diff --git a/groth16/dynamic/shared.nim b/groth16/dynamic/shared.nim
index f886d30..ef07a62 100644
--- a/groth16/dynamic/shared.nim
+++ b/groth16/dynamic/shared.nim
@@ -154,6 +154,67 @@ proc groupConvolveWithWVecBar*( D: Domain, gs: seq[G1] ): seq[G1] =
   inplaceScalarMulByFFTofWVecBar( gsHat )
   return inverseGroupFFT( gsHat , D)
 
+#---------------------------------------
+
+# only computes the result on a subgroup
+proc fieldConvolveWithWVecBarOnSubgroup*( sg: Subgroup, xs: seq[F] ): seq[F] =
+
+  let N = xs.len
+  assert( N == sg.bigDomain.domainSize )
+  let D = sg.bigDomain
+  let K = sg.smallDomain.domainSize
+  let ell = N div K
+
+  let fN   = intToFr( N )
+  let invN = invFr(  fN )
+  let fell    : F = intToFr(ell)
+  let invN_per_fell : F = invN / fell
+
+  let xsHat = forwardNTT( xs , D )
+  var ysHat : seq[F] = newSeq[F]( K )
+
+  let c0  = divBy2Fr(oneFr - fN) * invN_per_fell
+  for k in 0..<K:
+    var sum: F = zeroFr
+    for i in 0..<ell:
+      let kk = k + K*i
+      let kk_rev = (if kk==0: 0 else: N-kk)
+      let u = c0 + intToFr(kk_rev) * invN_per_fell
+      sum += u * xsHat[kk]
+    ysHat[k] = sum
+
+  return inverseNTT( ysHat , sg.smallDomain )
+
+#---------------------------------------
+
+proc groupConvolveWithWVecOnSubgroup*( sg: Subgroup, pts: seq[G1] ): seq[G1] =
+
+  let N = pts.len
+  assert( N == sg.bigDomain.domainSize )
+  let D = sg.bigDomain
+  let K = sg.smallDomain.domainSize
+  let ell = N div K
+
+  let fN   = intToFr( N )
+  let invN = invFr(  fN )
+  let fell    : F = intToFr(ell)
+  let invN_per_fell : F = invN / fell
+
+  var ptsHat = forwardGroupFFT( pts , D )
+
+  var small   : seq[G1] = newSeq[G1]( K ) 
+  var miniPts : seq[G1] = newSeq[G1]( ell )
+  var miniCfs : seq[F]  = newSeq[F] ( ell )
+  let c0  = divBy2Fr(oneFr - fN) * invN_per_fell
+  for k in 0..<K:
+    for i in 0..<ell:
+      let kk = k + K*i
+      miniCfs[i] = c0 + intToFr(kk) * invN_per_fell
+      miniPts[i] = ptsHat[kk]
+    small[k] = msmConstantineG1( miniCfs , miniPts )
+
+  return inverseGroupFFT( small , sg.smallDomain )
+
 #-------------------------------------------------------------------------------
 
 # computes the vectors A*z, B*z (but skips C*z)
diff --git a/groth16/dynamic/v1/finish.nim b/groth16/dynamic/v1/finish.nim
index 10feaf9..2a9e74e 100644
--- a/groth16/dynamic/v1/finish.nim
+++ b/groth16/dynamic/v1/finish.nim
@@ -9,8 +9,8 @@ import std/options
 
 import taskpools
 
-import constantine/math/arithmetic
-import constantine/named/properties_fields
+#import constantine/math/arithmetic
+#import constantine/named/properties_fields
 
 import groth16/bn128
 import groth16/bn128/arrays
@@ -20,9 +20,9 @@ import groth16/zkey_types
 import groth16/files/witness
 
 import groth16/math/domain
-import groth16/math/ntt
-import groth16/math/poly
 import groth16/math/convolution
+#import groth16/math/poly
+#import groth16/math/ntt
 
 import groth16/prover/types
 import groth16/prover/shared
@@ -30,7 +30,6 @@ import groth16/partial/finish
 
 import groth16/dynamic/shared
 import groth16/dynamic/v1/types
-import groth16/dynamic/v1/setup
 
 #-------------------------------------------------------------------------------
 
diff --git a/groth16/dynamic/v1/preprocess.nim b/groth16/dynamic/v1/preprocess.nim
index 3dab47a..744038d 100644
--- a/groth16/dynamic/v1/preprocess.nim
+++ b/groth16/dynamic/v1/preprocess.nim
@@ -10,12 +10,10 @@
 # this is V1, the version closest to the Dynark paper
 #
 
-import std/options
-
 import taskpools
 
 import constantine/math/arithmetic
-import constantine/named/properties_fields
+#import constantine/named/properties_fields
 
 import groth16/bn128
 import groth16/bn128/arrays
@@ -24,7 +22,6 @@ import groth16/misc
 import groth16/zkey_types
 
 import groth16/math/domain
-import groth16/math/convolution
 import groth16/math/poly
 
 import groth16/partial/precalc
@@ -32,7 +29,6 @@ import groth16/partial/precalc
 import groth16/dynamic/shared
 import groth16/dynamic/v1/types
 import groth16/dynamic/v1/setup
-import groth16/dynamic/v1/finish
 
 
 #-------------------------------------------------------------------------------
diff --git a/groth16/dynamic/v1/setup.nim b/groth16/dynamic/v1/setup.nim
index a37a533..4bceed8 100644
--- a/groth16/dynamic/v1/setup.nim
+++ b/groth16/dynamic/v1/setup.nim
@@ -6,8 +6,8 @@
 import taskpools
 
 import groth16/bn128
-import groth16/bn128/arrays
 import groth16/misc
+#import groth16/bn128/arrays
 
 import groth16/math/domain
 import groth16/math/group_fft
diff --git a/groth16/dynamic/v1/types.nim b/groth16/dynamic/v1/types.nim
index 9f0f4ef..af9f581 100644
--- a/groth16/dynamic/v1/types.nim
+++ b/groth16/dynamic/v1/types.nim
@@ -4,7 +4,7 @@
 #import constantine/named/properties_fields
 
 import groth16/bn128
-import groth16/math/domain
+#import groth16/math/domain
 #import groth16/math/ntt
 #import groth16/math/poly
 
diff --git a/groth16/dynamic/v3/preprocess.nim b/groth16/dynamic/v3/preprocess.nim
index 8fe09a4..0c6b224 100644
--- a/groth16/dynamic/v3/preprocess.nim
+++ b/groth16/dynamic/v3/preprocess.nim
@@ -26,22 +26,70 @@ import groth16/partial/precalc
 
 import groth16/dynamic/shared
 import groth16/dynamic/permute
-import groth16/dynamic/v2/types as v2types
+# import groth16/dynamic/v2/types as v2types
 import groth16/dynamic/v2/preprocess
 import groth16/dynamic/v3/types
 import groth16/dynamic/v3/setup
 
 #-------------------------------------------------------------------------------
 
-func convert_preprocess_V2_to_V3( v2: DynaPreprocessV2 ): DynaPreprocessV3 = 
-  return DynaPreprocessV3( unified: v2.unified )
+func projectionElementsV3*(setup: DynaSetupV3, As: seq[F] ): seq[G1] =
+  let sg = setup.imageSubgroup
+  let D  = sg.bigDomain
+  let N  = D.domainSize
+  let K  = sg.smallDomain.domainSize
+  let ell = N div K
 
-func convert_setup_V3_to_V2( v3: DynaSetupV3 ): DynaSetupV2 =
-  let fakePhi: seq[G1] = newSeq[G1]()
-  return DynaSetupV2( weightVec     : v3.weightVec     ,
-                      pointsDeltaLZ : v3.pointsDeltaLZ ,
-                      wConvDeltaLZ  : v3.wConvDeltaLZ  ,
-                      diagPhiPoints : fakePhi          )
+  var Us: seq[G1] = newSeq[G1]( K )
+
+  let fldN : F = intToFr( N ) 
+  let sumW : F = sumOfWVec( N ) 
+
+  let WBarStarA : seq[F]  = fieldConvolveWithWVecBarOnSubgroup( sg , As ) 
+  let ALstarW   : seq[G1] = groupConvolveWithWVecOnSubgroup( sg , pointwiseScaleG1( As , setup.pointsDeltaLZ ) )
+
+  # 2*d scalar multiplications + the group convolution above
+  for k in 0..<K: 
+ 
+    let ellk = ell*k
+    let cf : F = WBarStarA[k] - As[ellk] * sumW
+    Us[k] = ALstarW[k] - (As[ellk] ** setup.wConvDeltaLZ[ellk]) + (cf ** setup.pointsDeltaLZ[ellk])
+
+  return Us
+
+#-------------------------------------------------------------------------------
+
+proc dynaPreprocessV3*(zkey: ZKey, setup: DynaSetupV3, partialAB: PartialAB, partial_mask: seq[bool], zdeltaMask: seq[bool] ): DynaPreprocessV3 =
+  let sg  = setup.imageSubgroup
+  let D   = sg.bigDomain
+  let K   = sg.smallDomain.domainSize
+  let N   = D.domainSize
+  let ell = N div K
+
+  var projA_mini: seq[G1]
+  var projB_mini: seq[G1]
+  withMeasureTime(true," + the \"projection\" points U_k, V_k"):
+    projA_mini = projectionElementsV3( setup , partialAB.valuesAz )
+    projB_mini = projectionElementsV3( setup , partialAB.valuesBz )
+
+  let m = countTrues(zdeltaMask)
+  var Xs: seq[G1] = newSeq[G1]( m )
+
+  let points_C1 = compactifyPointsC1( zkey, partial_mask )
+
+  withMeasureTime(true," + the \"unified\" points X_j"):
+
+    let matABC = zkeyToSparseMatrices(zkey)
+    let colsA = selectTrues( zdeltaMask , matABC.A.columns )
+    let colsB = selectTrues( zdeltaMask , matABC.B.columns )
+
+    for j in 0..<m:
+      var h : G1 = points_C1[j]
+      for (k,a) in colsA[j].pairs: h += (a ** projB_mini[k div ell])
+      for (k,b) in colsB[j].pairs: h += (b ** projA_mini[k div ell])
+      Xs[j] = h
+
+  return DynaPreprocessV3( unified: Xs )
 
 #-------------------------------------------------------------------------------
 
@@ -84,15 +132,14 @@ proc dynaPreProofV3*(zkey: ZKey, setup: DynaSetupV3, partialWitness: PartialWitn
 
   partialProof.partial_pi_c += nonlin
 
-  var preprocessV3 : DynaPreprocessV3
+  var preprocess : DynaPreprocessV3
   withMeasureTime(printTimings,"precomputing the \"projection\" and then the unified points"):
-    let setupV2   = convert_setup_V3_to_V2( setup )
-    preprocessV3  = convert_preprocess_V2_to_V3( dynaPreprocessV2( zkey, setupV2, partialAB, partialMask, zdeltaMask ))
+    preprocess = dynaPreprocessV3( zkey, setup, partialAB, partialMask, zdeltaMask )
 
   return DynaPreProofV3( partialProof   : partialProof ,
                          deltaImages    : deltaImages  ,
                          dynaSetup      : setup        ,
-                         dynaPreprocess : preprocessV3 )
+                         dynaPreprocess : preprocess   )
 
 #-------------------------------------------------------------------------------
 
diff --git a/groth16/math/convolution.nim b/groth16/math/convolution.nim
index 3ab5648..120b2c8 100644
--- a/groth16/math/convolution.nim
+++ b/groth16/math/convolution.nim
@@ -32,12 +32,86 @@ func groupConvolution*(xs: seq[F], gs: seq[G1]): seq[G1] =
   assert( N == gs.len )
   let D = createDomain( N )
 
-  let us =      forwardNTT(xs, D) 
-  let hs = forwardGroupFFT(gs, D)
+  let us =       forwardNTT(xs, D) 
+  let hs =  forwardGroupFFT(gs, D)
   let rs = pointwiseScaleG1(us, hs)
 
   return inverseGroupFFT(rs, D)
 
+#-------------------------------------------------------------------------------
+
+func fieldConvolutionOnSubgroup*(sg: Subgroup, xs: seq[F], ys: seq[F]): seq[F] = 
+  let N = xs.len
+  assert( N == ys.len )
+  assert( N == sg.bigDomain.domainSize )
+  let D = sg.bigDomain
+  let K = sg.smallDomain.domainSize
+  let ell = N div K
+  let fell    : F = intToFr(ell)
+  let invfell : F = invFr(fell)
+
+  let us = forwardNTT(xs, D) 
+  let vs = forwardNTT(ys, D)
+
+  var ws: seq[F] = newSeq[F]( K )
+  for k in 0..<K:
+    var sum: F = zeroFr
+    for i in 0..<ell:
+      sum += us[k + i*K] * vs[k + i*K]
+    ws[k] = sum * invfell
+
+  return inverseNTT(ws, sg.smallDomain)
+
+#---------------------------------------
+
+# only computes the result in a subgroup
+func groupConvolutionOnSubgroup*(sg: Subgroup, xs: seq[F], gs: seq[G1]): seq[G1] =
+  let N = xs.len
+  assert( N == gs.len )
+  assert( N == sg.bigDomain.domainSize )
+  let D = sg.bigDomain
+  let K = sg.smallDomain.domainSize
+  let ell = N div K
+  let fell    : F = intToFr(ell)
+  let invfell : F = invFr(fell)
+
+  let us =      forwardNTT(xs, D) 
+  let hs = forwardGroupFFT(gs, D)
+  
+  var rs      : seq[G1] = newSeq[G1]( K   )
+  var miniPts : seq[G1] = newSeq[G1]( ell )
+  var miniCfs : seq[F]  = newSeq[F] ( ell )
+  for k in 0..<K:
+    for i in 0..<ell:
+      miniCfs[i] = us[k + K*i] * invfell
+      miniPts[i] = hs[k + K*i]
+    rs[k] = msmConstantineG1( miniCfs , miniPts )
+
+  return inverseGroupFFT(rs, sg.smallDomain)
+
+#---------------------------------------
+
+proc testFieldConvolutionOnSubgroup*(N: int, K: int): bool = 
+  let sg = createSubgroup( createDomain(N) , K )
+
+  let xs = randFrSeq(N)
+  let ys = randFrSeq(N)
+
+  let target = selectOnSubgroup( sg , fieldConvolution(xs, ys) )
+  let smart  = fieldConvolutionOnSubgroup( sg, xs, ys )
+  return isEqualFrSeq(target , smart)
+
+
+proc testGroupConvolutionOnSubgroup*(N: int, K: int): bool = 
+  let sg = createSubgroup( createDomain(N) , K )
+
+  let xs = randFrSeq(N)
+  let ps = randG1Seq(N)
+
+  let target = selectOnSubgroup( sg , groupConvolution(xs, ps) )
+  let smart  = groupConvolutionOnSubgroup( sg, xs, ps )
+  return isEqualG1Seq(target , smart)
+
 #-------------------------------------------------------------------------------
 # naive reference implementations for testing
 
diff --git a/tests/groth16/testDyna.nim b/tests/groth16/testDyna.nim
index b743697..742ffeb 100644
--- a/tests/groth16/testDyna.nim
+++ b/tests/groth16/testDyna.nim
@@ -17,9 +17,9 @@ import groth16/math/convolution
 
 import groth16/dynamic/types
 import groth16/dynamic/shared
-import groth16/dynamic/setup
-import groth16/dynamic/preprocess
-import groth16/dynamic/finish
+import groth16/dynamic/v1/preprocess
+#import groth16/dynamic/v1/setup
+#import groth16/dynamic/v1/finish
 
 #-------------------------------------------------------------------------------
 
@@ -30,7 +30,9 @@ func ones(N : int): seq[F] = constSeq(N, oneFr)
 suite "dynamic proof (Dynark) tests":
 
   let N : int = 128
-  let D = createDomain( N )
+  let K : int =  32
+  let D  = createDomain( N )
+  let sg = createSubgroup( D , K )
   let tau   : F = randFr()
   let delta : F = randFr()
 
@@ -75,6 +77,12 @@ suite "dynamic proof (Dynark) tests":
     let rhs = fieldConvolution( xs , calculateWVecBar(D) )
     check isEqualFrSeq( lhs , rhs )  
 
+  test "field convolution with WvecBar on a subgroup":
+    let xs  = randFrSeq( N )
+    let lhs = fieldConvolveWithWVecBarOnSubgroup( sg , xs ) 
+    let rhs = selectOnSubgroup( sg , fieldConvolution( xs , calculateWVecBar(D) ) )
+    check isEqualFrSeq( lhs , rhs )  
+
   test "group convolution with Wvec":
     let gs  = randG1Seq( N )
     let lhs = groupConvolveWithWVec( D , gs ) 
@@ -87,6 +95,12 @@ suite "dynamic proof (Dynark) tests":
     let rhs = groupConvolution( calculateWVecBar(D) , gs )
     check isEqualG1Seq( lhs , rhs )  
 
+  test "group convolution with Wvec on a subgroup":
+    let gs  = randG1Seq( N )
+    let lhs = groupConvolveWithWVecOnSubgroup( sg , gs ) 
+    let rhs = selectOnSubgroup( sg , groupConvolution( calculateWVec(D) , gs ) )
+    check isEqualG1Seq( lhs , rhs )  
+
   test "expansion of the product of Lagrange polynomials":
     var ok = true
     for i in 0..<N:
diff --git a/tests/groth16/testFFT.nim b/tests/groth16/testFFT.nim
index 39e7f0b..5a9e19c 100644
--- a/tests/groth16/testFFT.nim
+++ b/tests/groth16/testFFT.nim
@@ -84,4 +84,10 @@ suite "convolution checks":
   test "FFT group convolutions vs. slow reference":
     check isEqualG1Seq( groupConvolution(xs,gs) , naiveGroupConvolution(xs,gs) )
 
+  test "FFT field convolution evaluated on a subgroup":
+    check testFieldConvolutionOnSubgroup( 32 , 8 )
+
+  test "FFT group convolution evaluated on a subgroup":
+    check testGroupConvolutionOnSubgroup( 128 , 16 )
+
 #-------------------------------------------------------------------------------