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Proving the Correctness of Dependency Graph Transformation

Proving the Correctness of Dependency Graph Transformation. Ilja Tšahhirov (joint work with Peeter Laud and Keiko Nakata). Talk plan. Dependency graphs: some background Execution semantics formalization First step – graph fragments equivalence Next steps. A Security Protocol.

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Proving the Correctness of Dependency Graph Transformation

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  1. Proving the Correctness of Dependency Graph Transformation Ilja Tšahhirov (joint work with Peeter Laud and Keiko Nakata) Theory Days at Andu

  2. Talk plan • Dependency graphs: some background • Execution semantics formalization • First step – graph fragments equivalence • Next steps Theory Days at Andu

  3. A Security Protocol A B : { secret }KAB B  : OK Theory Days at Andu

  4. Dependency Graph ? M K D ok? E 1 Λ V ?? ?? Theory Days at Andu

  5. Dependency Graph Execution • Initialize the graph node values with ┴ / false, • Repeat{ Adversary sets the and -nodes Graph is evaluated Adversary is made aware of the values of -nodes } until the Adversary indicates to stop • Adversary’s goal in the game is to produce different output depending on the secret message ?? Theory Days at Andu

  6. Transforming Dependency Graph The game does not change if a graph is replaced with another graph, having the same semantics, for example: Can be replaced with Λ Λ Λ Theory Days at Andu

  7. GUI for executing transformations Theory Days at Andu

  8. The Goal of this Work • One has to be sure that the transformation preserves the semantics, before applying it • The analyzer already has tens of transformations encoded; some of them are quite complex (in terms of amount of nodes involved) • Need to have a way of formally ensuring that: • Two fragments are equivalent, • Procedure for applying the transformation preserves graph semantics when exchanging one fragment with another, equivalent, fragment Theory Days at Andu

  9. Talk plan • Dependency graphs: some background • Execution semantics formalization • First step – graph fragments equivalence • Next steps Theory Days at Andu

  10. Graph • A Graph is a set of nodes, each representing a computation • A Node is identified by • Its identity (label) • Its operation. An operation is either bitstring-valued or boolean-valued. The operation dictates which input ports the node has • Operations: RS, Nonce, Const, Keypair, Pubkey, SigVer, VerKey, SymKey, PubEnc, SymEnc, PubEncZ, SymEncZ, Signature, SignedMsg, Tuple, Proj, PubDec, SymDec, Send, Begin, End, Receive, Secret, Merge, Id, Error, And, Or, Req, True, False, IsOK, IsEq, IsNeq, TestSig, TestSigP InputB, InputS, OutputB, OutputS • A Graph is a set of nodes, each representing a computation • A Node is identified by • Its identity (label) • Its operation. An operation is either bitstring-valued or boolean-valued. The operation dictates which input ports the node has • Operations: RS, Nonce, Const, Keypair, Pubkey, SigVer, VerKey, SymKey, PubEnc, SymEnc, PubEncZ, SymEncZ, Signature, SignedMsg, Tuple, Proj, PubDec, SymDec, Send, Begin, End, Receive, Secret, Merge, Id, Error, And, Or, Req, True, False, IsOK, IsEq, IsNeq, TestSig, TestSigP, InputB, InputS, OutputB, OutputS Theory Days at Andu

  11. Configuration • During the graph evaluation a value is computed for each node • Graph itself is not changed during evaluation – the evaluation result is stored in the configuration: • Environment: Label  Value • Input environment: Label*  Value • Label: set of label of all nodes • Label*: set of label of InputB-nodes Theory Days at Andu

  12. Graph Evaluation Informally, the graph evaluation proceeds as following: • Initialize: • Initialize the input environment with external inputs • Initialize the environment to map every node to false • Repeat { for each node { Compute operation result (the values of operation inputs are taken from the environment or input environment) Store the computed value in the environment } } until no more changes are observed (for each node the computed value is equal to what is stored in the environment) Theory Days at Andu

  13. Graph Evaluation - Example 1:InputB ρ 1 = false ρ 1 = true φ 1 = true ρ 2 = false 2:True ρ 2 = true 3:And ρ 3 = false ρ 3 = true 4:OutputB ρ 4 = false ρ 4 = true Theory Days at Andu

  14. Graph: Theorem Prover Encoding Definitionlabel := nat. Inductive operation : Type :=| andop (ll: list label)| trueop| falseop| inputop| outputop (l: label). Inductive node : Type :=boolnode (l: label)(o: operation). Definition graph := list node. Definition g3' : graph := ( (boolnode 1 inputop) ::(boolnode 3 (andop (1::nil))) ::(boolnode 4 (outputop 3)) ::nil). 1:InputB 3:And 4: OutputB Theory Days at Andu

  15. Environment: Theorem Prover Encoding (*Definition – both for environment and input environment *) label := nat value := bool env := list (label * value) (* Access and update functions *) lookup (r:env)(l:label) : option bool uf (r:env)(l:nat)(v:bool) : option env Theory Days at Andu

  16. Operation Semantics Fixpoint bf (rho:env) (phi:env) (n:node) : option bool := match n with boolnode l o  match o with | trueop  Some true | falseop  Some false | andop ll  andbn rho ll | inputop  lookup phi l | outputop l1  lookup rho l1 end end. Theory Days at Andu

  17. Graph Evaluation Step Fixpoint evalstep (g:graph)(rho:env)(phi:env) {struct g} : option env := match g with | nil  Some rho | (boolnode l o ) :: tl let v := lookup rho l in match v with | None  None | Some b let v':= bf rho phi (boolnode l o) in match v' with | None  None | Some b'  if (bool_dec b b') then evalstep tl rho phi else uf rho l b' end end end. Theory Days at Andu

  18. Graph Evaluation Fixpoint eval (g:graph)(rho phi:env)(n:nat): option env := match n with | 0  Some rho | S n'  match (evalstep g rho phi) with | None  None | Some rho'  let n'' := ‌rho'‌ in if (beq_nat n n'') then Some rho else eval g rho' phi n' end end. Theory Days at Andu

  19. Talk plan • Dependency graphs: some background • Execution semantics formalization • First step – graph fragments equivalence • Next steps Theory Days at Andu

  20. Equivalence Definition – Example g1 g2 phi := (1,v)::nil rhoinit1 := (1,false)::(4:false)::nil rhoinit2 := (1,false)::(2,false)::(3, false)::(4:false)::nil rhofinal1 := eval g1rhoinit1 phi 2 rhofinal2 := eval g2 rhoinit2 phi 4 Equivalence means that lookup rhofinal1 4 = lookup rhofinal2 4 1:input 1:input 2:true 3:and 4:output 4:output Theory Days at Andu

  21. Equivalence Definition Given the graphs g1, g2, satisfying the following requirements: • Each node must have unique label • Both graphs must have same set of input and output nodes The equivalence of g1 and g2 holds if for every output node with label l on g1,lookup (eval g1 rho1 phi ‌rho1‌) l = lookup (eval g2 rho2 phi ‌rho2‌) l Theory Days at Andu

  22. Proving Equivalence Key Lemmas about Semantics Lemma eval_is_evalstep_fixpoint: forall g rho rho' phi, eval g rho phi ‌rho‌ = rho'  evalstep g rho' phi = rho'. Lemma evalstep_fixpoint_is_correct: forall g rho phi, evalstep g rho phi = Some rho  forall l op, node_in_graph (boolnode l op) g  lookup rho l = bf rho phi (boolnode l op). Theory Days at Andu

  23. Equivalence Proof Plan for two particular fragments • Given the graph definitions, limit the output equality to particular output nodes • For each of the two graphs: • Show that evaluation result is a fixed point of evalstep (using eval_is_evalstep_fixpoint) • Show that the environment (rho) holds for all the nodes their “final” value at evaluation result (using evalstep_fixpoint_is_correct) • For each node present its value as a function from input environment (phi) • For each of the output nodes: • Show that on the first and the second graphs the functional dependency of the node from the input environment is the same Theory Days at Andu

  24. Talk plan • Dependency graphs: some background • Execution semantics formalization • First step – graph fragments equivalence • Next steps Theory Days at Andu

  25. Next Steps • Graph equivalence definition and proof framework was the first significant step towards integration with automated analyzer tool • The remaining steps are: • Formally define exchanging the (sub-)graph on another graph • Show that if two sub-graphs, g1 and g2, are equivalent, then exchanging g1 with g2 on a graph, containing g1, results in the equivalent graph • Bring back the complete operations set • Bring in the support for the infinite fragments / graphs Theory Days at Andu

  26. Thank you! Theory Days at Andu

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