This repository contains the Coq formalisation of the paper:
- SSProve: A Foundational Framework for Modular Cryptographic Proofs in Coq. Carmine Abate, Philipp G. Haselwarter, Exequiel Rivas, Antoine Van Muylder, Théo Winterhalter, Cătălin Hrițcu, Kenji Maillard, and Bas Spitters. Cryptology ePrint Archive, Report 2021/397. March 2021.
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
).
- OCaml
>=4.05.0 & <4.12
- Coq
8.13.1
- Equations
1.2.4+8.13
- Mathcomp analysis
0.3.7
- Coq Extructures
0.2.2
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 coq.8.13.1 coq-equations.1.2.4+8.13 coq-mathcomp-analysis.0.3.7 coq-extructures.0.2.2
To 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.
A documentation is available in DOC.md.
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
.
Package laws, as introduced in the paper, are all stated and proven in pkg_composition.v directly on raw packages.
In Coq, we call link p1 p2
the sequential composition of p1
and p2
(written p1 ∘ p2
in the paper).
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,
valid_package L1 M E p1 →
valid_package L2 I M p2 →
valid_package (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 →
valid_package L1 I1 E1 p1 →
valid_package L2 I2 E2 p2 →
valid_package (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 →
valid_package 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,
valid_package L I E p →
flat I →
trimmed E p →
link p (ID I) = p.
Lemma id_link :
∀ L I E p,
valid_package 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).
As stated in the paper, we also formalised the KEM-DEM example, whose definition and proof can be found in examples/KEMDEM.v.
Figure 13 presents 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
.
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 |
asrt | r_assert' |
asrtL | r_assertL |
Finally the "bwhile" rule is proven as bounded_do_while_rule
in
rules/RulesStateProb.v.
We now list the lemmas and theorems about packages from the paper and in the case of Theorems 1 & 2 proven using our probabilistic relational program logic. The first two can be found in package/pkg_advantage.v, the other two in package/pkg_rhl.v.
Lemma 1 (Triangle Inequality)
Lemma Advantage_triangle :
∀ P Q R A,
AdvantageE P Q A <= AdvantageE P R A + AdvantageE R Q A.
Lemma 2 (Reduction)
Lemma Advantage_link :
∀ G₀ G₁ A P,
AdvantageE G₀ G₁ (A ∘ P) =
AdvantageE (P ∘ G₀) (P ∘ G₁) A.
Theorem 1
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
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.
In theories/Relational/OrderEnrichedCategory.v we introduce some abstract notions such as categories, functors, relative monads, lax morphisms of relative monads and isomorphisms of functors, all of which are order-enriched. The file theories/Relational/OrderEnrichedCategory.v instantiates all of these abstract notions.
Free monads are defined in rhl_semantics/free_monad/FreeProbProg.v.
In rhl_semantics/ChoiceAsOrd.v we introduce the category of
choice types (choiceType
) which are useful for sub-distributions:
they are basically the types from which we can sample.
They are one of the reasons why our monads are always relative.
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 theory for §5.2 is developed in the following files: rhl_semantics/only_prob/Couplings.v, rhl_semantics/only_prob/Theta_dens.v, (rhl_semantics/only_prob/Theta_exCP.v), rhl_semantics/only_prob/ThetaDex.v.
The theory for §5.3 is developed in the following files, divided in two lists, one for abstract results, and one for the instances we use.
Abstract (in rhl_semantics/more_categories/): OrderEnrichedRelativeAdjunctions.v, LaxMorphismOfRelAdjunctions.v, TransformingLaxMorph.v.
Instances (in rhl_semantics/state_prob/): OrderEnrichedRelativeAdjunctionsExamples.v, StateTransformingLaxMorph.v, StateTransfThetaDens.v, LiftStateful.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 = 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 mathcomp
s 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 Assumptions
command 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.