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SSProve

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).

Documentation

A documentation is available in DOC.md.

Additional material

  • 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)

Installation

Prerequisites

  • 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.opam

To build the dependency graph, you can optionally install graphviz. On macOS, gsed is additionally required for this.

Running verification

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 organisation

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.

Mapping between paper and formalisation

Package definition and laws

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.

Sequential composition

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.

Parallel composition

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.

Identity package

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).

Interchange between sequential and parallel composition

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.

Examples

PRF

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.

ElGamal

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).

KEM-DEM

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)).

Σ-protocols

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.

Probabilistic relational program logic

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.

More Lemmas and Theorems for packages

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.

Semantic model and soundness of rules

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.

5.1 Relational effect observation

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:

  1. A product of Type valued order-enriched relative monads, corresponding to pairs of effectful computations.
  2. 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.

5.2 The probabilistic relational effect observation

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.

5.3 The stateful and probabilistic relational effect observation

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.

CSF state transformer/ section 5.4 of journal version

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 state transformer on relative monads (i.e. on objects)

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).

The state transformer for lax morphisms (i.e. on arrows)

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.

Axioms

List of axioms

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 = g

We 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 : realType

One 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))

Other admits not used by results from the paper

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.

How to find axioms/admits

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.type

The 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.