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EVM Words

Module EVM-DATA

EVM uses bounded 256 bit integer words, and sometimes also bytes (8 bit words). Here we provide the arithmetic of these words, as well as some data-structures over them. Both are implemented using K's Int.

requires "krypto.k"

module EVM-DATA
    imports KRYPTO
    imports STRING-BUFFER
    imports MAP

    syntax KResult ::= Int

JSON Formatting

The JSON format is used extensively for communication in the Ethereum circles. Writing a JSON-ish parser in K takes 6 lines.

    syntax JSONList ::= List{JSON,","}
    syntax JSONKey  ::= String | Int
    syntax JSON     ::= String
                      | JSONKey ":" JSON
                      | "{" JSONList "}"
                      | "[" JSONList "]"
 // ------------------------------------

Utilities

Important Powers

Some important numbers that are referred to often during execution. These can be used for pattern-matching on the LHS of rules as well (macro attribute expands all occurances of these in rules).

    syntax Int ::= "pow256" /* 2 ^Int 256 */
                 | "pow255" /* 2 ^Int 255 */
                 | "pow160" /* 2 ^Int 160 */
                 | "pow16"  /* 2 ^Int 16  */
 // ----------------------------------------
    rule pow256 => 115792089237316195423570985008687907853269984665640564039457584007913129639936 [macro]
    rule pow255 => 57896044618658097711785492504343953926634992332820282019728792003956564819968  [macro]
    rule pow160 => 1461501637330902918203684832716283019655932542976 [macro]
    rule pow16  => 65536 [macro]

    syntax Int ::= "minSInt128"
                 | "maxSInt128"
                 | "minUInt8"
                 | "maxUInt8"
                 | "minUInt48"
                 | "maxUInt48"
                 | "minUInt128"
                 | "maxUInt128"
                 | "minUInt160"
                 | "maxUInt160"
                 | "minSInt256"
                 | "maxSInt256"
                 | "minUInt256"
                 | "maxUInt256"
                 | "minSFixed128x10"
                 | "maxSFixed128x10"
                 | "minUFixed128x10"
                 | "maxUFixed128x10"
 // --------------------------------
    rule minSInt128      => -170141183460469231731687303715884105728                                        [macro]  /*  -2^127      */
    rule maxSInt128      =>  170141183460469231731687303715884105727                                        [macro]  /*   2^127 - 1  */
    rule minSFixed128x10 => -1701411834604692317316873037158841057280000000000                              [macro]  /* (-2^127    ) * 10^10 */
    rule maxSFixed128x10 =>  1701411834604692317316873037158841057270000000000                              [macro]  /* ( 2^127 - 1) * 10^10 */
    rule minSInt256      => -57896044618658097711785492504343953926634992332820282019728792003956564819968  [macro]  /*  -2^255      */
    rule maxSInt256      =>  57896044618658097711785492504343953926634992332820282019728792003956564819967  [macro]  /*   2^255 - 1  */

    rule minUInt8        =>  0                                                                              [macro]
    rule maxUInt8        =>  255                                                                            [macro]
    rule minUInt48       =>  0                                                                              [macro]
    rule maxUInt48       =>  281474976710655                                                                [macro]  /*   2^48 -  1  */
    rule minUInt128      =>  0                                                                              [macro]
    rule maxUInt128      =>  340282366920938463463374607431768211455                                        [macro]  /*   2^128 - 1  */
    rule minUFixed128x10 =>  0                                                                              [macro]
    rule maxUFixed128x10 =>  3402823669209384634633746074317682114550000000000                              [macro]  /* ( 2^128 - 1) * 10^10 */
    rule minUInt160      =>  0                                                                              [macro]
    rule maxUInt160      =>  1461501637330902918203684832716283019655932542975                              [macro]  /*   2^160 - 1  */
    rule minUInt256      =>  0                                                                              [macro]
    rule maxUInt256      =>  115792089237316195423570985008687907853269984665640564039457584007913129639935 [macro]  /*   2^256 - 1  */
  • Range of types
    syntax Bool ::= #rangeSInt    ( Int , Int )
                  | #rangeUInt    ( Int , Int )
                  | #rangeSFixed  ( Int , Int , Int )
                  | #rangeUFixed  ( Int , Int , Int )
                  | #rangeAddress ( Int )
                  | #rangeBytes   ( Int , Int )
 // -------------------------------------------
    rule #rangeSInt    ( 128 ,      X ) => #range ( minSInt128      <= X <= maxSInt128      ) [macro]
    rule #rangeSInt    ( 256 ,      X ) => #range ( minSInt256      <= X <= maxSInt256      ) [macro]
    rule #rangeUInt    (  48 ,      X ) => #range ( minUInt48       <= X <= maxUInt48       ) [macro]
    rule #rangeUInt    ( 128 ,      X ) => #range ( minUInt128      <= X <= maxUInt128      ) [macro]
    rule #rangeUInt    ( 256 ,      X ) => #range ( minUInt256      <= X <= maxUInt256      ) [macro]
    rule #rangeSFixed  ( 128 , 10 , X ) => #range ( minSFixed128x10 <= X <= maxSFixed128x10 ) [macro]
    rule #rangeUFixed  ( 128 , 10 , X ) => #range ( minUFixed128x10 <= X <= maxUFixed128x10 ) [macro]
    rule #rangeAddress (            X ) => #range ( minUInt160      <= X <= maxUInt160      ) [macro]
    rule #rangeBytes   (  32 ,      X ) => #range ( minUInt256      <= X <= maxUInt256      ) [macro]

    syntax Bool ::= "#range" "(" Int "<"  Int "<"  Int ")"
                  | "#range" "(" Int "<"  Int "<=" Int ")"
                  | "#range" "(" Int "<=" Int "<"  Int ")"
                  | "#range" "(" Int "<=" Int "<=" Int ")"
 // ------------------------------------------------------
    rule #range ( LB <  X <  UB ) => LB  <Int X andBool X  <Int UB [macro]
    rule #range ( LB <  X <= UB ) => LB  <Int X andBool X <=Int UB [macro]
    rule #range ( LB <= X <  UB ) => LB <=Int X andBool X  <Int UB [macro]
    rule #range ( LB <= X <= UB ) => LB <=Int X andBool X <=Int UB [macro]
  • chop interperets an integer modulo $2^256$.
    syntax Int ::= chop ( Int ) [function, smtlib(chop)]
 // ----------------------------------------------------
    rule chop ( I:Int ) => I modInt pow256 [concrete, smt-lemma]

Boolean Conversions

Primitives provide the basic conversion from K's sorts Int and Bool to EVM's words.

  • bool2Word interperets a Bool as a Int.
  • word2Bool interperets a Int as a Bool.
    syntax Int ::= bool2Word ( Bool ) [function]
 // --------------------------------------------
    rule bool2Word( B:Bool ) => 1 requires B
    rule bool2Word( B:Bool ) => 0 requires notBool B

    syntax Bool ::= word2Bool ( Int ) [function]
 // --------------------------------------------
    rule word2Bool( W ) => false requires W  ==Int 0
    rule word2Bool( W ) => true  requires W =/=Int 0
  • sgn gives the twos-complement interperetation of the sign of a word.
  • abs gives the twos-complement interperetation of the magnitude of a word.
    syntax Int ::= sgn ( Int ) [function]
                 | abs ( Int ) [function]
 // -------------------------------------
    rule sgn(I) => -1 requires I >=Int pow255
    rule sgn(I) => 1  requires I <Int pow255

    rule abs(I) => 0 -Word I requires sgn(I) ==Int -1
    rule abs(I) => I         requires sgn(I) ==Int 1
  • #signed : uInt256 -> sInt256 (i.e., [minUInt256..maxUInt256] -> [minSInt256..maxSInt256])
  • #unsigned : sInt256 -> uInt256 (i.e., [minSInt256..maxSInt256] -> [minUInt256..maxUInt256])
    syntax Int ::= #signed ( Int ) [function]
 // -----------------------------------------
    rule #signed(DATA) => DATA
      requires 0 <=Int DATA andBool DATA <=Int maxSInt256

    rule #signed(DATA) => DATA -Int pow256
      requires maxSInt256 <Int DATA andBool DATA <=Int maxUInt256

    syntax Int ::= #unsigned ( Int ) [function]
 // -----------------------------------------
    rule #unsigned(DATA) => DATA
      requires 0 <=Int DATA andBool DATA <=Int maxSInt256

    rule #unsigned(DATA) => pow256 +Int DATA
      requires minSInt256 <=Int DATA andBool DATA <Int 0

Empty Account

  • .Account represents the case when an account ID is referenced in the yellowpaper, but the actual value of the account ID is the empty set. This is used, for example, when referring to the destination of a message which creates a new contract.
    syntax Account ::= ".Account" | Int

Symbolic Words

  • #symbolicWord generates a fresh existentially-bound symbolic word.
    syntax Int ::= "#symbolicWord" [function]
 // -----------------------------------------
    rule #symbolicWord => chop ( ?X:Int )

Word Operations

Low-Level

  • up/Int performs integer division but rounds up instead of down.

NOTE: Here, we choose to add I2 -Int 1 to the numerator beforing doing the division to mimic the C++ implementation. You could alternatively calculate I1 modInt I2, then add one to the normal integer division afterward depending on the result.

    syntax Int ::= Int "up/Int" Int [function]
 // ------------------------------------------
    rule I1 up/Int 0  => 0
    rule I1 up/Int 1  => I1
    rule I1 up/Int I2 => (I1 +Int (I2 -Int 1)) /Int I2 requires I2 >Int 1
  • log256Int returns the log base 256 (floored) of an integer.
    syntax Int ::= log256Int ( Int ) [function]
 // -------------------------------------------
    rule log256Int(N) => log2Int(N) /Int 8

The corresponding <op>Word operations automatically perform the correct modulus for EVM words.

    syntax Int ::= Int "+Word" Int [function]
                 | Int "*Word" Int [function]
                 | Int "-Word" Int [function]
                 | Int "/Word" Int [function]
                 | Int "%Word" Int [function]
 // -----------------------------------------
    rule W0 +Word W1 => chop( W0 +Int W1 )
    rule W0 -Word W1 => chop( W0 -Int W1 ) requires W0 >=Int W1
    rule W0 -Word W1 => chop( (W0 +Int pow256) -Int W1 ) requires W0 <Int W1
    rule W0 *Word W1 => chop( W0 *Int W1 )
    rule W0 /Word W1 => 0                    requires W1  ==Int 0
    rule W0 /Word W1 => chop( W0 /Int W1 )   requires W1 =/=Int 0
    rule W0 %Word W1 => 0                    requires W1  ==Int 0
    rule W0 %Word W1 => chop( W0 modInt W1 ) requires W1 =/=Int 0

Care is needed for ^Word to avoid big exponentiation. The helper powmod is a totalization of the operator _^%Int__ (which comes with K). _^%Int__ is not defined when the modulus (third argument) is zero, but powmod is.

    syntax Int ::= Int "^Word" Int       [function]
    syntax Int ::= powmod(Int, Int, Int) [function]
 // -----------------------------------------------
    rule W0 ^Word W1 => powmod(W0, W1, pow256)

    rule powmod(W0, W1, W2) => W0 ^%Int W1 W2  requires W2 =/=Int 0
    rule powmod(W0, W1, W2) => 0               requires W2  ==Int 0

/sWord and %sWord give the signed interperetations of /Word and %Word.

    syntax Int ::= Int "/sWord" Int [function]
                 | Int "%sWord" Int [function]
 // ------------------------------------------
    rule W0 /sWord W1 => #sgnInterp(sgn(W0) *Int sgn(W1) , abs(W0) /Word abs(W1))
    rule W0 %sWord W1 => #sgnInterp(sgn(W0)              , abs(W0) %Word abs(W1))

    syntax Int ::= #sgnInterp ( Int , Int ) [function]
 // --------------------------------------------------
    rule #sgnInterp( W0 , W1 ) => 0          requires W0 ==Int 0
    rule #sgnInterp( W0 , W1 ) => W1         requires W0 >Int 0
    rule #sgnInterp( W0 , W1 ) => 0 -Word W1 requires W0 <Int 0

Word Comparison

The <op>Word comparisons similarly lift K operators to EVM ones:

    syntax Int ::= Int "<Word"  Int [function]
                 | Int ">Word"  Int [function]
                 | Int "<=Word" Int [function]
                 | Int ">=Word" Int [function]
                 | Int "==Word" Int [function]
 // ------------------------------------------
    rule W0 <Word  W1 => bool2Word(W0 <Int  W1)
    rule W0 >Word  W1 => bool2Word(W0 >Int  W1)
    rule W0 <=Word W1 => bool2Word(W0 <=Int W1)
    rule W0 >=Word W1 => bool2Word(W0 >=Int W1)
    rule W0 ==Word W1 => bool2Word(W0 ==Int W1)
  • s<Word implements a less-than for Word (with signed interperetation).
    syntax Int ::= Int "s<Word" Int [function]
 // ------------------------------------------
    rule W0 s<Word W1 => W0 <Word W1           requires sgn(W0) ==K 1  andBool sgn(W1) ==K 1
    rule W0 s<Word W1 => bool2Word(false)      requires sgn(W0) ==K 1  andBool sgn(W1) ==K -1
    rule W0 s<Word W1 => bool2Word(true)       requires sgn(W0) ==K -1 andBool sgn(W1) ==K 1
    rule W0 s<Word W1 => abs(W1) <Word abs(W0) requires sgn(W0) ==K -1 andBool sgn(W1) ==K -1

Bitwise Operators

Bitwise logical operators are lifted from the integer versions.

    syntax Int ::= "~Word" Int       [function]
                 | Int "|Word"   Int [function]
                 | Int "&Word"   Int [function]
                 | Int "xorWord" Int [function]
 // -------------------------------------------
    rule ~Word W       => chop( W xorInt (pow256 -Int 1) )
    rule W0 |Word   W1 => chop( W0 |Int W1 )
    rule W0 &Word   W1 => chop( W0 &Int W1 )
    rule W0 xorWord W1 => chop( W0 xorInt W1 )
  • bit gets bit $N$ (0 being MSB).
  • byte gets byte $N$ (0 being the MSB).
    syntax Int ::= bit  ( Int , Int ) [function]
                 | byte ( Int , Int ) [function]
 // --------------------------------------------
    rule bit (N, _) => 0 requires notBool (N >=Int 0 andBool N <Int 256)
    rule byte(N, _) => 0 requires notBool (N >=Int 0 andBool N <Int  32)

    rule bit (N, W) => bitRangeInt(W , (255 -Int N)        , 1) requires N >=Int 0 andBool N <Int 256
    rule byte(N, W) => bitRangeInt(W , ( 31 -Int N) *Int 8 , 8) requires N >=Int 0 andBool N <Int  32
  • #nBits shifts in $N$ ones from the right.
  • #nBytes shifts in $N$ bytes of ones from the right.
  • _<<Byte_ shifts an integer 8 bits to the left.
    syntax Int ::= #nBits  ( Int )  [function]
                 | #nBytes ( Int )  [function]
                 | Int "<<Byte" Int [function]
 // ------------------------------------------
    rule #nBits(N)  => (1 <<Int N) -Int 1 requires N >=Int 0
    rule #nBytes(N) => #nBits(N *Int 8)   requires N >=Int 0
    rule N <<Byte M => N <<Int (8 *Int M)
  • signextend(N, W) sign-extends from byte $N$ of $W$ (0 being MSB).
    syntax Int ::= signextend( Int , Int ) [function]
 // -------------------------------------------------
    rule signextend(N, W) => W requires N >=Int 32 orBool N <Int 0
    rule signextend(N, W) => chop( (#nBytes(31 -Int N) <<Byte (N +Int 1)) |Int W ) requires N <Int 32 andBool N >=Int 0 andBool         word2Bool(bit(256 -Int (8 *Int (N +Int 1)), W))
    rule signextend(N, W) => chop( #nBytes(N +Int 1)                      &Int W ) requires N <Int 32 andBool N >=Int 0 andBool notBool word2Bool(bit(256 -Int (8 *Int (N +Int 1)), W))
  • keccak serves as a wrapper around the Keccak256 in KRYPTO.
    syntax Int ::= keccak ( WordStack ) [function, smtlib(smt_keccak)]
 // ------------------------------------------------------------------
    rule keccak(WS) => #parseHexWord(Keccak256(#unparseByteStack(WS))) [concrete]

Data-Structures over Word

A WordStack for EVM

As a cons-list

A cons-list is used for the EVM wordstack.

  • .WordStack serves as the empty worstack, and
  • _:_ serves as the "cons" operator.
    syntax WordStack [flatPredicate]
    syntax WordStack ::= ".WordStack" | Int ":" WordStack
 // -----------------------------------------------------
  • _++_ acts as WordStack append.
  • #take(N , WS) keeps the first $N$ elements of a WordStack (passing with zeros as needed).
  • #drop(N , WS) removes the first $N$ elements of a WordStack.
    syntax WordStack ::= WordStack "++" WordStack [function, right]
 // ---------------------------------------------------------------
    rule .WordStack ++ WS' => WS'
    rule (W : WS)   ++ WS' => W : (WS ++ WS')

    syntax WordStack ::= #take ( Int , WordStack ) [function]
 // ---------------------------------------------------------
    rule #take(0, WS)         => .WordStack
    rule #take(N, .WordStack) => 0 : #take(N -Int 1, .WordStack) requires N >Int 0
    rule #take(N, (W : WS))   => W : #take(N -Int 1, WS)         requires N >Int 0

    syntax WordStack ::= #drop ( Int , WordStack ) [function]
 // ---------------------------------------------------------
    rule #drop(0, WS)         => WS
    rule #drop(N, .WordStack) => .WordStack
    rule #drop(N, (W : WS))   => #drop(N -Int 1, WS) requires N >Int 0

Element Access

  • WS [ N ] accesses element $N$ of $WS$.
  • WS [ N .. W ] access the range of WS beginning with N of width W.
  • WS [ N := W ] sets element $N$ of $WS$ to $W$ (padding with zeros as needed).
    syntax Int ::= WordStack "[" Int "]" [function]
 // -----------------------------------------------
    rule (W0 : WS)   [N] => W0           requires N ==Int 0
    rule (.WordStack)[N] => 0            requires N >Int 0
    rule (W0 : WS)   [N] => WS[N -Int 1] requires N >Int 0

    syntax WordStack ::= WordStack "[" Int ".." Int "]" [function]
 // --------------------------------------------------------------
    rule WS [ START .. WIDTH ] => #take(WIDTH, #drop(START, WS))

    syntax WordStack ::= WordStack "[" Int ":=" Int "]" [function]
 // --------------------------------------------------------------
    rule (W0 : WS)  [ N := W ] => W  : WS                             requires N ==Int 0
    rule .WordStack [ N := W ] => 0  : (.WordStack [ N -Int 1 := W ]) requires N >Int 0
    rule (W0 : WS)  [ N := W ] => W0 : (WS [ N -Int 1 := W ])         requires N >Int 0
  • #sizeWordStack calculates the size of a WordStack.
  • _in_ determines if a Int occurs in a WordStack.
    syntax Int ::= #sizeWordStack ( WordStack )       [function, smtlib(sizeWordStack)]
                 | #sizeWordStack ( WordStack , Int ) [function, klabel(sizeWordStackAux), smtlib(sizeWordStackAux)]
 // ----------------------------------------------------------------------------------------------------------------
    rule #sizeWordStack ( WS ) => #sizeWordStack(WS, 0)
    rule #sizeWordStack ( .WordStack, SIZE ) => SIZE
    rule #sizeWordStack ( W : WS, SIZE )     => #sizeWordStack(WS, SIZE +Int 1)

    syntax Bool ::= Int "in" WordStack [function]
 // ---------------------------------------------
    rule W in .WordStack => false
    rule W in (W' : WS)  => (W ==K W') orElseBool (W in WS)
  • #padToWidth(N, WS) and #padRightToWidth make sure that a WordStack is the correct size.
    syntax WordStack ::= #padToWidth ( Int , WordStack ) [function]
 // ---------------------------------------------------------------
    rule #padToWidth(N, WS) => WS                     requires notBool #sizeWordStack(WS) <Int N [concrete]
    rule #padToWidth(N, WS) => #padToWidth(N, 0 : WS) requires #sizeWordStack(WS) <Int N         [concrete]

    syntax WordStack ::= #padRightToWidth ( Int , WordStack ) [function]
 // --------------------------------------------------------------------
    rule  #padRightToWidth(N, WS) => #padRightToWidthAux(N -Int #sizeWordStack(WS), WS, .WordStack)

    syntax WordStack ::= #padRightToWidthAux ( Int , WordStack , WordStack ) [function]
 // -----------------------------------------------------------------------------------
    rule #padRightToWidthAux(0, WS, ZEROS) => WS ++ ZEROS

    rule #padRightToWidthAux(N, WS, ZEROS) => #padRightToWidthAux(N -Int 1, WS, 0 : ZEROS)
      requires N >Int 0
  • WordStack2List converts a term of sort WordStack to a term of sort List.
    syntax List ::= WordStack2List ( WordStack ) [function]
 // -------------------------------------------------------
    rule WordStack2List(.WordStack) => .List
    rule WordStack2List(W : WS) => ListItem(W) WordStack2List(WS)

Byte Arrays

The local memory of execution is a byte-array (instead of a word-array).

  • #asWord will interperet a stack of bytes as a single word (with MSB first).
  • #asInteger will interperet a stack of bytes as a single arbitrary-precision integer (with MSB first).
  • #asAccount will interpret a stack of bytes as a single account id (with MSB first). Differs from #asWord only in that an empty stack represents the empty account, not account zero.
  • #asByteStack will split a single word up into a WordStack where each word is a byte wide.
    syntax Int ::= #asWord ( WordStack ) [function, smtlib(asWord)]
 // ---------------------------------------------------------------
    rule #asWord( .WordStack     ) => 0                                    // [concrete]
    rule #asWord( W : .WordStack ) => W                                    // [concrete]
    rule #asWord( W0 : W1 : WS   ) => #asWord(((W0 *Word 256) +Word W1) : WS) [concrete]

    syntax Int ::= #asInteger ( WordStack ) [function]
 // --------------------------------------------------
    rule #asInteger( .WordStack     ) => 0
    rule #asInteger( W : .WordStack ) => W
    rule #asInteger( W0 : W1 : WS   ) => #asInteger(((W0 *Int 256) +Int W1) : WS)

    syntax Account ::= #asAccount ( WordStack ) [function]
 // ------------------------------------------------------
    rule #asAccount( .WordStack ) => .Account
    rule #asAccount( W : WS     ) => #asWord(W : WS)

    syntax WordStack ::= #asByteStack ( Int )             [function]
                       | #asByteStack ( Int , WordStack ) [function, klabel(#asByteStackAux), smtlib(asByteStack)]
 // --------------------------------------------------------------------------------------------------------------
    rule #asByteStack( W ) => #asByteStack( W , .WordStack )                                        [concrete]
    rule #asByteStack( 0 , WS ) => WS                                                            // [concrete]
    rule #asByteStack( W , WS ) => #asByteStack( W /Int 256 , W modInt 256 : WS ) requires W =/=K 0 [concrete]

Addresses

  • #addr turns an Ethereum word into the corresponding Ethereum address (160 LSB).
    syntax Int ::= #addr ( Int ) [function]
 // ---------------------------------------
    rule #addr(W) => W %Word pow160
  • #newAddr computes the address of a new account given the address and nonce of the creating account.
  • #sender computes the sender of the transaction from its data and signature.
    syntax Int ::= #newAddr ( Int , Int ) [function]
 // ------------------------------------------------
    rule #newAddr(ACCT, NONCE) => #addr(#parseHexWord(Keccak256(#rlpEncodeLength(#rlpEncodeBytes(ACCT, 20) +String #rlpEncodeWord(NONCE), 192))))

    syntax Account ::= #sender ( Int , Int , Int , Account , Int , String , Int , WordStack , WordStack ) [function]
                     | #sender ( String , Int , String , String )                                         [function, klabel(#senderAux)]
                     | #sender ( String )                                                                 [function, klabel(#senderAux2)]
 // -------------------------------------------------------------------------------------------------------------------------------------
    rule #sender(TN, TP, TG, TT, TV, DATA, TW, TR, TS)
      => #sender(#unparseByteStack(#parseHexBytes(Keccak256(#rlpEncodeLength(#rlpEncodeWordStack(TN : TP : TG : .WordStack) +String #rlpEncodeAccount(TT) +String #rlpEncodeWord(TV) +String #rlpEncodeString(DATA), 192)))), TW, #unparseByteStack(TR), #unparseByteStack(TS))

    rule #sender(HT, TW, TR, TS) => #sender(ECDSARecover(HT, TW, TR, TS))

    rule #sender("")  => .Account
    rule #sender(STR) => #addr(#parseHexWord(Keccak256(STR))) requires STR =/=String ""
  • #blockHeaderHash computes the hash of a block header given all the block data.
    syntax Int ::= #blockHeaderHash( Int , Int , Int , Int , Int , Int , WordStack , Int , Int , Int , Int , Int , WordStack , Int , Int ) [function, klabel(blockHeaderHash), symbol]
                 | #blockHeaderHash(String, String, String, String, String, String, String, String, String, String, String, String, String, String, String) [function, klabel(#blockHashHeaderStr), symbol]
 // -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   rule #blockHeaderHash(HP, HO, HC, HR, HT, HE, HB, HD, HI, HL, HG, HS, HX, HM, HN)
         => #blockHeaderHash(#asWord(#parseByteStackRaw(HP)),
                             #asWord(#parseByteStackRaw(HO)),
                             #asWord(#parseByteStackRaw(HC)),
                             #asWord(#parseByteStackRaw(HR)),
                             #asWord(#parseByteStackRaw(HT)),
                             #asWord(#parseByteStackRaw(HE)),
                                     #parseByteStackRaw(HB) ,
                             #asWord(#parseByteStackRaw(HD)),
                             #asWord(#parseByteStackRaw(HI)),
                             #asWord(#parseByteStackRaw(HL)),
                             #asWord(#parseByteStackRaw(HG)),
                             #asWord(#parseByteStackRaw(HS)),
                                     #parseByteStackRaw(HX) ,
                             #asWord(#parseByteStackRaw(HM)),
                             #asWord(#parseByteStackRaw(HN)))

    rule #blockHeaderHash(HP, HO, HC, HR, HT, HE, HB, HD, HI, HL, HG, HS, HX, HM, HN)
         => #parseHexWord(Keccak256(#rlpEncodeLength(         #rlpEncodeBytes(HP, 32)
                                                      +String #rlpEncodeBytes(HO, 32)
                                                      +String #rlpEncodeBytes(HC, 20)
                                                      +String #rlpEncodeBytes(HR, 32)
                                                      +String #rlpEncodeBytes(HT, 32)
                                                      +String #rlpEncodeBytes(HE, 32)
                                                      +String #rlpEncodeString(#unparseByteStack(HB))
                                                      +String #rlpEncodeWordStack(HD : HI : HL : HG : HS : .WordStack)
                                                      +String #rlpEncodeString(#unparseByteStack(HX))
                                                      +String #rlpEncodeBytes(HM, 32)
                                                      +String #rlpEncodeBytes(HN, 8),
                                                    192)))

Word Map

Most of EVM data is held in finite maps. We are using the polymorphic Map sort for these word maps.

  • WM [ N := WS ] assigns a contiguous chunk of $WM$ to $WS$ starting at position $W$.
  • #asMapWordStack converts a WordStack to a Map.
  • #range(M, START, WIDTH) reads off $WIDTH$ elements from $WM$ beginning at position $START$ (padding with zeros as needed).
    syntax Map ::= Map "[" Int ":=" WordStack "]" [function]
 // --------------------------------------------------------
    rule WM[ N := .WordStack ] => WM
    rule WM[ N := W : WS     ] => (WM[N <- W])[N +Int 1 := WS]

    syntax Map ::= #asMapWordStack ( WordStack ) [function]
 // -------------------------------------------------------
    rule #asMapWordStack(WS:WordStack) => .Map [ 0 := WS ]

    syntax WordStack ::= #range ( Map , Int , Int )            [function]
    syntax WordStack ::= #range ( Map , Int , Int , WordStack) [function, klabel(#rangeAux)]
 // ----------------------------------------------------------------------------------------
    rule #range(WM, START, WIDTH) => #range(WM, START +Int WIDTH -Int 1, WIDTH, .WordStack)

    rule #range(WM,           END, WIDTH, WS) => WS                                           requires WIDTH ==Int 0
    rule #range(WM,           END, WIDTH, WS) => #range(WM, END -Int 1, WIDTH -Int 1, 0 : WS) requires (WIDTH >Int 0) andBool notBool END in_keys(WM)
    rule #range(END |-> W WM, END, WIDTH, WS) => #range(WM, END -Int 1, WIDTH -Int 1, W : WS) requires (WIDTH >Int 0)
  • #removeZeros removes any entries in a map with zero values.
    syntax Map ::= #removeZeros ( Map ) [function]
 // ----------------------------------------------
    rule #removeZeros( .Map )               => .Map
    rule #removeZeros( KEY |-> 0     REST ) => #removeZeros(REST)
    rule #removeZeros( KEY |-> VALUE REST ) => KEY |-> VALUE #removeZeros(REST) requires VALUE =/=K 0
  • #lookup looks up a key in a map and returns 0 if the key doesn't exist, otherwise returning its value.
    syntax Int ::= #lookup ( Map , Int ) [function]
 // -----------------------------------------------
    rule #lookup( (KEY |-> VAL) M, KEY ) => VAL
    rule #lookup(               M, KEY ) => 0 requires notBool KEY in_keys(M)

Parsing/Unparsing

The EVM test-sets are represented in JSON format with hex-encoding of the data and programs. Here we provide some standard parser/unparser functions for that format.

Parsing

These parsers can interperet hex-encoded strings as Ints, WordStacks, and Maps.

  • #parseHexWord interperets a string as a single hex-encoded Word.
  • #parseHexBytes interperets a string as a hex-encoded stack of bytes.
  • #parseByteStack interperets a string as a hex-encoded stack of bytes, but makes sure to remove the leading "0x".
  • #parseByteStackRaw inteprets a string as a stack of bytes.
  • #parseWordStack interperets a JSON list as a stack of Word.
  • #parseMap interperets a JSON key/value object as a map from Word to Word.
  • #parseAddr interperets a string as a 160 bit hex-endcoded address.
    syntax Int ::= #parseHexWord ( String ) [function]
                 | #parseWord    ( String ) [function]
 // --------------------------------------------------
    rule #parseHexWord("")   => 0
    rule #parseHexWord("0x") => 0
    rule #parseHexWord(S)    => String2Base(replaceAll(S, "0x", ""), 16) requires (S =/=String "") andBool (S =/=String "0x")

    rule #parseWord("") => 0
    rule #parseWord(S)  => #parseHexWord(S) requires lengthString(S) >=Int 2 andBool substrString(S, 0, 2) ==String "0x"
    rule #parseWord(S)  => String2Int(S) [owise]

    syntax WordStack ::= #parseHexBytes  ( String ) [function]
                       | #parseByteStack ( String ) [function]
                       | #parseByteStackRaw ( String ) [function]
 // ----------------------------------------------------------
    rule #parseByteStack(S) => #parseHexBytes(replaceAll(S, "0x", ""))
    rule #parseHexBytes("") => .WordStack
    rule #parseHexBytes(S)  => #parseHexWord(substrString(S, 0, 2)) : #parseHexBytes(substrString(S, 2, lengthString(S))) requires lengthString(S) >=Int 2

    rule #parseByteStackRaw(S) => ordChar(substrString(S, 0, 1)) : #parseByteStackRaw(substrString(S, 1, lengthString(S))) requires lengthString(S) >=Int 1
    rule #parseByteStackRaw("") => .WordStack

    syntax WordStack ::= #parseWordStack ( JSON ) [function]
 // --------------------------------------------------------
    rule #parseWordStack( [ .JSONList ] )            => .WordStack
    rule #parseWordStack( [ (WORD:String) , REST ] ) => #parseHexWord(WORD) : #parseWordStack( [ REST ] )

    syntax Map ::= #parseMap ( JSON ) [function]
 // --------------------------------------------
    rule #parseMap( { .JSONList                   } ) => .Map
    rule #parseMap( { _   : (VALUE:String) , REST } ) => #parseMap({ REST })                                                requires #parseHexWord(VALUE) ==K 0
    rule #parseMap( { KEY : (VALUE:String) , REST } ) => #parseMap({ REST }) [ #parseHexWord(KEY) <- #parseHexWord(VALUE) ] requires #parseHexWord(VALUE) =/=K 0

    syntax Int ::= #parseAddr ( String ) [function]
 // -----------------------------------------------
    rule #parseAddr(S) => #addr(#parseHexWord(S))

Unparsing

We need to interperet a WordStack as a String again so that we can call Keccak256 on it from KRYPTO.

  • #unparseByteStack turns a stack of bytes (as a WordStack) into a String.
  • #padByte ensures that the String interperetation of a Int is wide enough.
    syntax String ::= #unparseByteStack ( WordStack )                [function, klabel(unparseByteStack)]
                    | #unparseByteStack ( WordStack , StringBuffer ) [function, klabel(#unparseByteStackAux)]
 // ---------------------------------------------------------------------------------------------------------
    rule #unparseByteStack ( WS ) => #unparseByteStack(WS, .StringBuffer)

    rule #unparseByteStack( .WordStack, BUFFER ) => StringBuffer2String(BUFFER)
    rule #unparseByteStack( W : WS, BUFFER )     => #unparseByteStack(WS, BUFFER +String chrChar(W modInt (2 ^Int 8)))

    syntax String ::= #padByte( String ) [function]
 // -----------------------------------------------
    rule #padByte( S ) => S             requires lengthString(S) ==K 2
    rule #padByte( S ) => "0" +String S requires lengthString(S) ==K 1

Recursive Length Prefix (RLP)

RLP encoding is used extensively for executing the blocks of a transaction. For details about RLP encoding, see the YellowPaper Appendix B.

Encoding

  • #rlpEncodeWord RLP encodes a single EVM word.
  • #rlpEncodeString RLP encodes a single String.
    syntax String ::= #rlpEncodeWord ( Int )            [function]
                    | #rlpEncodeBytes ( Int , Int )     [function]
                    | #rlpEncodeWordStack ( WordStack ) [function]
                    | #rlpEncodeString ( String )       [function]
                    | #rlpEncodeAccount ( Account )     [function]
 // --------------------------------------------------------------
    rule #rlpEncodeWord(0) => "\x80"
    rule #rlpEncodeWord(WORD) => chrChar(WORD) requires WORD >Int 0 andBool WORD <Int 128
    rule #rlpEncodeWord(WORD) => #rlpEncodeLength(#unparseByteStack(#asByteStack(WORD)), 128) requires WORD >=Int 128

    rule #rlpEncodeBytes(WORD, LEN) => #rlpEncodeString(#unparseByteStack(#padToWidth(LEN, #asByteStack(WORD))))

    rule #rlpEncodeWordStack(.WordStack) => ""
    rule #rlpEncodeWordStack(W : WS)     => #rlpEncodeWord(W) +String #rlpEncodeWordStack(WS)

    rule #rlpEncodeString(STR) => STR                        requires lengthString(STR) ==Int 1 andBool ordChar(STR) <Int 128
    rule #rlpEncodeString(STR) => #rlpEncodeLength(STR, 128) [owise]

    rule #rlpEncodeAccount(.Account) => "\x80"
    rule #rlpEncodeAccount(ACCT)     => #rlpEncodeBytes(ACCT, 20) requires ACCT =/=K .Account

    syntax String ::= #rlpEncodeLength ( String , Int )          [function]
                    | #rlpEncodeLength ( String , Int , String ) [function, klabel(#rlpEncodeLengthAux)]
 // ----------------------------------------------------------------------------------------------------
    rule #rlpEncodeLength(STR, OFFSET) => chrChar(lengthString(STR) +Int OFFSET) +String STR requires lengthString(STR) <Int 56
    rule #rlpEncodeLength(STR, OFFSET) => #rlpEncodeLength(STR, OFFSET, #unparseByteStack(#asByteStack(lengthString(STR)))) requires lengthString(STR) >=Int 56
    rule #rlpEncodeLength(STR, OFFSET, BL) => chrChar(lengthString(BL) +Int OFFSET +Int 55) +String BL +String STR

Decoding

  • #rlpDecode RLP decodes a single String into a JSON.
  • #rlpDecodeList RLP decodes a single String into a JSONList, interpereting the string as the RLP encoding of a list.
    syntax JSON ::= #rlpDecode(String)               [function]
                  | #rlpDecode(String, LengthPrefix) [function, klabel(#rlpDecodeAux)]
 // ----------------------------------------------------------------------------------
    rule #rlpDecode(STR) => #rlpDecode(STR, #decodeLengthPrefix(STR, 0))
    rule #rlpDecode(STR, #str(LEN, POS))  => substrString(STR, POS, POS +Int LEN)
    rule #rlpDecode(STR, #list(LEN, POS)) => [#rlpDecodeList(STR, POS)]

    syntax JSONList ::= #rlpDecodeList(String, Int)               [function]
                      | #rlpDecodeList(String, Int, LengthPrefix) [function, klabel(#rlpDecodeListAux)]
 // ---------------------------------------------------------------------------------------------------
    rule #rlpDecodeList(STR, POS) => #rlpDecodeList(STR, POS, #decodeLengthPrefix(STR, POS)) requires POS <Int lengthString(STR)
    rule #rlpDecodeList(STR, POS) => .JSONList [owise]
    rule #rlpDecodeList(STR, POS, _:LengthPrefixType(L, P)) => #rlpDecode(substrString(STR, POS, L +Int P)) , #rlpDecodeList(STR, L +Int P)

    syntax LengthPrefixType ::= "#str" | "#list"
    syntax LengthPrefix ::= LengthPrefixType "(" Int "," Int ")"
                          | #decodeLengthPrefix ( String , Int )                                [function]
                          | #decodeLengthPrefix ( String , Int , Int )                          [function, klabel(#decodeLengthPrefixAux)]
                          | #decodeLengthPrefixLength ( LengthPrefixType , String , Int , Int ) [function]
                          | #decodeLengthPrefixLength ( LengthPrefixType , Int    , Int , Int ) [function, klabel(#decodeLengthPrefixLengthAux)]
 // --------------------------------------------------------------------------------------------------------------------------------------------
    rule #decodeLengthPrefix(STR, START) => #decodeLengthPrefix(STR, START, ordChar(substrString(STR, START, START +Int 1)))

    rule #decodeLengthPrefix(STR, START, B0) => #str(1, START)                                   requires B0 <Int 128
    rule #decodeLengthPrefix(STR, START, B0) => #str(B0 -Int 128, START +Int 1)                  requires B0 >=Int 128 andBool B0 <Int (128 +Int 56)
    rule #decodeLengthPrefix(STR, START, B0) => #decodeLengthPrefixLength(#str, STR, START, B0)  requires B0 >=Int (128 +Int 56) andBool B0 <Int 192
    rule #decodeLengthPrefix(STR, START, B0) => #list(B0 -Int 192, START +Int 1)                 requires B0 >=Int 192 andBool B0 <Int 192 +Int 56
    rule #decodeLengthPrefix(STR, START, B0) => #decodeLengthPrefixLength(#list, STR, START, B0) [owise]

    rule #decodeLengthPrefixLength(#str,  STR, START, B0) => #decodeLengthPrefixLength(#str,  START, B0 -Int 128 -Int 56 +Int 1, #asWord(#parseByteStackRaw(substrString(STR, START +Int 1, START +Int 1 +Int (B0 -Int 128 -Int 56 +Int 1)))))
    rule #decodeLengthPrefixLength(#list, STR, START, B0) => #decodeLengthPrefixLength(#list, START, B0 -Int 192 -Int 56 +Int 1, #asWord(#parseByteStackRaw(substrString(STR, START +Int 1, START +Int 1 +Int (B0 -Int 192 -Int 56 +Int 1)))))
    rule #decodeLengthPrefixLength(TYPE, START, LL, L) => TYPE(L, START +Int 1 +Int LL)
endmodule