implement quadratic field extension (unfortunately, for some reasons, the test framework doesn't work for the more interesting extension field operations...??)

circuit/field_params.circom

@@ -18,11 +18,11 @@ pragma circom 2.2.0;
 
 //------------------------------------------------------------------------------
 
-function SolinasExpoBig()   { return 64; }
-function SolinasExpoSmall() { return 32; }
+// function SolinasExpoBig()   { return 64; }
+// function SolinasExpoSmall() { return 32; }
 
-// function SolinasExpoBig()   { return 8; }
-// function SolinasExpoSmall() { return 4; }
+function SolinasExpoBig()   { return 8; }
+function SolinasExpoSmall() { return 4; }
 
 function FieldPrime()  { 
   return (2**SolinasExpoBig() - 2**SolinasExpoSmall() + 1); 

circuit/goldilocks.circom

@@ -191,7 +191,6 @@ template Mul() {
   C <== ReduceModP( SolinasExpoBig() )( A.val * B.val ); 
 }
 
-
 //
 // multiplication of 3 Goldilocks field elements
 // as this still fits into 192 < 254 bits, we can do it a bit more efficiently
@@ -209,6 +208,16 @@ template Mul3() {
   D <== ReduceModP( 2 * SolinasExpoBig() )( AB * C.val ); 
 }
 
+//--------------------------------------
+
+// Squaring (this is more interesting in the extension field case)
+//
+template Sqr() {
+  input  Goldilocks() A;
+  output Goldilocks() C;
+  C <== Mul()( A , A );
+}
+
 //------------------------------------------------------------------------------
 
 //

circuit/goldilocks_ext.circom

@@ -0,0 +1,140 @@
+
+//
+// the quadratic field extension `F[X] / (X^2 - 7)` over Goldilocks
+//
+// note: `X^2 - 7` is also irreducible over the field of size `2^8 - 2^4 + 1`
+// so can use it for testing too.
+//
+
+pragma circom 2.2.0;
+
+include "goldilocks.circom";
+// include "misc.circom";
+
+//------------------------------------------------------------------------------
+//
+// the type of quadratic field extension elements.
+//
+
+bus GoldilocksExt() {
+  Goldilocks() real;
+  Goldilocks() imag;
+}
+
+template GoldilocksToExt() {
+  input  Goldilocks()    inp;
+  output GoldilocksExt() out;
+  out.real     <== inp;
+  out.imag.val <== 0;
+}
+
+//------------------------------------------------------------------------------
+
+template NegExt() {
+  input  GoldilocksExt() A;
+  output GoldilocksExt() C;
+
+  C.real <== Neg()( A.real );
+  C.imag <== Neg()( A.imag );
+}
+
+template AddExt() {
+  input  GoldilocksExt() A,B;
+  output GoldilocksExt() C;
+
+  C.real <== Add()( A.real , B.real );
+  C.imag <== Add()( A.imag , B.imag );
+}
+
+template SubExt() {
+  input  GoldilocksExt() A,B;
+  output GoldilocksExt() C;
+
+  C.real <== Sub()( A.real , B.real );
+  C.imag <== Sub()( A.imag , B.imag );
+}
+
+//------------------------------------------------------------------------------
+
+template SevenTimesProduct() {
+  input  Goldilocks() inp1,inp2;
+  output Goldilocks() out;
+  out <== ReduceModP( 3 + SolinasExpoBig() )( 7 * inp1.val * inp2.val ); 
+}
+
+template SevenTimesSquare() {
+  input  Goldilocks() inp;
+  output Goldilocks() out;
+  out <== ReduceModP( 3 + SolinasExpoBig() )( 7 * inp.val * inp.val ); 
+}
+
+//--------------------------------------
+
+template MulExt() {
+  input  GoldilocksExt() A,B;
+  output GoldilocksExt() C;
+
+  Goldilocks() RR, IR, RI, IIx7;
+  RR   <== Mul()( A.real , B.real );
+  IR   <== Mul()( A.imag , B.real );
+  RI   <== Mul()( A.real , B.imag );
+  IIx7 <== SevenTimesProduct()( A.imag , B.imag );
+
+  C.real <== Add()( RR , IIx7 );
+  C.imag <== Add()( IR , RI   );
+}
+
+template SqrExt() {
+  input  GoldilocksExt() A;
+  output GoldilocksExt() C;
+
+  Goldilocks() RR, IR, IIx7;
+  RR   <== Sqr()( A.real );
+  IR   <== Mul()( A.imag , A.real );
+  IIx7 <== SevenTimesSquare()( A.imag );
+
+  C.real <== Add()( RR , IIx7 );
+  C.imag <== Add()( IR , IR   );
+
+  // log("\nsqrExt");
+  // log( "A = (", A.real.val, " , ", A.imag.val, ")" );
+  // log( "C = (", C.real.val, " , ", C.imag.val, ")" );
+  // log( "RR   = ", RR.val   );
+  // log( "IR   = ", IR.val   );
+  // log( "IRx7 = ", IIx7.val );
+}
+
+//------------------------------------------------------------------------------
+
+template InvExt() {
+  input  GoldilocksExt() A;
+  output GoldilocksExt() C;
+
+  Goldilocks() seven, RR, IIx7;
+  seven.val <== 7;
+  RR   <== Sqr()( A.real );
+  IIx7 <== SevenTimesSquare()( A.imag );
+
+  Goldilocks() denom     <== Sub()( RR , IIx7 );
+  Goldilocks() mult      <== Inv()( denom     );
+  Goldilocks() minusImag <== Neg()( A.imag );
+
+  C.real <== Mul()( A.real    , mult );
+  C.imag <== Mul()( minusImag , mult );
+}
+
+template DivExt() {
+  input  GoldilocksExt() A,B;
+  output GoldilocksExt() C;
+
+  GoldilocksExt() invB <== InvExt()( B );
+  C <== MulExt()( A , invB );
+
+  log("\ndivExt");
+  log( "A = (", A.real.val, " , ", A.imag.val, ")" );
+  log( "B = (", B.real.val, " , ", B.imag.val, ")" );
+  log( "C = (", C.real.val, " , ", C.imag.val, ")" );
+
+}
+
+//------------------------------------------------------------------------------

circuit/run.sh

@@ -2,13 +2,15 @@
 
 ORIG=`pwd`
 MAIN="testmain"
-INPUT="dummy.json"
+INPUT="input.json"
 
 cd build
 
 circom ../${MAIN}.circom --r1cs --wasm
 
-echo '{ "dummy": 666 }' >${INPUT}
+#echo '{ "dummy": 666 }' >${INPUT}
+#echo '{ "A": [66,111] }' >${INPUT}
+echo '{ "A": [57,137] , "B": [66,111] }' >${INPUT}
 
 cd ${MAIN}_js
 

circuit/test_wrapper.circom

@@ -65,6 +65,17 @@ template SubWrapper() {
 
 //------------------------------------------------------------------------------
 
+template SqrWrapper() {
+  signal input  A;
+  signal output C;
+
+  Goldilocks() A1; A1.val <== A;
+  Goldilocks() C1; 
+
+  C1 <== Sqr()( A1 );
+  C1.val ==> C;
+}
+
 template MulWrapper() {
   signal input  A,B;
   signal output C;
@@ -77,6 +88,8 @@ template MulWrapper() {
   C1.val ==> C;
 }
 
+//------------------------------------------------------------------------------
+
 template InvWrapper() {
   signal input  A;
   signal output C;

circuit/test_wrapper_ext.circom

@@ -0,0 +1,104 @@
+
+// wrappers around the Goldilocks templates, so that input and output are classic
+// signals, not Bus-es. The testing framework does not handle Bus inputs/outputs yet
+
+pragma circom 2.2.0;
+
+include "goldilocks.circom";
+include "goldilocks_ext.circom";
+
+//------------------------------------------------------------------------------
+
+template NegExtWrapper() {
+  signal input  A[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+
+  GoldilocksExt() C1 <== NegExt()( A1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+template AddExtWrapper() {
+  signal input  A[2],B[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+  GoldilocksExt() B1; B1.real.val <== B[0]; B1.imag.val <== B[1];
+
+  GoldilocksExt() C1 <== AddExt()( A1 , B1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+template SubExtWrapper() {
+  signal input  A[2],B[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+  GoldilocksExt() B1; B1.real.val <== B[0]; B1.imag.val <== B[1];
+
+  GoldilocksExt() C1 <== SubExt()( A1 , B1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+//------------------------------------------------------------------------------
+
+template SqrExtWrapper() {
+  signal input  A[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+
+  GoldilocksExt() C1 <== SqrExt()( A1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+template MulExtWrapper() {
+  signal input  A[2],B[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+  GoldilocksExt() B1; B1.real.val <== B[0]; B1.imag.val <== B[1];
+  
+  GoldilocksExt() C1 <== MulExt()( A1 , B1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+//------------------------------------------------------------------------------
+
+template InvExtWrapper() {
+  signal input  A[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+
+  GoldilocksExt() C1 <== InvExt()( A1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+template DivExtWrapper() {
+  signal input  A[2],B[2];
+  signal output C[2];
+
+  GoldilocksExt() A1; A1.real.val <== A[0]; A1.imag.val <== A[1];
+  GoldilocksExt() B1; B1.real.val <== B[0]; B1.imag.val <== B[1];
+
+  GoldilocksExt() C1 <== DivExt()( A1 , B1 );
+
+  C1.real.val ==> C[0];
+  C1.imag.val ==> C[1];
+}
+
+//------------------------------------------------------------------------------

circuit/testmain.circom

@@ -2,9 +2,15 @@
 pragma circom 2.2.0;
 
 include "test_wrapper.circom";
+include "test_wrapper_ext.circom";
+
 include "poseidon.circom";
 
 // component main { public [A,B] } = AddWrapper();
 // component main { public [A]   } = InvWrapper();
 
-component main { public [dummy] } = CheckPermutationKAT();
+// component main { public [dummy] } = CheckPermutationKAT();
+
+// component main { public [A,B] } = MulExtWrapper();
+// component main { public [A]   } = SqrExtWrapper();
+component main { public [A,B] } = DivExtWrapper();

tests/Main.hs

@@ -8,7 +8,8 @@ module Main where
 
 import R1CS
 
-import TestGoldilocks
+import qualified TestGoldilocks    as Gld
+import qualified TestGoldilocksExt as Ext
 
 --------------------------------------------------------------------------------
 
@@ -21,18 +22,39 @@ testGoldilocks' fld verbosity = runWithField fld $ \pxy -> do
   let runSpec     what = testSemantics     pxy what verbosity
   let runSpecMany what = testSemanticsMany pxy what verbosity
 
-  runSpec   specIsZero
-  runSpec   specToGoldi
-  runSpec $ specUnary  Neg semantics_neg
-  runSpec $ specBinary Add semantics_add
-  runSpec $ specBinary Sub semantics_sub
-  runSpec $ specUnary  Inv semantics_inv
+  runSpec   Gld.specIsZero
+  runSpec   Gld.specToGoldi
+  runSpec $ Gld.specUnary  Gld.Neg Gld.semantics_neg
+  runSpec $ Gld.specBinary Gld.Add Gld.semantics_add
+  runSpec $ Gld.specBinary Gld.Sub Gld.semantics_sub
+  runSpec $ Gld.specUnary  Gld.Sqr Gld.semantics_sqr
+  runSpec $ Gld.specUnary  Gld.Inv Gld.semantics_inv
 
   -- these are very slow so we don't do exhaustive testing
-  runSpec $ specBinarySmall Mul semantics_mul
-  runSpec $ specBinarySmall Div semantics_div
+  runSpec $ Gld.specBinarySmall Gld.Mul Gld.semantics_mul
+  runSpec $ Gld.specBinarySmall Gld.Div Gld.semantics_div
+
+--------------------------------------------------------------------------------
+
+testGoldilocksExt :: IO ()
+testGoldilocksExt = testGoldilocksExt' Field24 Silent
+
+testGoldilocksExt' :: FieldChoice -> Verbosity -> IO ()
+testGoldilocksExt' fld verbosity = runWithField fld $ \pxy -> do
+
+  let runSpec     what = testSemantics     pxy what verbosity
+  let runSpecMany what = testSemanticsMany pxy what verbosity
+
+  runSpec $ Ext.specUnary  Ext.Neg Ext.semantics_neg
+  runSpec $ Ext.specBinary Ext.Add Ext.semantics_add
+  runSpec $ Ext.specBinary Ext.Sub Ext.semantics_sub
+--  runSpec $ Ext.specUnary  Ext.Sqr Ext.semantics_sqr
+--  runSpec $ Ext.specBinary Ext.Mul Ext.semantics_mul
+--  runSpec $ Ext.specUnary  Ext.Inv Ext.semantics_inv
+--  runSpec $ Ext.specBinary Ext.Div Ext.semantics_div
 
 --------------------------------------------------------------------------------
 
 main = do
-  testGoldilocks
+  testGoldilocks
+  testGoldilocksExt

tests/Semantics.hs

@@ -1,13 +1,17 @@
 
+{-# LANGUAGE BlockArguments #-}
 module Semantics where
 
 --------------------------------------------------------------------------------
 
 import Prelude hiding (div)
+import Control.Monad
+import System.Random
 
 --------------------------------------------------------------------------------
 
-type F = Int
+type F    = Int
+type FExt = (F,F)
 
 fieldPrime :: Int
 fieldPrime = 2^8 - 2^4 + 1
@@ -54,3 +58,123 @@ div :: F -> F -> F
 div x y = mul x (inv y)
 
 --------------------------------------------------------------------------------
+
+negExt :: FExt -> FExt
+negExt (r,i) = (neg r, neg i)
+
+addExt :: FExt -> FExt -> FExt
+addExt (r1,i1) (r2,i2) = (add r1 r2, add i1 i2)
+
+subExt :: FExt -> FExt -> FExt
+subExt (r1,i1) (r2,i2) = (sub r1 r2, sub i1 i2)
+
+mulExt :: FExt -> FExt -> FExt
+mulExt (r1,i1) (r2,i2) = (r3,i3) where
+  r3 = (mul r1 r2) `add` (mul3 7 i1 i2) 
+  i3 = (mul r1 i2) `add` (mul i1 r2)
+
+sqrExt :: FExt -> FExt 
+sqrExt x = mulExt x x
+
+powExt :: FExt -> Int -> FExt
+powExt x0 expo
+  | expo < 0  = error "pow: negative exponent"
+  | expo == 0 = (1,0)
+  | otherwise = go (1,0) x0 expo
+  where
+    go acc s 0 = acc
+    go acc s e = case divMod e 2 of
+      (e', 0) -> go         acc    (sqrExt s) e'
+      (e' ,1) -> go (mulExt acc s) (sqrExt s) e'
+
+invExt :: FExt -> FExt
+invExt (r,i) = (r `mul` z , neg i `mul` z) where 
+  denom = (sqr r) `sub` (mul3 7 i i)
+  z = inv denom
+
+divExt :: FExt -> FExt -> FExt
+divExt x y = mulExt x (invExt y)
+
+--------------------------------------------------------------------------------
+-- some quick & dirty testing
+
+data Prop a 
+  = Prop1 String (a -> Bool)
+  | Prop2 String (a -> a -> Bool)
+  | Prop3 String (a -> a -> a -> Bool)
+
+propName :: Prop a -> String
+propName prop = case prop of
+  Prop1 name _ -> name
+  Prop2 name _ -> name
+  Prop3 name _ -> name
+
+some_properties :: [Prop F]
+some_properties = 
+  [ Prop2 "add-sub" \x y   -> add (sub x y) y == x 
+  , Prop2 "add-neg" \x y   -> sub x y == add x (neg y)
+  , Prop1 "inv-mul" \x     -> x == 0 || inv x `mul` x == 1
+  , Prop1 "inv-pow" \x     -> inv x == pow x (fieldPrime - 2)
+  , Prop2 "mul-div" \x y   -> y == 0 || mul (div x y) y == x 
+  , Prop3 "mul-add" \x y z -> mul (add x y) z == mul x z `add` mul y z 
+  , Prop1 "mul-pow" \x     -> mul (mul x x) x == pow x 3
+  ]
+
+some_ext_properties :: [Prop FExt]
+some_ext_properties = 
+  [ Prop2 "add-sub" \x y   -> addExt (subExt x y) y == x 
+  , Prop2 "add-neg" \x y   -> subExt x y == addExt x (negExt y)
+  , Prop1 "inv-mul" \x     -> x == (0,0) || invExt x `mulExt` x == (1,0)
+  , Prop1 "inv-pow" \x     -> invExt x == powExt x (fieldPrime^2 - 2)
+  , Prop2 "mul-div" \x y   -> y == (0,0) || mulExt (divExt x y) y == x 
+  , Prop3 "mul-add" \x y z -> mulExt (addExt x y) z == mulExt x z `addExt` mulExt y z 
+  , Prop1 "mul-pow" \x     -> mulExt (mulExt x x) x == powExt x 3
+  ]
+
+--------------------------------------------------------------------------------
+
+runTests :: IO ()
+runTests = do
+  let n = 1000
+  runTestsBase n
+  runTestsExt  n
+
+runTestsBase :: Int -> IO ()
+runTestsBase n = do
+  putStrLn $ "\nbase field properties"
+  forM_ some_properties $ \prop -> do
+    oks <- replicateM n (runPropF prop) 
+    let good = length (filter id oks)
+    let bad  = if good < n then " - FAILED!" else " - OK."
+    putStrLn $ "  - " ++ propName prop ++ ": " ++ show good ++ " / " ++ show n ++ " passed. " ++ bad 
+
+runTestsExt :: Int -> IO ()
+runTestsExt n = do
+  putStrLn $ "\nextension field properties"
+  forM_ some_ext_properties $ \prop -> do
+    oks <- replicateM n (runPropFExt prop) 
+    let good = length (filter id oks)
+    let bad  = if good < n then " - FAILED!" else " - OK."
+    putStrLn $ "  - " ++ propName prop ++ ": " ++ show good ++ " / " ++ show n ++ " passed. " ++ bad
+
+----------------------------------------
+
+rndF :: IO F
+rndF = randomRIO (0,fieldPrime-1)
+
+rndFExt :: IO FExt
+rndFExt = (,) <$> rndF <*> rndF
+
+runPropF :: Prop F -> IO Bool
+runPropF prop = case prop of
+  Prop1 _ f1 -> f1 <$> rndF
+  Prop2 _ f2 -> f2 <$> rndF <*> rndF
+  Prop3 _ f3 -> f3 <$> rndF <*> rndF <*> rndF
+
+runPropFExt :: Prop FExt -> IO Bool
+runPropFExt prop = case prop of
+  Prop1 _ f1 -> f1 <$> rndFExt
+  Prop2 _ f2 -> f2 <$> rndFExt <*> rndFExt
+  Prop3 _ f3 -> f3 <$> rndFExt <*> rndFExt <*> rndFExt
+
+--------------------------------------------------------------------------------

tests/TestGoldilocks.hs

@@ -18,6 +18,7 @@ data Op
   = Neg
   | Add
   | Sub
+  | Sqr
   | Mul
   | Inv
   | Div
@@ -34,6 +35,7 @@ mainComponent op =
     Neg -> unary  "Neg"
     Add -> binary "Add"
     Sub -> binary "Sub"
+    Sqr -> unary  "Sqr"
     Mul -> binary "Mul"
     Inv -> unary  "Inv"
     Div -> binary "Div"
@@ -81,6 +83,9 @@ semantics_add (x,y) = Expecting $ Semantics.add x y
 semantics_sub :: (F,F) -> Expected F
 semantics_sub (x,y) = Expecting $ Semantics.sub x y
 
+semantics_sqr :: F -> Expected F
+semantics_sqr x = Expecting $ Semantics.sqr x 
+
 semantics_mul :: (F,F) -> Expected F
 semantics_mul (x,y) = Expecting $ Semantics.mul x y
 

tests/TestGoldilocksExt.hs

@@ -0,0 +1,118 @@
+
+
+module TestGoldilocksExt where
+
+--------------------------------------------------------------------------------
+
+import Control.Monad
+import System.IO.Unsafe
+
+import Semantics
+import Common
+
+--------------------------------------------------------------------------------
+-- global parameters
+
+circomFile :: FilePath
+circomFile = circuitSourceDir </> "test_wrapper_ext.circom"
+
+data Op
+  = Neg
+  | Add
+  | Sub
+  | Sqr
+  | Mul
+  | Inv
+  | Div
+  deriving (Eq,Show,Bounded,Enum)
+
+enumerateOps :: [Op]
+enumerateOps = enumFromTo minBound maxBound
+
+----------------------------------------
+
+mainComponent :: Op -> MainComponent
+mainComponent op = 
+  case op of
+    Neg -> unary  "NegExt"
+    Add -> binary "AddExt"
+    Sub -> binary "SubExt"
+    Sqr -> unary  "SqrExt"
+    Mul -> binary "MulExt"
+    Inv -> unary  "InvExt"
+    Div -> binary "DivExt"
+  where
+
+    unary name = MainComponent 
+      { _templateName   = name ++ "Wrapper"
+      , _templateParams = []
+      , _publicInputs   = ["A"]
+      }
+
+    binary name = MainComponent 
+      { _templateName   = name ++ "Wrapper"
+      , _templateParams = []
+      , _publicInputs   = ["A","B"]
+      }
+
+--------------------------------------------------------------------------------
+-- test cases and expected semantics
+
+type TestCase1 =  (Int,Int)
+type TestCase2 = ((Int,Int),(Int,Int))
+
+type Output = (Int,Int)
+
+nNestCases = 10
+
+randomTestCasesUnary :: [TestCase1]
+randomTestCasesUnary = unsafePerformIO $ replicateM nNestCases rndFExt
+
+randomTestCasesBinary :: [TestCase2]
+randomTestCasesBinary = unsafePerformIO $ replicateM nNestCases $ do { x <- rndFExt ; y <- rndFExt ; return (x,y) }
+
+----------------------------------------
+
+semantics_neg :: FExt -> Expected FExt
+semantics_neg x = Expecting $ Semantics.negExt x
+
+semantics_add :: (FExt,FExt) -> Expected FExt
+semantics_add (x,y) = Expecting $ Semantics.addExt x y
+
+semantics_sub :: (FExt,FExt) -> Expected FExt
+semantics_sub (x,y) = Expecting $ Semantics.subExt x y
+
+semantics_sqr :: FExt -> Expected FExt
+semantics_sqr x = Expecting $ Semantics.sqrExt x
+
+semantics_mul :: (FExt,FExt) -> Expected FExt
+semantics_mul (x,y) = Expecting $ Semantics.mulExt x y
+
+semantics_inv :: FExt -> Expected FExt
+semantics_inv x = Expecting $ Semantics.invExt x
+
+semantics_div :: (FExt,FExt) -> Expected FExt
+semantics_div (x,y) = Expecting $ Semantics.divExt x y
+
+--------------------------------------------------------------------------------
+-- inputs and outputs
+
+inputsA :: TestCase1 -> Inputs Name Integer
+inputsA a = Inputs $  toMapping "A" a
+                    
+inputsAB :: TestCase2 -> Inputs Name Integer
+inputsAB (a,b) = Inputs $  toMapping "A" a
+                        <> toMapping "B" b
+
+outputsC :: Output -> Outputs Name Integer
+outputsC y = Outputs $ toMapping "C" y
+
+--------------------------------------------------------------------------------
+
+specUnary :: Op -> (FExt -> Expected FExt) -> TestSpec TestCase1 Output
+specUnary op semantics = TestSpec circomFile (mainComponent op) inputsA outputsC semantics randomTestCasesUnary
+
+specBinary :: Op -> ((FExt,FExt) -> Expected FExt) -> TestSpec TestCase2 Output
+specBinary op semantics = TestSpec circomFile (mainComponent op) inputsAB outputsC semantics randomTestCasesBinary
+
+--------------------------------------------------------------------------------