/Mon | External development coming from "Dijkstra Monads For All" |
| theories/Relational | External development coming from "The Next 700 Relational Program Logics" |
| theories/Crypt | This paper |
Unless specified with a full path, all files considered in this README can safely be assumed to be in theories/Crypt.
The formalisation of packages can be found in the package directory.
The definition of packages can be found in pkg_core_definition.v.
Herein, package L I E is the type of packages with set of locations L,
import interface I and export interface E. It is defined on top of
raw_package which does not contain the information about its interfaces
and the locations it uses.
Package laws, as introduced in the paper, are all stated and proven in pkg_composition.v directly on raw packages. This technical detail is not mentioned in the paper, but we are nonetheless only interested in these laws over proper packages whose interfaces match.
In Coq, we call link p1 p2 the sequential composition of p1 and p2
(written p1 ∘ p2 in the paper, but also in Coq thanks to notations).
Definition link (p1 p2 : raw_package) : raw_package.Linking is valid if the export and import match, and its set of locations
is the union of those from both packages (:|: denotes union of sets):
Lemma valid_link :
∀ L1 L2 I M E p1 p2,
ValidPackage L1 M E p1 →
ValidPackage L2 I M p2 →
ValidPackage (L1 :|: L2) I E (link p1 p2).Associativity is stated as follows:
Lemma link_assoc :
∀ p1 p2 p3,
link p1 (link p2 p3) =
link (link p1 p2) p3.It holds directly on raw packages, even if they are ill-formed.
In Coq, we write par p1 p2 for the parallel composition of p1 and p2
(written p1 || p2 in the paper).
Definition par (p1 p2 : raw_package) : raw_package.The validity of parallel composition can be proven with the following lemma:
Lemma valid_par :
∀ L1 L2 I1 I2 E1 E2 p1 p2,
Parable p1 p2 →
ValidPackage L1 I1 E1 p1 →
ValidPackage L2 I2 E2 p2 →
ValidPackage (L1 :|: L2) (I1 :|: I2) (E1 :|: E2) (par p1 p2).The Parable condition checks that the export interfaces are indeed disjoint.
We have commutativity as follows:
Lemma par_commut :
∀ p1 p2,
Parable p1 p2 →
par p1 p2 = par p2 p1.This lemma does not work on arbitrary raw packages, it requires that the packages implement disjoint signatures.
Associativity on the other hand is free from this requirement:
Lemma par_assoc :
∀ p1 p2 p3,
par p1 (par p2 p3) = par (par p1 p2) p3.The identity package is called ID in Coq and has the following type:
Definition ID (I : Interface) : raw_package.Its validity is stated as
Lemma valid_ID :
∀ L I,
flat I →
ValidPackage L I I (ID I).The extra flat I condition on the interface essentially forbids overloading:
there cannot be two procedures in I that share the same name, but have
different types. While our type of interface could in theory allow such
overloading, the way we build packages forbids us from ever implementing them,
hence the restriction.
The two identity laws are as follows:
Lemma link_id :
∀ L I E p,
ValidPackage L I E p →
flat I →
trimmed E p →
link p (ID I) = p.Lemma id_link :
∀ L I E p,
ValidPackage L I E p →
trimmed E p →
link (ID E) p = p.In both cases, we ask that the package we link the identity package with is
trimmed, meaning that it implements exactly its export interface and nothing
more. Packages created through our operations always verify this property
(as such it can be checked automatically on those).
Finally we prove a law involving sequential and parallel composition stating how we can interchange them:
Lemma interchange :
∀ A B C D E F L1 L2 L3 L4 p1 p2 p3 p4,
ValidPackage L1 B A p1 →
ValidPackage L2 E D p2 →
ValidPackage L3 C B p3 →
ValidPackage L4 F E p4 →
trimmed A p1 →
trimmed D p2 →
Parable p3 p4 →
par (link p1 p3) (link p2 p4) = link (par p1 p2) (par p3 p4).where the last line can be read as
(p1 ∘ p3) || (p2 ∘ p4) = (p1 || p2) ∘ (p3 || p4).
It once again requires some validity and trimming properties.
The PRF example is developed in examples/PRF.v. The security theorem is the following:
Theorem security_based_on_prf :
∀ LA A,
ValidPackage LA
[interface val #[i1] : 'word → 'word × 'word ] A_export A →
fdisjoint LA (IND_CPA false).(locs) →
fdisjoint LA (IND_CPA true).(locs) →
Advantage IND_CPA A <=
prf_epsilon (A ∘ MOD_CPA_ff_pkg) +
statistical_gap A +
prf_epsilon (A ∘ MOD_CPA_tt_pkg).As we claim in the paper, it bounds the advantage of any adversary to the
game pair IND_CPA by the sum of the statistical gap and the advantages against
MOD_CPA.
Note that we require some state separation hypotheses here, as such disjointness of state is not required by our package definitions and laws.
The ElGamal example is developed in examples/ElGamal.v. The security theorem is the following:
Theorem ElGamal_OT :
∀ LA A,
ValidPackage LA [interface val #[challenge_id'] : 'plain → 'cipher] A_export A →
fdisjoint LA (ots_real_vs_rnd true).(locs) →
fdisjoint LA (ots_real_vs_rnd false).(locs) →
Advantage ots_real_vs_rnd A <= AdvantageE DH_rnd DH_real (A ∘ Aux).The KEM-DEM case-study can be found in examples/KEMDEM.v.
The single key lemma is identified by single_key_a and single_key_b,
corresponding to the two inequalities of the paper. Their statements are
really verbose because of a lot of side-conditions pertaining to the validity
of the composed packages so we refer the user to the file.
The invariant used to prove perfect indistinguishability is given by
Notation inv := (
heap_ignore KEY_loc ⋊
triple_rhs pk_loc k_loc ek_loc PKE_inv ⋊
couple_lhs pk_loc sk_loc (sameSomeRel PkeyPair)
).We one again refer the use to the commented file for details. Said perfect indistinguishability is stated as
Lemma PKE_CCA_perf :
∀ b, (PKE_CCA KEM_DEM b) ≈₀ Aux b.while the final security theorem is the following:
Theorem PKE_security :
∀ LA A,
ValidPackage LA PKE_CCA_out A_export A →
fdisjoint LA PKE_CCA_loc →
fdisjoint LA Aux_loc →
Advantage (PKE_CCA KEM_DEM) A <=
Advantage KEM_CCA (A ∘ (MOD_CCA KEM_DEM) ∘ par (ID KEM_out) (DEM true)) +
Advantage DEM_CCA (A ∘ (MOD_CCA KEM_DEM) ∘ par (KEM false) (ID DEM_out)) +
Advantage KEM_CCA (A ∘ (MOD_CCA KEM_DEM) ∘ par (ID KEM_out) (DEM false)).The Σ-protocols case-study is divided over two files: examples/SigmaProtocol.v and examples/Schnorr.v.
The security theorem for hiding of commitment scheme from Σ-protocols is:
Theorem commitment_hiding :
∀ LA A eps,
ValidPackage LA [interface
val #[ HIDING ] : chInput → chMessage
] A_export A →
(∀ B,
ValidPackage (LA :|: Com_locs) [interface
val #[ TRANSCRIPT ] : chInput → chTranscript
] A_export B →
ɛ_SHVZK B <= eps
) →
AdvantageE (Hiding_real ∘ Sigma_to_Com ∘ SHVZK_ideal) (Hiding_ideal ∘ Sigma_to_Com ∘ SHVZK_ideal) A <=
(ɛ_hiding A) + eps + eps.And the corresponding theorem for binding:
Theorem commitment_binding :
∀ LA A LAdv Adv,
ValidPackage LA [interface
val #[ SOUNDNESS ] : chStatement → 'bool
] A_export A →
ValidPackage LAdv [interface] [interface
val #[ ADV ] : chStatement → chSoundness
] Adv →
fdisjoint LA (Sigma_locs :|: LAdv) →
AdvantageE (Com_Binding ∘ Adv) (Special_Soundness_f ∘ Adv) A <=
ɛ_soundness A Adv.Combining the above theorems with the instatiation of Schnorr's protocol we get a commitment scheme given by:
Theorem schnorr_com_hiding :
∀ LA A,
ValidPackage LA [interface
val #[ HIDING ] : chInput → chMessage
] A_export A →
fdisjoint LA Com_locs →
fdisjoint LA Sigma_locs →
AdvantageE (Hiding_real ∘ Sigma_to_Com ∘ SHVZK_ideal) (Hiding_ideal ∘ Sigma_to_Com ∘ SHVZK_ideal) A <= 0.and
Theorem schnorr_com_binding :
∀ LA A LAdv Adv,
ValidPackage LA [interface
val #[ SOUNDNESS ] : chStatement → 'bool
] A_export A →
ValidPackage LAdv [interface] [interface
val #[ ADV ] : chStatement → chSoundness
] Adv →
fdisjoint LA (Sigma_locs :|: LAdv) →
AdvantageE (Com_Binding ∘ Adv) (Special_Soundness_f ∘ Adv) A <= 0.The paper version (CSF: Figure 13, journal: section 4.1) introduces a selection
of rules for our probabilistic relational program logic.
Most of them can be found in package/pkg_rhl.v which provides an interface for
using these rules directly with code.
We separate by a slash (/) rule names that differ in the CSF (left) and journal
(right) version.
| Rule in paper | Rule in Coq |
|---|---|
| reflexivity | rreflexivity_rule |
| seq | rbind_rule |
| swap | rswap_rule |
| eqDistrL | rrewrite_eqDistrL |
| symmetry | rsymmetry |
| for-loop | for_loop_rule |
| uniform | r_uniform_bij |
| dead-sample | r_dead_sample |
| sample-irrelevant | r_const_sample |
| asrt / assert | r_assert' |
| asrtL / assertL | r_assertL |
| assertD | r_assertD |
| put-get | r_put_get |
| async-get-lhs | r_get_remember_lhs |
| async-get-lhs-rem | r_get_remind_lhs |
| async-put-lhs | r_put_lhs |
| restore-pre-lhs | r_restore_lhs |
Finally the "bwhile" / "do-while" rule is proven as
bounded_do_while_rule in rules/RulesStateProb.v.
We now list the lemmas and theorems about packages from the paper. Theorems 1 and 2 (CSF) / Theorems 2.4 and 4.1 (journal) were proven using our probabilistic relational program logic. The first two lemmas below can be found in package/pkg_advantage.v, the other two in package/pkg_rhl.v.
Lemma 1 / 2.2 (Triangle Inequality)
Lemma Advantage_triangle :
∀ P Q R A,
AdvantageE P Q A <= AdvantageE P R A + AdvantageE R Q A.Lemma 2 / 2.3 (Reduction)
Lemma Advantage_link :
∀ G₀ G₁ A P,
AdvantageE G₀ G₁ (A ∘ P) =
AdvantageE (P ∘ G₀) (P ∘ G₁) A.Theorem 1 / 2.4
Lemma eq_upto_inv_perf_ind :
∀ {L₀ L₁ LA E} (p₀ p₁ : raw_package) (I : precond) (A : raw_package)
`{ValidPackage L₀ Game_import E p₀}
`{ValidPackage L₁ Game_import E p₁}
`{ValidPackage LA E A_export A},
INV' L₀ L₁ I →
I (empty_heap, empty_heap) →
fdisjoint LA L₀ →
fdisjoint LA L₁ →
eq_up_to_inv E I p₀ p₁ →
AdvantageE p₀ p₁ A = 0.Theorem 2 / 4.1
Lemma Pr_eq_empty :
∀ {X Y : ord_choiceType}
{A : pred (X * heap_choiceType)} {B : pred (Y * heap_choiceType)}
Ψ ϕ
(c1 : FrStP heap_choiceType X) (c2 : FrStP heap_choiceType Y)
⊨ ⦃ Ψ ⦄ c1 ≈ c2 ⦃ ϕ ⦄ →
Ψ (empty_heap, empty_heap) →
(∀ x y, ϕ x y → (A x) ↔ (B y)) →
\P_[ θ_dens (θ0 c1 empty_heap) ] A =
\P_[ θ_dens (θ0 c2 empty_heap) ] B.This part of the mapping corresponds to section 5. Once again, we refer to results in the paper like so: CSF numbering/journal version numbering.
In our framework, a relational effect observation is defined as some kind of lax morphism between order-enriched relative monads. This general definition as well as the ingredients it requires are provided in theories/Relational/OrderEnrichedCategory.v. There we introduce categories, functors, relative monads, lax morphisms of relative monads and isomorphisms of functors, all of which are order-enriched.
Relational effect observations are lax morphisms between the following special cases of order-enriched relative monads:
- A product of Type valued order-enriched relative monads, corresponding to pairs of effectful computations.
- A relational specification monad
To build the above computation part (1) of an effect observation, the file theories/Relational/OrderEnrichedRelativeMonadExamples.v equips Type with a structure of order-enriched category. Often we use free monads to package effectful computations. Those are defined in rhl_semantics/free_monad/.
Since a relational specification monad as in (2) is by definition an order-enriched monad with codomain PreOrder, the latter category has to be endowed with an order-enrichment. This is done in theories/Relational/OrderEnrichedRelativeMonadExamples.v.
More basic categories can be found in the directory rhl_semantics/more_categories/, namely in the files RelativeMonadMorph_prod.v, LaxComp.v, LaxFunctorsAndTransf.v and InitialRelativeMonad.v.
The files of interest are mainly contained in the rhl_semantics/only_prob/ directory.
This relational effect observation is called
thetaDex in the development and is defined in the
file rhl_semantics/only_prob/ThetaDex.v as a composition:
FreeProb² ---unary_theta_dens²---> SDistr² ---θ_morph---> Wrelprop
The first part unary_theta_dens² consists in intepreting pairs
of probabilistic programs into pairs of actual subdistributions.
This unary semantics for probabilistic programs unary_theta_dens
is defined in rhl_semantics/only_prob/Theta_dens.v.
It is defined by pattern matching on the given probabilistic program
(which can be viewed as a tree).
The free relative monad over a probabilistic signature is defined
in rhl_semantics/free_monad/FreeProbProg.v.
The codomain of unary_theta_dens is defined in
rhl_semantics/only_prob/SubDistr.v.
Since subdistributions SDistr(A) only make sense
when A is a choiceType, both the domain and codomain
of unary_theta_dens are relative monads over
appropriate inclusion functors choiceType -> Type.
The required order-enrichment for the category of choiceTypes
and this inclusion are defined in the file rhl_semantics/ChoiceAsOrd.v.
The second part θ_morph is conceptually more important.
It is defined in the file rhl_semantics/only_prob/Theta_exCP.v.
θ_morph is "really" lax: it satisfies the morphism laws only
up to inequalities.
The definition of θ_morph relies on the notion of couplings,
defined in this file rhl_semantics/only_prob/Couplings.v.
The prove that it constitutes a lax morphism depends on lemmas
for couplings that can be found in the same file.
The important files are contained in this directory: rhl_semantics/state_prob/.
Again the effect observation is defined as a composition:
thetaFstdex: FrStP² → stT(Frp²) → stT(Wrel).
See file StateTransformingLaxMorph.v.
The first part uses unaryIntState: FrStP → stT(Frp)
from the same file which interprets state related instructions
as actual state manipulating functions S → Frp( - x S ).
Probabilistic instructions are left untouched by this morphism.
The more interesting part is the second one (same file)
stT_thetaDex: stT(Frp²) → stT(Wrel).
This morphism is obtained by state-transforming the
relational effect observation thetaDex from the previous section.
More details about the state transformer implementation are provided in the next section.
For the definition of relative monad (Def 5.1 journal), see section "5.1 Relational effect observation" of the present file.
The general definitions and theorems regarding the state transformer can be found in rhl_semantics/more_categories/: OrderEnrichedRelativeAdjunctions.v, LaxMorphismOfRelAdjunctions.v, TransformingLaxMorph.v.
On the other hand our instances can be found in rhl_semantics/state_prob/: OrderEnrichedRelativeAdjunctionsExamples.v, StateTransformingLaxMorph.v, StateTransfThetaDens.v, LiftStateful.v.
The concerned file is OrderEnrichedRelativeAdjunctions.v,
section TransformationViaRelativeAdjunction.
There we transform an arbitrary order-enriched relative monad
using a "transforming adjunction" (Thm 5.5 journal). The notion of transforming
adjunction (Def 5.4 journal) is a generalization of the notion of state adjunction.
State adjunctions for transforming computations/specifications are built in OrderEnrichedRelativeAdjunctionsExamples.v.
All of our adjunctions are left relative adjunctions (Def 5.2 journal). This notion is defined and studied in OrderEnrichedRelativeAdjunctions.v and this includes Kleisli adjunctions of relative monads (Def 5.3 journal).
See file TransformingLaxMorph.v. Given a lax morphism of relative monads θ : M1 → M2, both M1 and M2 factor through their Kleisli and give rise to Kleisli adjunctions. θ induces a lax morphism Kl(θ) between those Kleisli adjunctions. Kl(θ) is a lax morphism between left relative adjunctions, (see LaxMorphismOfRelAdjunctions.v) and we can transform such morphisms of adjunctions using the theory developped in TransformingLaxMorph.v. Finallly, out of this transformed morphism of adjunctions we can extract a lax morphism between monads Tθ : T M1 → T M2, as expected. This last step is also performed in TransformingLaxMorph.v.
In our development we rely on the following standard axioms: functional extensionality, proof irrelevance, and propositional extensionality, as listed below.
ax_proof_irrel : ClassicalFacts.proof_irrelevance
propositional_extensionality : ∀ P Q : Prop, P ↔ Q → P = Q
functional_extensionality_dep :
∀ (A : Type) (B : A → Type) (f g : ∀ x : A, B x),
(∀ x : A, f x = g x) → f = gWe also rely on the constructive indefinite description axiom, whose use
we inherit transitively from the mathcomp-analysis library.
boolp.constructive_indefinite_description :
∀ (A : Type) (P : A → Prop), (∃ x : A, P x) → {x : A | P x}The mathcomp-analysis library also uses an axiom to abstract away from any
specific construction of the reals:
R : realTypeOne could plug in any real number construction: Cauchy, Dedekind, ...
In mathcomps Rstruct.v an instance is built from any instance of the
abstract stdlib reals. An instance of the latter is built from the
(constructive) Cauchy reals in Coq.Reals.ClassicalConstructiveReals.
Finally, by using mathcomp-analysis we also inherit an admitted lemma they have:
interchange_psum :
∀ (R : realType) (T U : choiceType) (S : T → U → R),
(∀ x : T, summable (T:=U) (R:=R) (S x)) →
summable (T:=T) (R:=R) (λ x : T, psum (λ y : U, S x y)) →
psum (λ x : T, psum (λ y : U, S x y)) =
psum (λ y : U, psum (λ x : T, S x y))Our development also contains a few new work-in-progress results that are admitted, but none of them is used to show the results from the paper above.
We use the Print Assumptionscommand of Coq to list the axioms/admits on which
a definition, lemma, or theorem depends. In Main.v we run this
command on all the results above at once:
Print Assumptions results_from_the_paper.which yields
Axioms:
boolp.propositional_extensionality : forall P Q : Prop, P <-> Q -> P = Q
realsum.interchange_psum
: forall (R : reals.Real.type) (T U : choice.Choice.type)
(S : choice.Choice.sort T -> choice.Choice.sort U -> reals.Real.sort R),
(forall x : choice.Choice.sort T, realsum.summable (T:=U) (R:=R) (S x)) ->
realsum.summable (T:=T) (R:=R)
(fun x : choice.Choice.sort T =>
realsum.psum (fun y : choice.Choice.sort U => S x y)) ->
realsum.psum
(fun x : choice.Choice.sort T =>
realsum.psum (fun y : choice.Choice.sort U => S x y)) =
realsum.psum
(fun y : choice.Choice.sort U =>
realsum.psum (fun x : choice.Choice.sort T => S x y))
boolp.functional_extensionality_dep
: forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
(forall x : A, f x = g x) -> f = g
FunctionalExtensionality.functional_extensionality_dep
: forall (A : Type) (B : A -> Type) (f g : forall x : A, B x),
(forall x : A, f x = g x) -> f = g
boolp.constructive_indefinite_description
: forall (A : Type) (P : A -> Prop), (exists x : A, P x) -> {x : A | P x}
SPropBase.ax_proof_irrel : ClassicalFacts.proof_irrelevance
Axioms.R : reals.Real.typeThe ElGamal example is parametrized by a cyclic group using a Coq functor. To print its axioms we have to provide an instance of this functor, and for simplicity we chose to use ℤ₃ as an instance even if it is not realistic. The axioms we use do not depend on the instance itself. We do something similar for Schnorr's protocol.