Fiat-Crypto: Synthesizing Correct-by-Construction Code for Cryptographic Primitives

Building

This repository requires Coq 8.9 or later. Note that if you install Coq from Ubuntu aptitude packages, you need libcoq-ocaml-dev in addition to coq.

Git submodules are used for some dependencies. If you did not clone with --recursive, run

git submodule update --init --recursive

To build (if your COQPATH variable is empty):

   make

To build:

   export COQPATH="$(pwd)/rewriter/src:$(pwd)/coqprime/src:$(pwd)/bedrock2/bedrock2/src:$(pwd)/bedrock2/deps/coqutil/src${COQPATH:+:}$COQPATH"
   make

Usage

The Coq development builds binary compilers that generate code using some implementation strategy. The parameters (modulus, hardware multiplication input bitwidth, etc.) are are specified on the command line of the compiler. The generated C code is written to standard output.

A collection of C files for popular curves can be made with

make c-files

The C files will appear in the top-level directory.

Just the compilers generating these C files can be made with

make standalone

or make standalone-haskell or make standalone-ocaml for binaries generated with just one compiler. The binaries are located in src/ExtractionOcaml/ and src/ExtractionHaskell respectively.

There is a separate compiler binary for each implementation strategy:

• saturated_solinas
• unsaturated_solinas
• word_by_word_montgomery

Passing no arguments, or passing -h or --help (or any other invalid arguments) will result in a usage message being printed. These binaries output C code on stdout.

Here are some examples of ways to invoke the binaries (from the directories that they live in):

# Generate code for 2^255-19
./unsaturated_solinas '25519' '5' '2^255 - 19' '64' carry_mul carry_square carry_scmul121666 carry add sub opp selectznz to_bytes from_bytes > curve25519_64.c
./unsaturated_solinas '25519' '10' '2^255 - 19' '32' carry_mul carry_square carry_scmul121666 carry add sub opp selectznz to_bytes from_bytes > curve25519_32.c

# Generate code for NIST-P256 (2^256 - 2^224 + 2^192 + 2^96 - 1)
./word_by_word_montgomery 'p256' '2^256 - 2^224 + 2^192 + 2^96 - 1' '32' > p256_32.c
./word_by_word_montgomery 'p256' '2^256 - 2^224 + 2^192 + 2^96 - 1' '64' > p256_64.c

You can find more examples in the Makefile.

Reading About The Code

• Andres Erbsen, Jade Philipoom, Jason Gross, Robert Sloan, Adam Chlipala. Simple High-Level Code For Cryptographic Arithmetic -- With Proofs, Without Compromises. To Appear in Proceedings of the IEEE Symposium on Security & Privacy 2019 (S&P'19). May 2019.. This paper describes multiple field arithmetic implementations, and an older version of the compilation pipeline (preserved here). It is somewhat space-constrained, so some details are best read about in theses below.
• Jade Philipoom. Correct-by-Construction Finite Field Arithmetic in Coq. MIT Master's Thesis. February 2018. Chapters 3 and 4 contain a detailed walkthrough of the field arithmetic implementations (again, targeting the previous compilation pipeline).
• Andres Erbsen. Crafting Certified Elliptic CurveCryptography Implementations in Coq. MIT Master's Thesis. June 2017. Section 3 contains a whirlwind introduction to synthesizing field arithmetic code using coq, without assuming Coq skills, but covering a tiny fraction of the overall library. Sections 5 and 6 contain the only write-up on the ellitpic-curve library in this repository.
• The newest compilation pipeline does not have a separate document yet, but this README does go over it in some detail.

Reading The Code

Demo of Synthesis

The idea of the synthesis process is demoed in src/Demo.v. We strongly recommend reading this before studying the full-scale system.

Proofs About Elliptic Curves

We have some about elliptic curves, for example:

• src/Curves/Edwards/AffineProofs.v,
• src/Curves/Edwards/XYZT/Basic.v,
• src/Curves/Montgomery/AffineProofs.v,
• src/Curves/Montgomery/XZProofs.v.

Actual Synthesis Pipeline

The entry point for clients of the PHOAS expressions we use is Language/API.v. Refer to comments in that file for an explanation of the interface; the following text describes how the expressions are generated, not how to interact with them.

The ordering of files (eliding *Proofs.v files) is:

Language/*.v
    ↑
    ├────────────────────────────────┬───────────────────────┬───────────────────────┐
AbstractInterpretation/*.v     MiscCompilerPasses.v    Rewriter/*.v     PushButtonSynthesis/ReificationCache.v      Arithmetic.v
    ↑                                ↑                       ↑                       ↑                                   ↑
Stringification/*.v                  │                       │                       │                        COperationSpecifications.v
    ↑                                │                       │                       │                                   ↑
    └────────────┬───────────────────┴───────────────────────┴────────┬──────────────┘                                   │
           BoundsPipeline.v                                  CompilersTestCases.v                                        │
                 ↑                                                                                                       │
                 └────────────┬──────────────────────────────────────────────────────────────────────────────────────────┘
                     PushButtonSynthesis/*.v
                              ↑
                   ┌──────────┴────────────────┐
                  CLI.v                SlowPrimeSynthesisExamples.v
                   ↑
        ┌──────────┴────────────────┐
StandaloneHaskellMain.v   StandaloneOCamlMain.v
        ↑                           ↑
ExtractionHaskell.v          ExtractionOCaml.v

Within each directory, the dependency graphs (again eliding *Proofs.v and related files) are:

Within Language/:

  Pre.v ←──────────────────────────────────────────────────────────────────────── IdentifierParameters.v
    ↑                                                                                        ↑
Language.v ←── IdentifiersBasicLibrary.v ←──── IdentifiersBasicGenerate.v ←── IdentifiersBasicGENERATED.v ←───────────────────────────── API.v
    ↑                        ↑                                                               ↑
    ├────────────────┐       └────────────────────────────┐                                  │
UnderLets.v    IdentifiersLibrary.v ←──────────── IdentifiersGenerate.v ←─────── IdentifiersGENERATED.v
                     ↑                                       ↑                               ↑
              IdentifiersLibraryProofs.v ←─── IdentifiersGenerateProofs.v ←─ IdentifersGENERATEDProofs.v

Within Stringification/:

Language.v
    ↑
   IR.v
    ↑
 ┌──┴───────┐
C.v       Rust.v

We will come back to the Rewriter/* files shortly.

The files contain:

• Arithmetic.v: All of the high-level field arithmetic stuff

• COperationSpecifications.v: The specifications for the various operations to be synthesized. TODO: This file should probably be renamed.

• AbstractInterpretation/*.v: type-code-based ZRange definitions, abstract interpretation of identifiers (which does let-lifting, for historical reasons, and the dependency on UnderLets should probably be removed), defines the passes:

  • PartialEvaluateWithBounds
  • PartialEvaluateWithListInfoFromBounds
  • CheckPartialEvaluateWithBounds

• MiscCompilerPasses.v: Defines the passes:

  • EliminateDead (dead code elimination)
  • Subst01 (substitute let-binders used 0 or 1 times)

• Rewriter/*.v: rewrite rules, rewriting. See below for actual stucture of files. Defines the passes:

  • RewriteNBE
  • RewriteArith
  • RewriteArithWithCasts
  • RewriteStripLiteralCasts
  • RewriteToFancy
  • RewriteToFancyWithCasts
  • PartialEvaluate (which is just a synonym for RewriteNBE)

• Inside Language/:

  • Pre.v: A few definitions which are used in writing out rewrite rules and the interpretations of PHOAS identifiers, e.g., ident.cast, ident.eagerly, Thunked.list_rect, etc

  • Language.v: Defines parts of the PHOAS basic infrastructure parameterized over base types and identifiers including: . PHOAS . reification . denotation/intepretation . utilities for inverting PHOAS exprs . default/dummy values of PHOAS exprs . default instantiation of generic PHOAS types . gallina reification of ground terms . Flat/indexed syntax trees, and conversions to and from PHOAS

    Defines the passes: . ToFlat . FromFlat . GeneralizeVar

  • API.v: Specializes the type of PHOAS expressions to the particular identifiers we're using, and defines convenience notations, tactics, and definitions for some of the specialized versions.

  • IdentifierParameters.v: Defines a couple of definitions determining the identifiers and types used by the language. These are used as input for the generation of identifier definitions.

  • IdentifiersBasicLibrary.v: Defines the package type holding basic identifier definitions.

  • IdentifiersBasicGenerate.v: Defines the tactics that generate all of the identifier-list-specific definitions used by the PHOAS machinery, in addition to defining the tactics that do reification based on the generated package.

  • IdentifiersBasicGENERATED.v: Basically autogenerated file that defines the inductives of base type codes and identifier codes (the first hand-written because it's short; the latter copy-pasted from a tactic that prints out the inductive), and calls the package-generation-tactic from IdentifiersBasicGenerate.v.

  • UnderLets.v: the UnderLets monad, a pass that does substitution of var-like things, a pass that inserts let-binders in the next-to-last line of code, substituting away var-like things (this is used to ensure that when we output C code, aliasing the input and the output arrays doesn't cause issues). Defines the passes: . SubstVar . SubstVarLike . SubstVarOrIdent

  The following files in Language/ are used only by the rewriter:

  • IdentifiersLibrary.v: Some definitions about identifiers and pattern identifiers and raw identifiers. Some of these definitions take generated definitions as arguments. Also defines a package record to hold all of the generated definitions.

  • IdentifiersGenerate.v: Tactics to generate definitions about untyped and pattern versions of identifiers for the rewriter. Culminates in a tactic which inhabits the package type defined in IdentifiersLibrary.v

  • IdentifiersLibraryProofs.v: proofs about definitions in IdentifiersLibrary. Also defines a package to hold generated proofs that require destructing inductives not yet defined in this file.

  • IdentifiersGenerateProofs.v: tactics to prove lemmas to inhabit the package defined in IdentifiersLibraryProofs.v

  • IdentifiersGENERATE.v: identifiers / inductives and definitions generated by IdentifiersGenerate.

  • IdentifiersGENERATEProofs.v: proofs generated by IdentifiersGenerateProofs, about definitions in IdentifiersGENERATE

• Inside Stringification/:

  • Language.v: defines a printer (Show instance) for the PHOAS language, defines some common language-independent utilities for conversion to output code, and defines the spec/API of conversion from PHOAS to code in a language as strings. (Depends on AbstractInterpretation.v for ZRange utilities.) Defines the passes: . ToString.LinesToString . ToString.ToFunctionLines

  • IR.v: Defines a common IR for C and Rust (and maybe eventually other languages), and builds most of the infrastructure necessary for instantiating the LanguageSpecification API for a language with pointers and function calls

  • C.v: conversion to C code as strings. Instantiates the API defined in Stringification.Language.

  • Rust.v: conversion to Rust code as strings. Instantiates the API defined in Stringification.Language.

• CompilersTestCases.v: Various test cases to ensure everything is working

• BoundsPipeline.v: Assemble the various compiler passes together into a composed pipeline. It is the final interface for the compiler. Also contains some tactics for applying the BoundsPipeline correctness lemma.

• PushButtonSynthesis/ReificationCache.v: Defines the cache that holds reified versions of operations, as well as the tactics that reify and apply things from the cache.

• PushButtonSynthesis/*: Reifies the various operations from Arithmetic.v, definies the compositions of the BoundsPipeline with these operations, proves that their interpretations satisfies the specs from COperationSpecifications.v, assembles the reified post-bounds operations into synthesis targets. These are the files that CLI.v depends on:

  • ReificationCache.v: Defines the cache of pre-reified terms. Splitting up reification from uses of the pipeline allows us to not have to re-reify big terms every time we change the pipeline or intermediate stages thereof.
  • InvertHighLow.v: Defines some common definitions, around plitting apart high and low bits of things, for Barrett and FancyMontgomeryReduction.
  • Primitives.v: Specializes the pipeline to basic "primitive" operations such as cmovznz, addcarryx, subborrowx, etc.
  • SmallExamples.v: Some small examples of using the pipeline. Nothing depends on this file; it is for demonstration purposes only.
  • *ReificationCache.v: Holds the reified versions of the definitions used in the corresponding file.
  • BarrettReduction.v, FancyMontgomeryReduction.v, SaturatedSolinas.v, UnsaturatedSolinas.v, WordByWordMontgomery.v: Holds the instantiation of the pipeline to the corresponding implementation choice, as well as any relevant correctness proofs (such as that things assemble into a ring).

• SlowPrimeSynthesisExamples.v: Additional uses of the pipeline for primes that are kind-of slow, which I don't want extraction blocking on. Also contains some debugging examples.

• CLI.v: Setting up all of the language-independent parts of extraction; relies on having a list of strings-or-error-messages for each pipeline, and on the arguments to that pipeline, and builds a parser for command line arguments for that.

• StandaloneHaskellMain.v, StandaloneOCamlMain.v, ExtractionHaskell.v, ExtractionOCaml.v: Extraction of pipeline to various languages

The files defined in Rewriter/ are split up into the following dependency graph (including some files from Language/ at the top):

IdentifiersLibrary.v ←───────────────────────── IdentifiersGenerate.v ←──────────────────── IdentifiersGENERATED.v
    ↑ ↑                                                   ↑                                        ↑
    │ └──────────────── IdentifiersLibraryProofs.v ←──────┴─ IdentifiersGenerateProofs.v ←─ IdentifersGENERATEDProofs.v
    │                                     ↑                                                        ↑
    │                                     │                                                        │
    │                                     │                                                        │
    │                                     │                                                        │
    │                                     │                                                        │
Rewriter.v ←────────────────────── ProofsCommon.v ←──────────────────── ProofsCommonTactics.v      │
    ↑                                 ↗        ↖                                ↑                  │
Reify.v ←──────────────┐           Wf.v   InterpProofs.v                        │                  │
                       │              ↖        ↗                                │                  │
Rules.v                └──────────── AllTactics.v ──────────────────────────────┘                  │
    ↑                                      ↑       ┌───────────────────────────────────────────────┘
RulesProofs.v                         AllTacticsExtra.v
    ↑                                      ↑
    ├────────┬─────────────┬───────────────┴────────┬─────────────────────────────┐
    │   Passes/NBE.v    Passes/Arith.v    Passes/ArithWithCasts.v    Passes/StripLiteralCasts.v
    │        ↑             ↑                        ↑                             ↑
    │        └─────────────┴────────────────────────┴─────────────────────────────┴─────────────┐
    │                                                                                           │
    └────────┬──────────────────────────┐                                                       │
      Passes/ToFancy.v      Passes/ToFancyWithCasts.v                                           │
             ↑                          ↑                                                       │
             └───────┬──────────────────┴───────────────────────────────────────────────────────┘
                     │
                   All.v

• Rules.v: Defines the list of types of the rewrite rules that will be reified. Largely independent of the expression language.

• RulesProofs.v: Proves all of the Gallina versions of the rewrite rules correct.

• Rewriter.v: Defines the rewriter machinery. In particular, all of the rewriter definitions that have non-rewrite-rule-specific proofs about them are found in this file.

• RewrierReify.v: Defines reification of rewrite rules, continuing on from Rewriter.v, and culminates in the tactic RewriteRules.Tactic.Build_RewriterT and the tactic notation make_Rewriter which define a package of type RewriteRules.GoalType.RewriterT. The Build_* tactic returns a constr, while the make_* tactic notation refines that constr in the goal. Both tactics take two arguments: first a boolean include_interp which specifies whether (true) or not (false) to prefix the list of rewrite rules with the reduction-to-literal rewrite rules; and second a list of bool * Prop which is the list of rewrite rule types to reify, each paired with a boolean saying whether or not to try rewriting again in the output of the replacement for that rule.

• ProofsCommon.v: Defines the notion of interp-goodness and wf-goodness for rewrite rules, defines tactics to prove these notions, and contains a semi-arbitrary collection of proofs and definitions that are mostly shared between the wf proofs and the interp proofs. Importantly, this file defines everything needed to state and prove that specific rewrite rules are correct. Additionally defines a package RewriteRules.GoalType.VerifiedRewriter which describes the type of the overall specialized rewriter together with its Wf and Interp proofs. (This package should perhaps move to another file?)

• ProofsCommonTactics.v: Defines the actual tactics used to prove that specific rewrite rules are correct, and to inhabit the packages defined in ProofsCommon.v.

• Wf.v: Proves wf-preservation of the generic rewriter, taking in wf-goodness of rewrite rules as a hypothesis.

• InterpProofs.v: Proves interp-correctness of the generic rewriter, taking in interp-goodness of rewrite rules as a hypothesis.

• AllTactics.v: Defines the tactic RewriteRules.Tactic.make_rewriter (and a similar tactic notation) which build the entire VerifiedRewriter. They take in the include_interp as in Rewriter.v tactics, as well as an hlist of proofs of rewrite rules indexed over a list (bool * Prop) of rewrite rule types. This is the primary interface for defining a rewriter from a list of rewrite rules.

• AllTacticsExtra.v: Specializes AllTactics.v to what's defined in Identifier.v

• {NBE, Arith, ArithWithCasts, StripLiteralCasts, ToFancy, ToFancyWithCasts}.v: Use the tactic from AllTactics.v together with the proven list of rewrite rules from RulesProofs.v to reify and reduce the corresponding pass and generate a rewriter.

• All.v: Definitionless file that Exports the rewriters defined in Rewriter/*.v

Proofs files: For Language.v, there is a semi-arbitrary split between two files Language.Inversion and Language.Wf.

• Inversion.v:

  • classifies equality of type codes and exprs
  • type codes have decidable equality
  • correctness of the various type-transport definitions
  • correctness lemmas for the various expr.invert_* definitions
  • correctness lemmas for the various reify_* definitions in Language.v
  • inversion_type, which inverts equality of type codes
  • type_beq_to_eq, which converts boolean equality of types to Leibniz equality
  • rewrite_type_transport_correct, which rewrites with the correctness lemmas of the various type-transport definitions
  • type.invert_one e which does case analysis on any inductive type indexed over type codes, in a way that preserves information about the type of e, and generally works even when the goal is dependently typed over e and/or its type
  • ident.invert, which does case-anaylsis on idents whose type has structure (i.e., is not a var)
  • ident.invert_match, which does case-analysis on idents appearing as the discriminee of a match in the goal or in any hypothesis
  • expr.invert, which does case-anaylsis on exprs whose type has structure (i.e., is not a var)
  • expr.invert_match, which does case-analysis on exprs appearing as the discriminee of a match in the goal or in any hypothesis
  • expr.invert_subst, which does case-analysis on exprs which show up in hypotheses of the form expr.invert_* _ = Some _
  • expr.inversion_expr, which inverts equalities of exprs

• Wf.v: Depends on Inversion.v Defines:

  • expr.wf, expr.Wf, expr.wf3, expr.Wf3
  • GeneralizeVar.Flat.wf
  • expr.inversion_wf (and variants), which invert wf hypotheses
  • expr.wf_t (and variants wf_unsafe_t and wf_safe_t) which make progress on wf goals; wf_safe_t should never turn a provable goal into an unprovable one, while wf_unsafe_t might.
  • expr.interp_t (and variants), which should make progress on equivalence-of-interp hypotheses and goals, but is not used much (mainly because I forgot I had defined it)
  • prove_Wf, which proves wf goals on concrete syntax trees in a more optimized way than repeat constructor Proves:
  • funext → (type.eqv ↔ Logic.eq) (eqv_iff_eq_of_funext)
  • type.related and type.eqv are PERs
  • Proper instances for type.app_curried, type.and_for_each_lhs_of_arrow
  • type.is_not_higher_order → Reflexive (type.and_for_each_lhs_of_arrow type.eqv)
  • iff between type.related{,_hetero} and related of type.app_curried
  • various properties of type.and{,b_bool}for_each_lhs_of_arrow
  • various properties of type.eqv and ident.{gen_,}interp
  • various properties of ident.cast
  • various properties of expr.wf (particularly of things defined in Language.v)
  • interp and wf proofs for the passes to/from Flat

• UnderLetsProofs.v: wf and interp lemmas for the various passes defined in UnderLets.v

• MiscCompilerPassesProofs.v: wf and interp lemmas for the various passes defined in MiscCompilerPasses.v

• AbstractInterpretation/ZRangeProofs.v: Proves correctness lemmas of the per-operation zrange-bounds-analysis functions

• AbstractInterpretation/Wf.v: wf lemmas for the AbstractInterpretation pass

• AbstractInterpretation/Proofs.v: interp lemmas for the AbstractInterpretation pass, and also correctness lemmas that combine Wf and interp