Verifying Low-level Code for Security and Correctness using F*

Program verification: Shall the twain ever meet?

• In the left corner: Very expressive dependently-typed logics,
  but only purely functional programming

• In the right: effectful programming, SMT-based automation,
  but only first-order logic

Bridging the gap: F*

• Functional programming language with effects

  • like OCaml, F#, Haskell, …
  • F* extracted to OCaml or F# by default
  • Subset of F* compiled to efficient C code

• Semi-automated verification system using SMT

  • like Dafny, FramaC, Why3, …

• With an expressive core language based on dependent type theory

  • like Coq, Lean, Agda, Nuprl, …

• A metaprogramming and tactic framework for interactive proof and user-defined automation

  • like Coq, Isabelle, Lean, Nuprl, etc.

A first taste

• Write ML-like code

  let rec factorial n = 
    if n = 0 then 1 
    else n * factorial (n - 1)

• Give it a specification, claiming that factorial is a total function from non-negative to positive integers.

  val factorial: n:int{n >= 0} -> i:int{i >= 1}

• Ask F* to check it

  fstar factorial.fst
  Verified module: Factorial
  All verification conditions discharged successfully

Running F* Programs

Some uses of F*

• Project Everest: verify and deploy new, efficient HTTPS stack
  • miTLS: Verified reference implementation of TLS (1.2 and 1.3)
  • EverParse: Verified parsers and formatter generators
  • EverCrypt: Agile Cryptographic Provider
  • HACL*: High-Assurance Cryptographic Library
  • Vale: Verified Assembly Language for Everest
• Verified Everest code deployed in
  • Firefox (Mozilla NSS crypto)
  • Windows (WinQUIC)
  • Azure Confidential Consortium (Verified Merkle tree for the blockchain)
  • WireGuard VPN
  • Zinc crypto library for Linux
  • Tezos and Concordium blockchains

EverCrypt: A No-Excuses, Verified Cryptographic Provider

• 180KLOC of code & proof, producing 40KLOC of verified C & assembly
  • Memory safe
  • Functionally correct
  • Side-channel resistant

AES-GCM: Authenticated Encryption for 90% of Secure Internet Traffic

The Current Everest and F* team

• many former members, interns, external contributors, … you?

This talk

• Need to write low-level code for performance

  • Cryptographic routines are heavily optimized in C and assembly
  • Networking protocols are inherently effectful

• Need to write low-level code for security

  • Hard to control side-channels with a GC, JIT etc.

• Specifying and verifying effectful programs is hard

  • Much effort involved in designing and structuring specifications, perhaps more effort than programming effectful code itself

  • Typical code to proof ratio for functional correctness and security proofs is ~ 1 : 5 (more like 1 : 20 without SMT)

• Pays to structure specifications well … functional programming can help

Computations indexed by familiar FP structures

Core F* design principle:

• Computation types are indexed by their specifications

• Specs equipped with some algebraic structure for formal manipulation

Later this week

2. Modeling and proving cryptographic security
3. Implementing efficient, verified cryptographic algorithms in Low* (C)
4. Faster, lower, still correct & secure: Verified assembly programs (Vale)?
   4. Relational equivalences in F*?

Try the tutorial: https://fstar-lang.org/tutorial

The Functional Core of F*

Informally, like Coq or Agda, but with an extensional type theory (~Nuprl)

• Refinement types

  let nat = x:int{x>=0}
  

• Recursive functions

  let rec factorial (n:nat) : nat = if n = 0 then 1 else n * (factorial (n - 1))

Refinement types

type nat = x:int{x>=0}

• Refinements introduced by type annotations (code unchanged)

  let rec factorial (n:nat) : nat = if n = 0 then 1 else n * (factorial (n - 1))
  

• Logical obligations discharged by SMT (simplified)

  n >= 0, n <> 0 |= n - 1 >= 0
  n >= 0, n <> 0, factorial (n - 1) >= 0 |= n * (factorial (n - 1)) >= 0

• Refinements eliminated by subtyping: nat<:int

  let i : int = factorial 42
  let f : x:nat{x>0} -> int = factorial

Dependent types

• Dependent function types (), here together with refinements:

  val incr : x:int -> y:int{x < y}
  let incr x = x + 1

• Can express pre- and post- conditions of pure functions

  val incr : x:int -> y:int{y = x + 1}

• Exercise: Can you find other types for incr?

Inductive types and pattern matching

• Inductive datatypes (immutable) and pattern matching

  type list a =
    | Nil  : list a
    | Cons : hd:a -> tl:list a -> list a
  
  let rec map (f:a -> b) (x:list a) : list b =
    match x with
    | [] -> []
    | h :: t -> f h :: map f t

• Lambdas (unnamed, first-class functions)

  map (fun x -> x + 42) [1;2;3]

Inductive types and pattern matching

• Recursive functions over inductive datatypes

  type vec a : nat -> Type =
    | Nil : vec a 0
    | Cons : #n:nat -> hd:a -> tl:vec a n -> vec a (n + 1)
  
  let rec append (v1:vec a n) (v2:vec a m) 
    : vec a (n + m) =
    match v1 with
    | Nil -> v2
    | Cons hd tl -> Cons hd (append tl v2)

• Higher order

    let rec map (f:a -> b) (v:vec a n) 
      : vec b n 
      = match v with
        | Nil -> Nil
        | Cons hd tl -> Cons (f hd) (map f tl)
  
    map (fun x -> x + 42) (Cons 1 Nil)

Total functions in F*

• The F* functions we saw so far were all total

• Tot effect (default) = no side-effects, terminates on all inputs

  • a -> b is really shorthand for a -> Tot b

  val factorial : nat -> nat ~ nat -> Tot nat
  let rec factorial n = (if n = 0 then 1 else n * (factorial (n - 1)))

• Quiz: How about giving this weak type to factorial?

  val factorial : int -> Tot int

  let rec factorial n = (if n = 0 then 1 else n * (factorial (n - 1)))
                                                              ^^^^^
  Subtyping check failed; expected type (x:int{(x << n)}); got type int

Semantic termination checking

• Based on well-founded ordering on expressions (<<)
  • naturals related by < (negative integers unrelated)
  • inductives related by subterm ordering
  • lex tuples %[a;b;c] with lexicographic ordering
  • result of a total function smaller than the function itself (f x << f)
• order constraints discharged by the SMT solver
• arbitrary total expression as decreases metric

  let rec ackermann (m n:nat) : Tot nat (decreases %[m;n]) =
    if m=0 then n + 1
    else if n = 0 then ackermann (m - 1) 1
    else ackermann (m - 1) (ackermann m (n - 1))

• default metric is lex ordering of all (non-function) args

  val ackermann: m:nat -> n:nat -> Tot nat

Values and Computations

• “Values” aka unconditionally total terms

• Two classes of types

  • Value types (t): int, list int, …
  • Computation types (C): conditionally pure, divergent, stateful, …

• F* effect system encapsulates effectful code

  • Pure code cannot call effectful code
  • But, via sub-effecting, pure code can be used in effectful contexts

• Dependent function types of the form: x:t -> C

  • F* is call-by-value
  • argument can't have side-effects, so value type

• Two forms of refinement types

  • Refined value types: x:t{p}
  • Refined computation types …

Refined Computation Types

Monadic effects in F*

• A classic state monad:

type st a = s -> (a * s)
let return x h = x,h
let bind f g h = let x,h' = f h in g x h'
let get () h = h,h
let put h _ = (),h

• Turned into an abstract “computation type” and can be primitively implemented under the hood or not

  new_effect STATE { st; return; bind; get; put }

• Can now program effectful computations directly:

  let double () = put (get () + get ())

• And F* infers double : unit -> STATE unit w

Specifying Effectful Programs

• Type inference in F* includes inference of weakest preconditions

• e : STATE a w means

  forall post h0.        //for all postconditions and initial states
   w post h0 ==>       //given the weakest precondition is valid for post and h0
   e h0 ~>* (v, h1) /\ //the computation reduces to a value and final state
   post (v, h1)        //that validate the postcondition

• What is a weakest precondition predicate transformer?

  let st_wp a : st_post a -> st_pre  //transforms postconditions to preconditions
  where st_post a : (a * s) -> prop //postconditions relate results to memories
  and   st_pre : s -> prop //preconditions are memory predicates

• Dijkstra monad for state

WP inference CPS tranform

let st_wp a : st_post a -> st_pre  //transforms postconditions to preconditions
where st_post a : (a * s) -> prop //postconditions relate results to memories
and   st_pre : s -> prop //preconditions are memory predicates

• Hey, wait a minute … isn't that like the continuation monad?

cont r a = (a -> r) -> r

• Take the continuation monad with result type prop

  s -> cont prop (a * s)
  = s -> ((a * s) -> prop) -> prop
  ~ ((a * s) -> prop) -> s -> prop
  = st_post a -> st_pre
  = st_wp a

• WPs for state are just the state monad transformer applied to the continuation monad with result type prop

  stateT m a = s -> m (a * s)
  st_wp a = stateT (cont prop) a

Auto-generating WP calculi for monads

• F* automatically derives a weakest precondition calculus for m-effectful computations whose WPs have the form

  m_wp a = mT (cont prop) a

  • the m transformer on the continuation monad with result type prop

• And infers computations types of the form M a (wp : m_wp a)

• E.g., for exceptions

  ex_wp a = ex_t (cont prop) a
        = (option a -> prop) -> prop

• For state + exceptions

  st_ex_wp a = state_t (ex_t (cont prop)) a
           = s -> (ex_t (cont prop) a * s)
           = s -> ((option a * state) -> prop) -> prop
           ~ ((option a * state) -> prop) -> s -> prop
           = st_ex_post a -> st_ex_pre

Derived Specifications: Hoare triples

• Reasoning about continuations: great for a core logic / not for a human

• Hoare logic-style pre-conditions and post-conditions

  ST a (pre: s -> prop) (post: s -> a -> s -> prop) =
  STATE a (fun k s0 -> pre s0 /\ (forall x s1. post s0 x s1 ==> k (x, s1)))

• stateful pre-condition is predicate on initial states

• post-condition is relation on initial states, results, and final states

val get ()
  : ST s
    (requires fun s -> True)
    (ensures fun s0 result s1 -> s0 == result /\ result == s1)
    
val put s
  : ST unit
    (requires fun _ -> True)
    (ensures fun _ _ s1 -> s1 == s0)

Richer Model of Mutable Memory

with Lens-indexed Imperative Lenses

Richer memory models

• Program libraries to model memory, e.g., the ML heap

• Derive effectful actions for primitive operations (e.g., !, := etc.)

• Write effectful programs against these libraries and verify them with refined computation types

• Extract them to programs in ML or C with primtive effects

• F*:

  let incr (r:ref int) : ST unit (requires (fun _ -> True))
           (ensures (fun h0 _ h1 -> modifies !{r} h0 h1 /\ sel h2 r = sel h1 r + 1))
      = r := !r + 1

  ML:

  let incr (r:ref int) : unit = r := !r + 1

  C:

  void incr (int *r) { *r = *r + 1; }

Modeling the ML heap

A sketch of FStar.Heap:

module Heap
  let addr = nat
  abstract let heap = {
    next_addr: addr;
    map: addr -> option (a:Type & v:a) {
       forall a. h > next_addr ==> map a == None
    }
  }
  abstract let ref t = addr  
  let contains h (r:ref t) = r < h.next_addr /\ h.map r == Some (t, _)
  let sel h (r:ref t{h `contains` r}) = let Some (_, v) = h.map r in v
  let upd h (r:ref t{h `contains` r}) v = ...

More on modeling heaps

Deriving ML-like Effectful Operations

• Reading references

  let (!) #t (r:ref t)
    : ST t
        (requires fun h -> h `contains` r)
        (ensures fun h0 x h1 -> h0 == h1 /\ x = sel h1 r) =
    sel (get()) r

• Writing references

  let (:=) #t (r:ref t) (v:t)
    : ST (ref t)
        (requires fun h -> h `contains` r)
        (ensures fun h0 x h1 -> h1 == upd h0 r v) =
    put (upd (get()) r v); r

• Allocating and freeing references …

Freeing references

let free #r (r:ref t)
  : ST unit
    (requires fun h -> h `contains` r)
    (ensures fun h0 () h1 -> modifies !{r} h0 h1)
  = let h = get () in
    put ({h with map = fun a -> if a = r then None else h.map a})

Bidirectional data access, abstractly with lenses

type lens a b = {
    get : a -> b;
    put : b -> a -> a
}

• sel and upd form a lens

  let ref_lens : lens (heap * ref a) a  = {
    get = fun (h, r) -> sel h r;
    put = fun v (h, r) -> upd h r v
  }

Imperative lenses

• A lens-indexed computation type:

  type st_lens inv (l:lens (heap * a) b) = {
   st_get : x:a
         -> ST b
            (requires fun h -> inv h x)
            (ensures fun h0 y h1 -> h0==h1 /\ y == l.get (h0, x));
  
   st_put : y:b 
         -> x:a 
         -> ST a
            (requires fun h -> inv h x)
            (ensures fun h0 x' h1 -> h1, x' == l.put y (h0, x))
  }

• (!) and (:=) are imperative lenses

  let iref_lens : st_lens contains ref_lens = {
    st_get = (!);
    st_put = (:=);
  }

More on lens-indexed imperative lenses

Manipulating Binary-formatted Data

with the EverParse parser combinator library

Networking Protocols: Many performance and security-critical parsers

• TLS 1.3 message format, with variable-length data

  uint16 ProtocolVersion; opaque Random[32]; uint8 CipherSuite[2];    ​
  ​
  struct {​
       ProtocolVersion legacy_version = 0x0303;​
       Random          random;​
       opaque          legacy_session_id<0..32>;​
       CipherSuite     cipher_suites<2..2^16-2>;​
       opaque          legacy_compression_methods<1..2^8-1>;​
       Extension       extensions<8..2^16-1>;​
  } ClientHello;

• De facto: Hand-written parsers and formatters, for performance, but also many vulnerabilities

  • Heartbleed was a parsing bug

EverParse: A Verified Low-level Parser Generator

• Given a type description (e.g., ClientHello)

• Produces a parser and a formatter indexed by its (inverse) parser

let parser t = b:bytes -> option (t * n:nat{n <= length b})
let format (p:parser t) = f:(t -> bytes) {
    forall x. p (f x) == Some (x, length (f x))
}

• And a low-level zero-copy validator indexed by the parser

  let validator (p:parser t) = 
    bs:array uint8 ->
    pos:u32 { pos <= length bs } ->
    ST u32
    (requires fun h0 -> h0 `contains` bs)
    (ensures fun h0 res h1 ->
        h0 == h1 /\
        (if res < ERROR_CODE
         then exists v. p (as_bytes bs h0 pos) = Some (v, res) (* parsing succeeds *)
         else p (as_bytes bs h0 pos) = None (* parsing fails *)))

Combinator library for validators

• Mirroring the structure of parser combinators

val return : validator unit_parser
val bind : validator p -> (t -> validator q) -> validator (p `Parser.bind` q)
val seq: validator p -> validator s -> validator (p `Parser.seq` s)
val map : f:(t -> s)  -> validator p -> -> validator (Parser.map f p)
...

• Client-hello validator

  protocolVersion_validator​
  `seq` random_validator​
  `seq` sessionID_validator​
  `seq` clientHello_cipher_suites_validator​
  `seq` clientHello_compression_method_validator​
  `seq` clientHelloExtensions_validator​

• Imperative code refines pure parser by construction

Data accessors

• Structured access into validated binary data, specified by parsers and lenses

  let accessor (p1:parser t1) (p2:parser t2) (l:lens t1 t2) =
    bs:array uint8 ->
    pos:u32 { pos <= length bs } ->
    ST u32
    (requires fun h0 ->
        h0 `contains` bs /\
        Some? (p1 (as_bytes bs h0 pos))) (* bs contains a valid t1 *))
    (ensures fun h0 res h1 ->
        h0 == h1 /\
        let Some (v1, _) = p1 (as_bytes bs h0 pos) in
        p2 (as_bytes bs h0 res) == Some (l.get v1, _)) (* returns offset to t2 *)
    

• Abstractly, structured data access into a (representation of) t1

• Concretely, just performs arithmetic into the binary formatted data

  • Effectively, reads directly from computed offsets into network packets
  • Would be crazy to try this directly in C!

Extracted to efficient C code

• After partial evaluation and proof erasure …

uint32_t Parsers_ClientHello_clientHello_validator(LowParse_Low_Base_slice input, uint32_t pos)​
{​
  uint32_t pos10 = Parsers_ProtocolVersion_protocolVersion_validator(input, pos);​
  uint32_t pos11;​
  if (pos10 > ERROR_CODE)​
    pos11 = pos10;​
  else​
    pos11 = Parsers_Random_random_validator(input, pos10);​
  uint32_t pos12;​
  if (pos11 > ERROR_CODE)​
    pos12 = pos11;​
  else​
    pos12 = Parsers_SessionID_sessionID_validator(input, pos11);​
  uint32_t pos13;​
  if (pos12 > ERROR_CODE)​
    pos13 = pos12;​
  else​
    pos13 = Parsers_ClientHello_cipher_suites_clientHello_cipher_suites_validator(input, pos12);​
  uint32_t pos1;​
  ...
}​

Performance of extracted C code

Wrapping up

• Write low-level code if you must

• But, program it tastefully in a proof assistant, not directly in C or asm

• Thoughtful structuring of imperative coding patterns can make reasoning about imperative programs similar to functional programming

• And with proofs, partial evaluation and proof erasure, the resulting code can be significantly faster than hand-written C

• Homework: Try Chapters 1 – 7 of https://fstar-lang.org/tutorial

Bonus: From F* to SMT

SMT encoding

• F* verification conditions

  • classical, dependently typed, higher-order logic

• We encode this to SMT

  • FOL + equality and function symbols + arithmetic + e-matching
  • goals: soundness, predictability, efficiency, scalability
  • pragmatic balance between completeness and practical tractability

• The encoding

  • preserves types
  • combines a deep and a shallow embedding of F* terms
  • allows bounded unrolling of recursive and inductive definitions
  • eliminates first-class functions by lambda lifting
  • uses SMT-patterns extensively to guide instantiation of quantifiers
  • currently targets only Z3 (but some early experiments with CVC4)

Encoding primitive operations and types

• Primitive types (booleans, integers…) are boxed

  (declare-sort Term)
  
  (declare-fun BoxInt (Int) Term)
  (declare-fun BoxInt_proj (Term) Int)
  (assert (forall ((@x Int)) (= (BoxInt_proj (BoxInt @x)) @x))) 
  
  (declare-fun Prims.int () Type)
  (assert (forall ((@x Int)) (HasType (BoxInt @x) Prims.int))))

  Operators are given shallow semantics:

   (assert (forall ((@x Term) (@y Term))
     (= (op_Subtraction @x @y) (BoxInt (- (BoxInt_proj @x) (BoxInt_proj @y))))))

Typechecking factorial

type nat = x:int{x>=0}
let rec factorial (n:nat) : nat = if n = 0 then 1 else n * (factorial (n - 1))

• Encoding type nat:

  (assert (forall ((@x Term))
  (iff (HasType @x nat) (and (HasType @x int) (>= (BoxInt_proj @x) 0)))))

• Proving termination:

  (declare-fun n () Term)
  (assert (not (implies (and (HasType n nat) (not (= n (BoxInt 0))))
                        (Valid (Precedes (op_Subtraction n (BoxInt 1)) n)))))

• Can be proved because of this axiom:

  (assert (forall ((@x Term) (@y Term))
           (implies (and (HasType @x nat) (HasType @y nat)
                         (< (BoxInt_proj @x) (BoxInt_proj @y)))
                    (Valid (Precedes @x @y)))))

Typechecking factorial (2)

type nat = x:int{x>=0}
let rec factorial (n:nat) : nat = if n = 0 then 1 else n * (factorial (n - 1))

• Return value is nat

  (declare-fun n () Term)
  (assert (HasType n nat))
  (assert (forall ((@x Term))
                (implies (and (HasType @x nat) (Valid (Precedes @x n)))
                         (HasType (factorial @x) nat))))

  (assert (not (ite (BoxBool_proj (op_Equality int n (BoxInt 0)))
                  (>= 1 0)
                  (>= (BoxInt_proj (op_Multiply n
                        (factorial (op_Subtraction n (BoxInt 1))))) 0 ))))

Allowing SMT solver to do bounded unrolling

type nat = x:int{x>=0}
let rec factorial (n:nat) : nat = if n = 0 then 1 else n * (factorial (n - 1))

(declare-datatypes () ((Fuel (ZFuel) (SFuel (prec Fuel)))))
(declare-fun MaxFuel () Fuel)
(assert (= MaxFuel (SFuel (SFuel ZFuel))))

(assert (forall ((@f Fuel) (x Term))
        (implies (HasType x nat)
                 (= (factorial_fuel (SFuel @f) x)
                    (ite (op_Equality int x (BoxInt 0))
                         (BoxInt 1)
                         (op_Multiply x
                                (factorial_fuel @f
                                     (op_Subtraction x (BoxInt 1)))))))))

(assert (forall ((@x Term)) (= (factorial @x) (factorial_fuel MaxFuel @x))))