17 files changed, 1634 insertions(+), 378 deletions(-)

M cmd/tallyard/main.go
M election/election.go
M election/event.go
M election/map.go
M election/msg.go
M election/result.go
M election/version.go
M election/voter.go
M go.mod
M go.sum
A math/lagrange.go
A math/lagrange_test.go
D math/linalg.go
D math/linalg_test.go
M math/poly.go
A math/zk.go
M ui/tui.go

@@ 115,7 115,7 @@ func main() {
 			// user likely hit C-c
 			return
 		}
-		el.LocalVoter.Poly = math.NewRandomPoly(uint(len(*el.FinalJoinIDs)-1), 1024, *ballot)
+		el.LocalVoter.Poly = math.NewRandomPoly(*ballot, len(*el.FinalJoinIDs), &el.LocalVoter.Input)
 		el.Save()
 	}
 


@@ 172,6 172,7 @@ var electionFilter = &mautrix.Filter{
 				election.CreateElectionMessage,
 				election.JoinElectionMessage,
 				election.StartElectionMessage,
+				election.KeysMessage,
 				election.EvalsMessage,
 				election.SumMessage,
 			},

@@ 17,7 17,7 @@ type Election struct {
 	FinalJoinIDs *[]id.EventID         `json:"final_voters,omitempty"`
 	Joins        map[id.EventID]*Voter `json:"joins"`
 	LocalVoter   *LocalVoter           `json:"local_voter,omitempty"`
-	Result       *[]byte               `json:"result,omitempty"`
+	Result       *[32]byte             `json:"result,omitempty"`
 	RoomID       id.RoomID             `json:"room_id"`
 	StartID      *id.EventID           `json:"start_id,omitempty"`
 	Title        string                `json:"title"`

@@ 55,6 55,8 @@ func (store *EventStore) UnmarshalJSON(b []byte) error {
 			event.Type = JoinElectionMessage
 		case StartElectionMessage.Type:
 			event.Type = StartElectionMessage
+		case KeysMessage.Type:
+			event.Type = KeysMessage
 		case EvalsMessage.Type:
 			event.Type = EvalsMessage
 		case SumMessage.Type:


@@ 82,6 84,11 @@ type StartEvent struct {
 	*StartElectionContent
 }
 
+type KeysEvent struct {
+	*event.Event
+	*KeysMessageContent
+}
+
 type EvalsEvent struct {
 	*event.Event
 	*EvalsMessageContent


@@ 137,6 144,21 @@ func (store *EventStore) GetStartEvent(roomID id.RoomID, startID id.EventID) *St
 	}
 }
 
+func (store *EventStore) GetKeysEvent(roomID id.RoomID, keysID id.EventID) *KeysEvent {
+	evt, err := store.getAndHandleEvent(roomID, keysID, KeysMessage)
+	if err != nil {
+		log.Warnf("an error occurred getting keys event '%s': %s", keysID, err)
+		return nil
+	}
+	if evt == nil {
+		return nil
+	}
+	return &KeysEvent{
+		evt,
+		evt.Content.Parsed.(*KeysMessageContent),
+	}
+}
+
 func (store *EventStore) GetEvalsEvent(roomID id.RoomID, evalsID id.EventID) *EvalsEvent {
 	evt, err := store.getAndHandleEvent(roomID, evalsID, EvalsMessage)
 	if err != nil {

@@ 37,7 37,7 @@ type ElectionsMap struct {
 	save       func(*ElectionsMap)         `json:"-"`
 }
 
-const electionsMapVersion = 5
+const electionsMapVersion = 6
 
 func NewElectionsMap(userID id.UserID, save func(*ElectionsMap)) *ElectionsMap {
 	return &ElectionsMap{

@@ 1,18 1,24 @@
 package election
 
 import (
+	"bytes"
 	"encoding/base64"
 	"fmt"
-	"math/big"
 	"reflect"
 	"time"
 
+	"github.com/consensys/gnark/backend/groth16"
+	"github.com/consensys/gnark/crypto/hash/mimc/bls381"
+	"github.com/consensys/gnark/frontend"
+	"github.com/consensys/gurvy"
+	"github.com/consensys/gurvy/bls381/fr"
 	log "github.com/sirupsen/logrus"
 	"golang.org/x/crypto/nacl/box"
 	"golang.org/x/mod/semver"
 	"maunium.net/go/mautrix"
 	"maunium.net/go/mautrix/event"
 	"maunium.net/go/mautrix/id"
+	"tallyard.xyz/math"
 )
 
 var (


@@ 32,6 38,11 @@ var (
 		Type:  "xyz.tallyard.start",
 		Class: event.MessageEventType,
 	}
+	// contains keys needed to do zero knowledge proofs
+	KeysMessage = event.Type{
+		Type:  "xyz.tallyard.keys",
+		Class: event.MessageEventType,
+	}
 	// indicate's user's evaluations of their polynomial using others' inputs
 	EvalsMessage = event.Type{
 		Type:  "xyz.tallyard.evals",


@@ 57,6 68,7 @@ type JoinElectionContent struct {
 	CreateID id.EventID `json:"create_id"`
 	Input    string     `json:"input"`
 	PubKey   string     `json:"pub_key"`
+	HashSeed string     `json:"hash_seed"`
 }
 
 type StartElectionContent struct {


@@ 66,12 78,32 @@ type StartElectionContent struct {
 	JoinIDs  []id.EventID `json:"join_ids"`
 }
 
+type KeysMessageContent struct {
+	Version string `json:"version"`
+
+	EvalProvingKeyURI id.ContentURI `json:"eval_proving_key_uri"`
+	JoinID            id.EventID    `json:"join_id"`
+	StartID           id.EventID    `json:"start_id"`
+	SumProvingKeyURI  id.ContentURI `json:"sum_proving_key_uri"`
+}
+
 type EvalsMessageContent struct {
 	Version string `json:"version"`
 
-	Evals   map[id.EventID]string `json:"evals"`
-	JoinID  id.EventID            `json:"join_id"`
-	StartID id.EventID            `json:"start_id"`
+	Evals   []Eval       `json:"evals"`
+	JoinID  id.EventID   `json:"join_id"`
+	KeysIDs []id.EventID `json:"keys_ids"`
+}
+
+type Eval struct {
+	// encrypted for specific voter
+	Output      string `json:"output"`
+	// public; used by everyone in summation proofs
+	OutputHash  string `json:"output_hash"`
+	// encrypted for specific voter
+	POutputHash string `json:"poutput_hash"`
+	// encrypted for specific voter
+	Proof       string `json:"proof"`
 }
 
 type SumMessageContent struct {


@@ 80,20 112,14 @@ type SumMessageContent struct {
 	EvalsIDs []id.EventID `json:"evals_ids"`
 	JoinID   id.EventID   `json:"join_id"`
 	Sum      string       `json:"sum"`
-}
-
-type ResultMessageContent struct {
-	Version string `json:"version"`
-
-	JoinID id.EventID   `json:"join_id"`
-	Result string       `json:"result"`
-	SumIDs []id.EventID `json:"sums"`
+	Proofs   []string     `json:"proofs"`
 }
 
 func init() {
 	event.TypeMap[CreateElectionMessage] = reflect.TypeOf(CreateElectionContent{})
 	event.TypeMap[JoinElectionMessage]   = reflect.TypeOf(JoinElectionContent{})
 	event.TypeMap[StartElectionMessage]  = reflect.TypeOf(StartElectionContent{})
+	event.TypeMap[KeysMessage]           = reflect.TypeOf(KeysMessageContent{})
 	event.TypeMap[EvalsMessage]          = reflect.TypeOf(EvalsMessageContent{})
 	event.TypeMap[SumMessage]            = reflect.TypeOf(SumMessageContent{})
 }


@@ 113,7 139,8 @@ func (elections *ElectionsMap) SetupEventHooks(client *mautrix.Client, syncer ma
 		eventStore.Processing = make(map[id.EventID]struct{})
 	}
 
-	wrapper := func(f func(*event.Event) bool) func(*event.Event) bool {
+	// TODO: this is a freaking mess
+	wrapper := func(f func(*event.Event, *mautrix.Client) bool) func(*event.Event) bool {
 		return func(evt *event.Event) bool {
 			if evt.Unsigned.RedactedBecause != nil {
 				log.Debugf("event %s was redacted", evt.ID)


@@ 140,7 167,7 @@ func (elections *ElectionsMap) SetupEventHooks(client *mautrix.Client, syncer ma
 				eventStore.Processing[evt.ID] = struct{}{}
 				eventStore.Unlock()
 			}
-			success := f(evt)
+			success := f(evt, client)
 			eventStore.Lock()
 			// see EventStore doc for success explanation
 			if success {


@@ 153,12 180,18 @@ func (elections *ElectionsMap) SetupEventHooks(client *mautrix.Client, syncer ma
 			return success
 		}
 	}
+	wrapperNoClient := func(f func(*event.Event) bool) func(*event.Event) bool {
+		return wrapper(func(evt *event.Event, _ *mautrix.Client) bool {
+			return f(evt)
+		})
+	}
 
-	eventHandlers[CreateElectionMessage] = wrapper(elections.onCreateElectionMessage)
-	eventHandlers[JoinElectionMessage]   = wrapper(elections.onJoinElectionMessage)
-	eventHandlers[StartElectionMessage]  = wrapper(elections.onStartElectionMessage)
-	eventHandlers[EvalsMessage]          = wrapper(elections.onEvalsMessage)
-	eventHandlers[SumMessage]            = wrapper(elections.onSumMessage)
+	eventHandlers[CreateElectionMessage] = wrapperNoClient(elections.onCreateElectionMessage)
+	eventHandlers[JoinElectionMessage]   = wrapperNoClient(elections.onJoinElectionMessage)
+	eventHandlers[StartElectionMessage]  = wrapperNoClient(elections.onStartElectionMessage)
+	eventHandlers[KeysMessage]           = wrapper(elections.onKeysMessage)
+	eventHandlers[EvalsMessage]          = wrapperNoClient(elections.onEvalsMessage)
+	eventHandlers[SumMessage]            = wrapperNoClient(elections.onSumMessage)
 
 	for eventType, handler := range eventHandlers {
 		func(handler func(*event.Event) bool) {


@@ 206,6 239,11 @@ func (elections *ElectionsMap) onCreateElectionMessage(evt *event.Event) (succes
 		return
 	}
 
+	if len(content.Candidates) < 2 || len(content.Candidates) > 5 {
+		warnf("the number of candidates must be between 2 and 5 (inclusive), but was %s", len(content.Candidates))
+		return
+	}
+
 	elections.AddElection(NewElection(
 		content.Candidates,
 		*evt,


@@ 216,6 254,8 @@ func (elections *ElectionsMap) onCreateElectionMessage(evt *event.Event) (succes
 	return true
 }
 
+var zero *fr.Element = new(fr.Element).SetZero()
+
 func (elections *ElectionsMap) onJoinElectionMessage(evt *event.Event) (success bool) {
 	errorf, warnf, debugf := logFuncs(fmt.Sprintf("ignoring %s's join msg (%s) since %s", evt.Sender, evt.ID, "%s"))
 


@@ 225,37 265,55 @@ func (elections *ElectionsMap) onJoinElectionMessage(evt *event.Event) (success 
 		return
 	}
 
+	// Version
 	if incompatibleVersion(content.Version) {
 		debugf("the version is incompatible")
 		return
 	}
 
+	// CreateID
 	createEvt := elections.EventStore.GetCreateEvent(evt.RoomID, content.CreateID)
 	if createEvt == nil {
 		debugf("we couldn't get the create event, %s", content.CreateID)
 		return
 	}
 
-	bytes, err := base64.StdEncoding.DecodeString(content.Input)
+	// HashSeed
+	if content.HashSeed == "" {
+		warnf("the hash seed is empty")
+		return
+	}
+	hashSeed, err := base64.StdEncoding.DecodeString(content.HashSeed)
 	if err != nil {
-		warnf("we couldn't decode their input: %s", err)
+		warnf("we couldn't decode their hash seed: %s", err)
+		return
+	}
+	if len(hashSeed) < 32 {
+		warnf("their hash seed is fewer than 32 bytes")
 		return
 	}
-	input := new(big.Int).SetBytes(bytes)
 
+	// Input
+	byts, err := base64.StdEncoding.DecodeString(content.Input)
+	if err != nil {
+		warnf("we couldn't decode their input: %s", err)
+		return
+	}
+	input := new(fr.Element).SetBytes(byts)
 	// zero is not allowed
-	if input.Cmp(new(big.Int)) == 0 {
+	if input.Cmp(zero) == 0 {
 		warnf("their input is zero")
 		return
 	}
 
-	bytes, err = base64.StdEncoding.DecodeString(content.PubKey)
+	// PubKey
+	byts, err = base64.StdEncoding.DecodeString(content.PubKey)
 	if err != nil {
 		warnf("we couldn't decode their public key: %s", err)
 		return
 	}
 	var pubKey [32]byte
-	copy(pubKey[:], bytes)
+	copy(pubKey[:], byts)
 
 	el := elections.GetElection(createEvt.ID)
 	if el == nil {


@@ 269,7 327,7 @@ func (elections *ElectionsMap) onJoinElectionMessage(evt *event.Event) (success 
 	defer el.Save()
 	defer el.Unlock()
 
-	el.Joins[evt.ID] = NewVoter(input, evt, &pubKey)
+	el.Joins[evt.ID] = NewVoter(input, evt, &pubKey, hashSeed)
 
 	return true
 }


@@ 308,7 366,7 @@ func (elections *ElectionsMap) onStartElectionMessage(evt *event.Event) (success
 	}
 
 	if createEvt.Sender != evt.Sender {
-		warnf("they didn't create the election")
+		warnf("they didn't create the election; %s did", createEvt.Sender)
 		return
 	}
 


@@ 331,6 389,126 @@ func (elections *ElectionsMap) onStartElectionMessage(evt *event.Event) (success
 
 	el.StartID = &evt.ID
 	el.FinalJoinIDs = &content.JoinIDs
+	for i, joinID := range *el.FinalJoinIDs {
+		// TODO: gross
+		joinIndex := i
+		el.Joins[joinID].JoinIDIndex = &joinIndex
+	}
+
+	return true
+}
+
+func (elections *ElectionsMap) onKeysMessage(evt *event.Event, client *mautrix.Client) (success bool) {
+	errorf, warnf, debugf := logFuncs(fmt.Sprintf("ignoring %s's keys msg (%s) since %s", evt.Sender, evt.ID, "%s"))
+
+	content, ok := evt.Content.Parsed.(*KeysMessageContent)
+	if !ok {
+		warnf("we couldn't cast message content to KeysMessageContent")
+		return
+	}
+
+	// Version
+	if incompatibleVersion(content.Version) {
+		debugf("the version is incompatible")
+		return
+	}
+
+	joinEvt := elections.EventStore.GetJoinEvent(evt.RoomID, content.JoinID)
+	if joinEvt == nil {
+		debugf("we couldn't get the join event, %s", content.JoinID)
+		return
+	}
+	if evt.Sender != joinEvt.Sender {
+		warnf("they did not send the join event; %s did", joinEvt.Sender)
+		return
+	}
+
+	// process the start event so that JoinIDIndex is availabe below
+	if elections.EventStore.GetStartEvent(evt.RoomID, content.StartID) == nil {
+		debugf("we couldn't process the start event, %s", content.StartID)
+		return
+	}
+
+	el := elections.GetElection(joinEvt.CreateID)
+	if el == nil {
+		// should never happen because we retrieved the join event above
+		errorf("election %s doesn't exist", joinEvt.ID)
+		return
+	}
+
+	var evalProvingKey groth16.ProvingKey
+	{
+		// yes, I know we should be using Download here
+		byts, err := client.DownloadBytes(content.EvalProvingKeyURI)
+		if err != nil {
+			errorf("couldn't download eval proving key: %s", err)
+			return
+		}
+	tryEvalPk:
+		var buf bytes.Buffer
+		_, err = buf.Write(byts)
+		if err != nil {
+			errorf("couldn't write to eval bytes buffer: %s")
+			return
+		}
+		evalProvingKey = groth16.NewProvingKey(gurvy.BLS381)
+		_, err = evalProvingKey.ReadFrom(&buf)
+		if err != nil {
+			log.Errorf("couldn't read eval key from downloaded eval proving keys file: %s", err)
+			log.Debug("trying again...")
+			goto tryEvalPk
+		}
+	}
+
+	var sumProvingKey groth16.ProvingKey
+	{
+		byts, err := client.DownloadBytes(content.SumProvingKeyURI)
+		if err != nil {
+			errorf("couldn't download sum proving key: %s", err)
+			return
+		}
+	trySumPk:
+		var buf bytes.Buffer
+		_, err = buf.Write(byts)
+		if err != nil {
+			errorf("couldn't write to sum bytes buffer: %s")
+			return
+		}
+		sumProvingKey = groth16.NewProvingKey(gurvy.BLS381)
+		_, err = sumProvingKey.ReadFrom(&buf)
+		if err != nil {
+			log.Errorf("couldn't read sum key from downloaded sum keys file: %s", err)
+			log.Debug("trying again...")
+			goto trySumPk
+		}
+	}
+
+	var voter *Voter
+	{
+		var exists bool
+		// join should exist because we processed the start event above
+		if voter, exists = el.Joins[content.JoinID]; !exists {
+			warnf("the join event (%s) does not point to the election", content.JoinID)
+			return
+		}
+		if voter.JoinIDIndex == nil {
+			warnf("the join ID (%s) was not included in the start event (%s)", content.JoinID, content.StartID)
+			return
+		}
+	}
+
+	if voter.KeysID != nil {
+		warnf("voter already sent a keys message")
+		return
+	}
+
+	el.Lock()
+	defer el.Save()
+	defer el.Unlock()
+
+	voter.EvalProvingKey = &evalProvingKey
+	voter.SumProvingKey = &sumProvingKey
+	voter.KeysID = &evt.ID
 
 	return true
 }


@@ 344,46 522,52 @@ func (elections *ElectionsMap) onEvalsMessage(evt *event.Event) (success bool) {
 		return
 	}
 
+	// Version
 	if incompatibleVersion(content.Version) {
 		debugf("the version is incompatible")
 		return
 	}
 
-	startEvt := elections.EventStore.GetStartEvent(evt.RoomID, content.StartID)
-	if startEvt == nil {
-		debugf("we couldn't get the start event, %s", content.StartID)
-		return
-	}
-
+	// JoinID
 	joinEvt := elections.EventStore.GetJoinEvent(evt.RoomID, content.JoinID)
 	if joinEvt == nil {
 		debugf("we couldn't get the join event, %s", content.JoinID)
 		return
 	}
+	if evt.Sender != joinEvt.Sender {
+		warnf("they did not send the join event; %s did", joinEvt.Sender)
+		return
+	}
 
-	// ensure keys of Evals are in JoinIDs of start event
-	if len(content.Evals) != len(startEvt.JoinIDs) {
-		warnf("the number of evals is wrong (%s instead of %s)",
-			len(content.Evals), len(startEvt.JoinIDs))
+	el := elections.GetElection(joinEvt.CreateID)
+	if el == nil {
+		// should never happen because we got the start event above
+		errorf("election %s doesn't exist", joinEvt.CreateID)
+		return
+	}
+
+	// KeysIDs
+	if len(content.KeysIDs) != len(*el.FinalJoinIDs) {
+		warnf("the number of keys IDs is wrong (%s instead of %s)",
+			len(content.KeysIDs), len(*el.FinalJoinIDs))
 		return
 	}
-	for _, joinID := range startEvt.JoinIDs {
-		if _, exists := content.Evals[joinID]; !exists {
-			warnf("it doesn't include an eval for join %s", joinID)
+	for i, keysID := range content.KeysIDs {
+		keysEvent := elections.EventStore.GetKeysEvent(evt.RoomID, keysID)
+		if keysEvent == nil {
+			debugf("we couldn't get the keys event, %s", keysID)
+			return
+		}
+		if keysEvent.JoinID != (*el.FinalJoinIDs)[i] {
+			warnf("the join ID (%s) of a key (%s) does not match the start event's join ID (%s)",
+				keysEvent.JoinID, keysID, (*el.FinalJoinIDs)[i])
 			return
 		}
-	}
-	// is the voter's join ID in the start event's list of join IDs?
-	if _, exists := content.Evals[joinEvt.ID]; !exists {
-		warnf("the join event %s is not listed in the start event's join IDs", joinEvt.ID)
-		return
 	}
 
-	el := elections.GetElection(startEvt.CreateID)
-	if el == nil {
-		// should never happen because we retrieved the start/join
-		// events above
-		errorf("election %s doesn't exist", startEvt.CreateID)
+	if len(content.Evals) != len(*el.FinalJoinIDs) {
+		warnf("the number of evals is wrong (%s instead of %s)",
+			len(content.Evals), len(*el.FinalJoinIDs))
 		return
 	}
 


@@ 393,54 577,147 @@ func (elections *ElectionsMap) onEvalsMessage(evt *event.Event) (success bool) {
 
 	voter := el.Joins[joinEvt.ID]
 	if voter == nil {
-		// should never happen because we called getJoinEvent above
+		// should never happen because we called GetJoinEvent above
 		errorf("voter %s doesn't exist", joinEvt.ID)
 		return
 	}
-
+	if voter.JoinIDIndex == nil {
+		warnf("the voter's join ID was not included in the start event")
+		return
+	}
 	if voter.EvalsID != nil {
 		warnf("voter submitted multiple evals events")
 		return
 	}
-
 	voter.EvalsID = &evt.ID
 
+	// we're not participating in this election
 	if el.LocalVoter == nil {
 		return true
+	} else if el.LocalVoter.JoinIDIndex == nil {
+		warnf("we didn't join the election in time (or the election creator excluded us)")
+		return
 	}
+	ourEval := content.Evals[*el.LocalVoter.JoinIDIndex]
 
-	encodedEncryptedOutput, exists := content.Evals[el.LocalVoter.JoinEvt.ID]
-	// If our ID doesn't exist in the keys of evalsEvtContent.Evals, our
-	// JoinID wasn't included in the startEvtContent.JoinIDs (since we
-	// checked that the two are equivalent above).  I'm checking membership
-	// in Evals instead of startEvt.JoinIDs because maps are easier than
-	// slices for that.
-	if !exists {
-		debugf("we didn't join the election in time (or the election creator excluded us)")
-		return
+	// decrypt output
+	var output *fr.Element
+	{
+		encryptedOutput, err := base64.StdEncoding.DecodeString(ourEval.Output)
+		if err != nil {
+			warnf("couldn't decode the encrypted output for us: %s", err)
+			return
+		}
+		var decryptNonce [24]byte
+		copy(decryptNonce[:], encryptedOutput[:24])
+		decryptedOutput, ok := box.Open(nil, encryptedOutput[24:], &decryptNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+		if !ok {
+			warnf("couldn't decrypt eval output for us")
+			return
+		}
+		output = new(fr.Element).SetBytes(decryptedOutput)
 	}
-	encryptedOutput, err := base64.StdEncoding.DecodeString(encodedEncryptedOutput)
-	if err != nil {
-		warnf("couldn't decode %s's encrypted output: %s", evt.Sender, err)
-		return
+
+	// decrypt proof
+	var proof groth16.Proof
+	{
+		encryptedProof, err := base64.StdEncoding.DecodeString(ourEval.Proof)
+		if err != nil {
+			warnf("couldn't decode the encrypted proof for us: %s", err)
+			return
+		}
+		var decryptNonce [24]byte
+		copy(decryptNonce[:], encryptedProof[:24])
+		decryptedProofBytes, ok := box.Open(nil, encryptedProof[24:], &decryptNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+		if !ok {
+			warnf("couldn't decrypt eval proof for us")
+			return
+		}
+		proof = groth16.NewProof(gurvy.BLS381)
+		_, err = proof.ReadFrom(bytes.NewBuffer(decryptedProofBytes))
+		if err != nil {
+			errorf("couldn't read proof from bytes buffer: %s", err)
+			return
+		}
 	}
-	var decryptNonce [24]byte
-	copy(decryptNonce[:], encryptedOutput[:24])
-	decryptedOutput, ok := box.Open(nil, encryptedOutput[24:], &decryptNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
-	if !ok {
-		warnf("couldn't decrypt eval for us")
-		return
+
+	// decrypt poutputHash
+	var poutputHash []byte
+	{
+		encryptedPOutputHash, err := base64.StdEncoding.DecodeString(ourEval.POutputHash)
+		if err != nil {
+			warnf("couldn't decode the encrypted poutput hash for us: %s", err)
+			return
+		}
+		var decryptNonce [24]byte
+		copy(decryptNonce[:], encryptedPOutputHash[:24])
+		poutputHash, ok = box.Open(nil, encryptedPOutputHash[24:], &decryptNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+		if !ok {
+			warnf("couldn't decrypt poutput hash for us")
+			return
+		}
+	}
+
+	outputHashes := make([][]byte, len(*el.FinalJoinIDs))
+	var err error
+	for i, eval := range content.Evals {
+		outputHashes[i], err = base64.StdEncoding.DecodeString(eval.OutputHash)
+		if err != nil {
+			warnf("we couldn't decode the %d th output hash", i)
+			return
+		}
+	}
+
+	// verify proof
+	{
+		var publicCircuit math.EvalCircuit
+		publicCircuit.Coeffs = make([]frontend.Variable, len(*el.FinalJoinIDs)-1)
+		publicCircuit.BallotBits = make([]frontend.Variable, len(el.Candidates)*len(el.Candidates))
+		publicCircuit.Input.Assign(&el.LocalVoter.Input)
+		publicCircuit.Output.Assign(output)
+		publicCircuit.PInput.Assign(&voter.Input)
+		publicCircuit.POutputHash.Assign(poutputHash)
+		publicCircuit.HashSeed = el.LocalVoter.HashSeed
+		if el.LocalVoter.EvalVerifyingKey == nil {
+			// should never happen because we processed all keys
+			// events above
+			errorf("our evals verifying key is nil")
+			return
+		}
+		err = groth16.Verify(proof, *el.LocalVoter.EvalVerifyingKey, &publicCircuit)
+		if err != nil {
+			warnf("poly eval proof verification failed: %s", err)
+			return
+		}
 	}
-	voter.Eval = new(big.Int).SetBytes(decryptedOutput)
 
+	// ensure our output hashes to the given hash
+	{
+		outputBytes := output.Bytes()
+		outputHash, err := bls381.Sum(el.LocalVoter.HashSeed, outputBytes[:])
+		if err != nil {
+			errorf("couldn't hash output: %s", err)
+			return
+		}
+		// el.LocalVoter.JoinIDIndex checked for nilness above
+		if bytes.Compare(outputHashes[*el.LocalVoter.JoinIDIndex], outputHash) != 0 {
+			warnf("our given output did not hash to the given hash")
+			return
+		}
+	}
+
+	voter.Output = output
+	voter.OutputHashes = &outputHashes
+
+	// calculate our sum if we have everyone's outputs
 	for _, voterID := range *el.FinalJoinIDs {
-		if el.Joins[voterID].Eval == nil {
+		if el.Joins[voterID].Output == nil {
 			return true
 		}
 	}
-	sum := big.NewInt(0)
+	sum := new(fr.Element).SetZero()
 	for _, voterID := range *el.FinalJoinIDs {
-		sum.Add(sum, el.Joins[voterID].Eval)
+		sum.Add(sum, el.Joins[voterID].Output)
 	}
 	el.LocalVoter.Sum = sum
 


@@ 456,6 733,7 @@ func (elections *ElectionsMap) onSumMessage(evt *event.Event) (success bool) {
 		return
 	}
 
+	// Version
 	if incompatibleVersion(content.Version) {
 		debugf("the version is incompatible")
 		return


@@ 466,50 744,35 @@ func (elections *ElectionsMap) onSumMessage(evt *event.Event) (success bool) {
 		debugf("we couldn't get the join event, %s", content.JoinID)
 		return
 	}
+	if evt.Sender != joinEvt.Sender {
+		warnf("they did not send the join event; %s did", joinEvt.Sender)
+		return
+	}
 
 	if len(content.EvalsIDs) == 0 {
 		warnf("evals length is zero")
 		return
 	}
 
-	joinIDs := make(map[id.EventID]id.EventID)
-	var startID id.EventID
+	el := elections.GetElection(joinEvt.CreateID)
+	if el == nil {
+		// should never happen because we retrieved the start/join
+		// events above
+		errorf("election %s doesn't exist", joinEvt.CreateID)
+		return
+	}
 
-	for _, evalsID := range content.EvalsIDs {
+	for i, evalsID := range content.EvalsIDs {
 		evalsEvt := elections.EventStore.GetEvalsEvent(evt.RoomID, evalsID)
 		if evalsEvt == nil {
 			debugf("we couldn't get an evals event, %s", evalsID)
 			return
 		}
-
-		// ensure all evals have the same start ID
-		if startID == "" {
-			startID = evalsEvt.StartID
-		} else if evalsEvt.StartID != startID {
-			warnf("at least two evals have different startIDs (%s, %s, ...)", startID, evalsEvt.StartID)
-			return
-		}
-
-		// ensure no two evals use the same join ID.  This'll also catch
-		// duplicates in EvalsIDs
-		if same, exists := joinIDs[evalsEvt.JoinID]; exists {
-			warnf("at least two evals use the same join ID (%s, %s, ...)", same, evalsID)
+		if evalsEvt.JoinID != (*el.FinalJoinIDs)[i] {
+			warnf("the join ID (%s) of an evals (%s) does not match the start event's join ID (%s)",
+				evalsEvt.JoinID, evalsID, (*el.FinalJoinIDs)[i])
 			return
 		}
-
-		joinIDs[evalsEvt.JoinID] = evalsID
-
-		// Remember, we don't need to check that the evals's join ID is
-		// in the start event because we successfully retrieved the
-		// evals event above.  Recursion is wonderful.
-	}
-
-	el := elections.GetElection(joinEvt.CreateID)
-	if el == nil {
-		// should never happen because we retrieved the start/join
-		// events above
-		errorf("election %s doesn't exist", joinEvt.CreateID)
-		return
 	}
 
 	el.Lock()


@@ 528,14 791,84 @@ func (elections *ElectionsMap) onSumMessage(evt *event.Event) (success bool) {
 		return
 	}
 
-	bytes, err := base64.StdEncoding.DecodeString(content.Sum)
-	if err != nil {
-		warnf("we couldn't decode their sum: %s",  err)
-		return
+	var sum *fr.Element
+	{
+		sumBytes, err := base64.StdEncoding.DecodeString(content.Sum)
+		if err != nil {
+			warnf("we couldn't decode their sum: %s",  err)
+			return
+		}
+		sum = new(fr.Element).SetBytes(sumBytes)
+	}
+
+	var proof groth16.Proof
+	{
+		encrypted, err := base64.StdEncoding.DecodeString(content.Proofs[*el.LocalVoter.JoinIDIndex])
+		if err != nil {
+			warnf("couldn't decode sum proof for us: %s", err)
+			return
+		}
+		var nonce [24]byte
+		copy(nonce[:], encrypted[:24])
+		decrypted, ok := box.Open(nil, encrypted[24:], &nonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+		if !ok {
+			warnf("couldn't decrypt sum proof for us: %s", err)
+			return
+		}
+		var buf bytes.Buffer
+		_, err = buf.Write(decrypted)
+		if err != nil {
+			errorf("couldn't write decrypted sum proof to buffer: %s", err)
+			return
+		}
+		proof = groth16.NewProof(gurvy.BLS381)
+		_, err = proof.ReadFrom(&buf)
+		if err != nil {
+			errorf("couldn't read decrypted sum proof from buffer: %s", err)
+			return
+		}
+	}
+
+	// verify proof
+	{
+		var publicCircuit math.SumCircuit
+		n := len(*el.FinalJoinIDs)
+
+		hashSeeds := make([]string, n)
+		hashSelects := make([]frontend.Variable, n)
+		outputHashes := make([]frontend.Variable, n)
+		voterJoinIDIndex := *voter.JoinIDIndex
+		for i, joinID := range *el.FinalJoinIDs {
+			evaler := el.Joins[joinID]
+			hashSeeds[i] = evaler.HashSeed
+			if i == voterJoinIDIndex {
+				hashSelects[i].Assign(1)
+			} else {
+				hashSelects[i].Assign(0)
+			}
+			outputHashes[i].Assign((*evaler.OutputHashes)[voterJoinIDIndex])
+		}
+		publicCircuit.HashSeeds = hashSeeds
+		publicCircuit.HashSelects = hashSelects
+		publicCircuit.OutputHashes = outputHashes
+		publicCircuit.Sum.Assign(sum)
+		publicCircuit.Outputs = make([]frontend.Variable, n)
+
+		if el.LocalVoter.SumVerifyingKey == nil {
+			// should never happen because we processed all keys
+			// events above
+			errorf("our sum verifying key is nil")
+			return
+		}
+		err := groth16.Verify(proof, *el.LocalVoter.SumVerifyingKey, &publicCircuit)
+		if err != nil {
+			warnf("sum proof verification failed: %s", err)
+			return
+		}
 	}
 
 	voter.SumID = &evt.ID
-	voter.Sum = new(big.Int).SetBytes(bytes)
+	voter.Sum = sum
 
 	// if we've recieved all sums, calculate the election result!
 	for _, voterID := range *el.FinalJoinIDs {

@@ 2,7 2,6 @@ package election
 
 import (
 	"fmt"
-	"math/big"
 	"sort"
 	"time"
 


@@ 13,16 12,14 @@ import (
 func (el *Election) calculateResult() {
 	// this assumes all sums have been received and successfully validated
 	// and processed
-	M := constructPolyMatrix(el)
-	M.RREF()
-	constant := M[0][len(M[0])-1]
-	if !constant.IsInt() {
-		panic("constant term is not an integer")
+
+	points := make([]math.Point, len(*el.FinalJoinIDs))
+	for i, joinID := range *el.FinalJoinIDs {
+		points[i][0] = &el.Joins[joinID].Input
+		points[i][1] = el.Joins[joinID].Sum
 	}
-	result := constant.Num().Bytes()
-	// number of bytes we need to insert at the front since they're zero
-	diff := (len(el.Candidates) * len(el.Candidates)) - len(result)
-	result = append(make([]byte, diff), result...)
+
+	result := math.LagrangeEvalAtZero(points).Bytes()
 	el.Result = &result
 }
 


@@ 33,16 30,25 @@ func (el *Election) PrintResults() {
 			time.Sleep(time.Millisecond * 200)
 		}
 	}
-	result := *el.Result
 	candidates := el.Candidates
 
-	log.Debugf("result: %v", result)
+	var getRank func(i, j int) byte
+	{
+		result := *el.Result
+		log.Debugf("result: %v", result)
+		n := len(candidates)
+		l := len(result)
+		getRank = func(i, j int) byte {
+			return result[l - n*n + i*n+j]
+		}
+	}
+
 	fmt.Println("=== Results ===")
-	n := len(candidates)
+
 	for i, cand := range candidates {
 		for j, vs := range candidates {
 			if i != j {
-				fmt.Printf("%s over %s: %d\n", cand, vs, result[i*n+j])
+				fmt.Printf("%s over %s: %d\n", cand, vs, getRank(i, j))
 			}
 		}
 	}


@@ 69,8 75,8 @@ func (el *Election) PrintResults() {
 		p[i] = make([]int, len(candidates))
 		for j := range candidates {
 			if i != j {
-				if result[i*n+j] > result[j*n+i] {
-					p[i][j] = int(result[i*n+j])
+				if getRank(i, j) > getRank(j, i) {
+					p[i][j] = int(getRank(i, j))
 				} else {
 					p[i][j] = 0
 				}


@@ 113,23 119,3 @@ func (el *Election) PrintResults() {
 		}
 	}
 }
-
-func constructPolyMatrix(el *Election) math.Matrix {
-	mat := make(math.Matrix, len(*el.FinalJoinIDs))
-
-	i := 0
-	for _, voterJoinId := range *el.FinalJoinIDs {
-		voter := el.Joins[voterJoinId]
-		mat[i] = make([]big.Rat, len(mat)+1) // includes column for sum
-		row := mat[i]
-		row[0].SetInt64(1)
-		var j int64
-		for j = 1; j <= int64(len(*el.FinalJoinIDs)-1); j++ {
-			row[j].SetInt(new(big.Int).Exp(&voter.Input, big.NewInt(j), nil))
-		}
-		row[j].SetInt(voter.Sum)
-		i++
-	}
-
-	return mat
-}

@@ 1,3 1,3 @@
 package election
 
-const Version string = "v0.3.0"
+const Version string = "v0.4.0"

@@ 1,13 1,19 @@
 package election
 
 import (
+	"bytes"
 	"crypto/rand"
 	"encoding/base64"
 	"errors"
 	"fmt"
 	"io"
-	"math/big"
 
+	"github.com/consensys/gnark/backend/groth16"
+	"github.com/consensys/gnark/backend/r1cs"
+	"github.com/consensys/gnark/crypto/hash/mimc/bls381"
+	"github.com/consensys/gnark/frontend"
+	"github.com/consensys/gurvy"
+	"github.com/consensys/gurvy/bls381/fr"
 	"golang.org/x/crypto/nacl/box"
 	"maunium.net/go/mautrix"
 	"maunium.net/go/mautrix/event"


@@ 16,27 22,39 @@ import (
 )
 
 type Voter struct {
-	Input   big.Int     `json:"input"`
-	JoinEvt event.Event `json:"join_evt"`
-	PubKey  [32]byte    `json:"pub_key"`
-
-	Eval    *big.Int    `json:"eval,omitempty"`
-	EvalsID *id.EventID `json:"evals_id,omitempty"`
-	Sum     *big.Int    `json:"sum,omitempty"`
-	SumID   *id.EventID `json:"sum_id,omitempty"`
+	Input    fr.Element  `json:"input"`
+	JoinEvt  event.Event `json:"join_evt"`
+	PubKey   [32]byte    `json:"pub_key"`
+	HashSeed string      `json:"hash_seed"`
+
+	JoinIDIndex  *int        `json:"join_id_index,omitempty"`
+	Output       *fr.Element `json:"output,omitempty"`
+	OutputHashes *[][]byte   `json:"output_hashes,omitempty"`
+	KeysID       *id.EventID `json:"keys_id,omitempty"`
+	EvalsID      *id.EventID `json:"evals_id,omitempty"`
+	Sum          *fr.Element `json:"sum,omitempty"`
+	SumID        *id.EventID `json:"sum_id,omitempty"`
+
+	EvalProvingKey *groth16.ProvingKey `json:"eval_proving_key"`
+	SumProvingKey  *groth16.ProvingKey `json:"sum_proving_key"`
 }
 
 type LocalVoter struct {
 	*Voter
-	Poly    *math.Poly `json:"poly,omitempty"`
-	PrivKey [32]byte   `json:"priv_key"`
+
+	PrivKey          [32]byte             `json:"priv_key"`
+
+	EvalVerifyingKey *groth16.VerifyingKey `json:"eval_verifying_key"`
+	Poly             *math.Poly            `json:"poly,omitempty"`
+	SumVerifyingKey  *groth16.VerifyingKey `json:"sum_verifying_key"`
 }
 
-func NewVoter(input *big.Int, joinEvt *event.Event, pubKey *[32]byte) *Voter {
+func NewVoter(input *fr.Element, joinEvt *event.Event, pubKey *[32]byte, hashSeed []byte) *Voter {
 	return &Voter{
-		Input:   *new(big.Int).Set(input),
-		JoinEvt: *joinEvt,
-		PubKey:  *pubKey,
+		Input:    *new(fr.Element).Set(input),
+		JoinEvt:  *joinEvt,
+		PubKey:   *pubKey,
+		HashSeed: string(hashSeed),
 	}
 }
 


@@ 71,23 89,30 @@ func (el *Election) JoinElection(client *mautrix.Client, eventStore *EventStore)
 		return err
 	}
 
-	input, err := math.RandomBigInt(1024, false)
+	input, err := new(fr.Element).SetRandom()
 	if err != nil {
 		return err
 	}
+	inputBytes := input.Bytes()
+
+	var seedBytes [32]byte
+	if _, err := io.ReadFull(rand.Reader, seedBytes[:]); err != nil {
+		return fmt.Errorf("couldn't read random bytes: %s", err)
+	}
 
 	resp, err := client.SendMessageEvent(el.RoomID, JoinElectionMessage, JoinElectionContent{
-		Version:  Version,
+		Version: Version,
+
 		CreateID: el.CreateEvt.ID,
-		Input:    base64.StdEncoding.EncodeToString(input.Bytes()),
+		Input:    base64.StdEncoding.EncodeToString(inputBytes[:]),
 		PubKey:   base64.StdEncoding.EncodeToString((*pubKey)[:]),
+		HashSeed: base64.StdEncoding.EncodeToString(seedBytes[:]),
 	})
 	if err != nil {
-		return err
+		return fmt.Errorf("couldn't send join messages: %s", err)
 	}
 
-	joinEvt := eventStore.GetJoinEvent(el.RoomID, resp.EventID)
-	if joinEvt == nil {
+	if eventStore.GetJoinEvent(el.RoomID, resp.EventID) == nil {
 		return fmt.Errorf("couldn't process our own join event, %s", resp.EventID)
 	}
 


@@ 95,7 120,7 @@ func (el *Election) JoinElection(client *mautrix.Client, eventStore *EventStore)
 	defer el.Save()
 	defer el.Unlock()
 
-	el.LocalVoter = NewLocalVoter(el.Joins[joinEvt.ID], privKey)
+	el.LocalVoter = NewLocalVoter(el.Joins[resp.EventID], privKey)
 
 	return nil
 }


@@ 134,14 159,133 @@ func (el *Election) StartElection(client *mautrix.Client, eventStore *EventStore
 		return err
 	}
 
-	startEvt := eventStore.GetStartEvent(el.RoomID, resp.EventID)
-	if startEvt == nil {
+	if eventStore.GetStartEvent(el.RoomID, resp.EventID) == nil {
 		return fmt.Errorf("couldn't process our own start event, %s", resp.EventID)
 	}
 
 	return nil
 }
 
+func (el *Election) SendProvingKeys(client *mautrix.Client, eventStore *EventStore) error {
+	// eval proving key
+	var (
+		evalProvingKeyURI id.ContentURI
+		evalVerifyingKey  groth16.VerifyingKey
+	)
+	{
+		var evalCircuit math.EvalCircuit
+		evalCircuit.Coeffs = make([]frontend.Variable, len(*el.FinalJoinIDs)-1)
+		evalCircuit.BallotBits = make([]frontend.Variable, len(el.Candidates)*len(el.Candidates))
+		evalCircuit.HashSeed = el.LocalVoter.HashSeed
+		r1cs, err := frontend.Compile(gurvy.BLS381, &evalCircuit)
+		if err != nil {
+			return fmt.Errorf("couldn't compile eval circuit: %s", err)
+		}
+		var evalProvingKey groth16.ProvingKey
+		evalProvingKey, evalVerifyingKey, err = groth16.Setup(r1cs)
+		if err != nil {
+			return fmt.Errorf("couldn't setup r1cs: %s", err)
+		}
+
+		var evalPkBytes bytes.Buffer
+		// TODO: change back to WriteTo.  There seems to be a bug in
+		// gnark/gurvy (or a misunderstanding on my part) where
+		// decompression fails
+		// ignoring ...'s keys msg ($tsSEYHiouPdJlaTqAOG2q_4t9MZa0OGHtJPD95jOvAg) since couldn't read sum key from bytes buffer: invalid compressed coordinate: square root doesn't exist
+		// The file was successfully processed on one side (who
+		// coincidentally was the sender) but not on the other
+		// NOTE: it's plausible this was fixed in gurvy (now gnark-crypto) 0.4.0:
+		//
+		//     point.SetBytes can now be called concurently with same byte slice input
+		//
+		// https://github.com/ConsenSys/gnark-crypto/releases/tag/v0.4.0
+		evalProvingKey.WriteRawTo(&evalPkBytes)
+		uploadResp, err := client.UploadMedia(mautrix.ReqUploadMedia{
+			Content:       &evalPkBytes,
+			ContentLength: int64(evalPkBytes.Len()),
+			ContentType:   "application/octet-stream",
+		})
+		if err != nil {
+			return fmt.Errorf("couldn't upload eval proving key: %s", err)
+		}
+		evalProvingKeyURI = uploadResp.ContentURI
+	}
+
+	// sum proving key
+	var (
+		sumProvingKeyURI id.ContentURI
+		sumVerifyingKey  groth16.VerifyingKey
+	)
+	{
+		var sumCircuit math.SumCircuit
+		numVoters := len(*el.FinalJoinIDs)
+		hashSeeds := make([]string, numVoters)
+		for i, joinID := range *el.FinalJoinIDs {
+			hashSeeds[i] = el.Joins[joinID].HashSeed
+		}
+		sumCircuit.HashSeeds = hashSeeds
+		sumCircuit.HashSelects = make([]frontend.Variable, numVoters)
+		sumCircuit.OutputHashes = make([]frontend.Variable, numVoters)
+		sumCircuit.Outputs = make([]frontend.Variable, numVoters)
+
+		r1cs, err := frontend.Compile(gurvy.BLS381, &sumCircuit)
+		if err != nil {
+			return fmt.Errorf("couldn't compile sum circuit: %s", err)
+		}
+
+		var sumProvingKey groth16.ProvingKey
+		sumProvingKey, sumVerifyingKey, err = groth16.Setup(r1cs)
+		if err != nil {
+			return fmt.Errorf("couldn't setup r1cs: %s", err)
+		}
+
+		var sumPkBytes bytes.Buffer
+		// TODO: see above
+		sumProvingKey.WriteRawTo(&sumPkBytes)
+		uploadResp, err := client.UploadMedia(mautrix.ReqUploadMedia{
+			Content:       &sumPkBytes,
+			ContentLength: int64(sumPkBytes.Len()),
+			ContentType:   "application/octet-stream",
+		})
+		if err != nil {
+			return fmt.Errorf("couldn't upload sum proving key: %s", err)
+		}
+		sumProvingKeyURI = uploadResp.ContentURI
+	}
+
+	resp, err := client.SendMessageEvent(el.RoomID, KeysMessage, KeysMessageContent{
+		Version:        Version,
+
+		EvalProvingKeyURI: evalProvingKeyURI,
+		JoinID:            el.LocalVoter.JoinEvt.ID,
+		StartID:           *el.StartID,
+		SumProvingKeyURI:  sumProvingKeyURI,
+	})
+	if err != nil {
+		return fmt.Errorf("couldn't send keys message event: %s", err)
+	}
+
+	if eventStore.GetKeysEvent(el.RoomID, resp.EventID) == nil {
+		return fmt.Errorf("couldn't process our own keys event, %s", resp.EventID)
+	}
+
+	if el.LocalVoter.EvalProvingKey == nil {
+		return fmt.Errorf("our eval proving key wasn't set during processing")
+	}
+	if el.LocalVoter.SumProvingKey == nil {
+		return fmt.Errorf("our sum proving key wasn't set during processing")
+	}
+
+	el.Lock()
+	defer el.Save()
+	defer el.Unlock()
+
+	el.LocalVoter.EvalVerifyingKey = &evalVerifyingKey
+	el.LocalVoter.SumVerifyingKey = &sumVerifyingKey
+
+	return nil
+}
+
 func (el *Election) SendEvals(client *mautrix.Client, eventStore *EventStore) error {
 	el.RLock()
 	if el.StartID == nil {


@@ 149,24 293,77 @@ func (el *Election) SendEvals(client *mautrix.Client, eventStore *EventStore) er
 		return errors.New("SendEvals called before election started")
 	}
 
-	evals := make(map[id.EventID]string)
-	for _, joinID := range *el.FinalJoinIDs {
+	evals := make([]Eval, len(*el.FinalJoinIDs))
+	keysIDs := make([]id.EventID, len(*el.FinalJoinIDs))
+	for i, joinID := range *el.FinalJoinIDs {
 		voter := el.Joins[joinID]
-		eval := el.LocalVoter.Poly.Eval(&voter.Input)
-		var nonce [24]byte
-		if _, err := io.ReadFull(rand.Reader, nonce[:]); err != nil {
+
+		output, poutputHash, proof := el.LocalVoter.Poly.EvalAndProve(&voter.Input, *voter.EvalProvingKey, voter.HashSeed)
+
+		// encrypt output
+		var outputBytes [32]byte
+		var encryptedOutput []byte
+		{
+			var outputNonce [24]byte
+			if _, err := io.ReadFull(rand.Reader, outputNonce[:]); err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't read random bytes for output nonce: %s", err)
+			}
+			outputBytes = output.Bytes()
+			encryptedOutput = box.Seal(outputNonce[:], outputBytes[:], &outputNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+		}
+
+		// hash output
+		outputHash, err := bls381.Sum(voter.HashSeed, outputBytes[:])
+		if err != nil {
 			el.RUnlock()
-			return err
+			return fmt.Errorf("couldn't hash output: %s", err)
+		}
+
+		// hash and encrypt poutput
+		var encryptedPOutputHash []byte
+		{
+			var poutputNonce [24]byte
+			if _, err := io.ReadFull(rand.Reader, poutputNonce[:]); err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't read random bytes for poutput nonce: %s", err)
+			}
+			encryptedPOutputHash = box.Seal(poutputNonce[:], poutputHash, &poutputNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+		}
+
+		// encrypt proof
+		var encryptedProof []byte
+		{
+			var proofNonce [24]byte
+			if _, err := io.ReadFull(rand.Reader, proofNonce[:]); err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't read random bytes for proof nonce: %s", err)
+			}
+			var proofBuffer bytes.Buffer
+			_, err := proof.WriteTo(&proofBuffer)
+			if err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't write evals proof to buffer: %s", err)
+			}
+			encryptedProof = box.Seal(proofNonce[:], proofBuffer.Bytes(), &proofNonce, &voter.PubKey, &el.LocalVoter.PrivKey)
 		}
-		encrypted := box.Seal(nonce[:], eval.Bytes(), &nonce, &voter.PubKey, &el.LocalVoter.PrivKey)
-		evals[voter.JoinEvt.ID] = base64.StdEncoding.EncodeToString(encrypted)
+
+		evals[i] = Eval{
+			Output:      base64.StdEncoding.EncodeToString(encryptedOutput),
+			OutputHash:  base64.StdEncoding.EncodeToString(outputHash),
+			POutputHash: base64.StdEncoding.EncodeToString(encryptedPOutputHash),
+			Proof:       base64.StdEncoding.EncodeToString(encryptedProof),
+		}
+
+		keysIDs[i] = *voter.KeysID
 	}
 
 	resp, err := client.SendMessageEvent(el.RoomID, EvalsMessage, EvalsMessageContent{
 		Version: Version,
+
 		Evals:   evals,
 		JoinID:  el.LocalVoter.JoinEvt.ID,
-		StartID: *el.StartID,
+		KeysIDs: keysIDs,
 	})
 	el.RUnlock()
 	if err != nil {


@@ 188,17 385,88 @@ func (el *Election) SendSum(client *mautrix.Client, eventStore *EventStore) erro
 		return errors.New("SendSum called before local sum was calculated")
 	}
 
-	var evalsIDs []id.EventID
-	for _, joinID := range *el.FinalJoinIDs {
+	numVoters := len(*el.FinalJoinIDs)
+
+	evalsIDs := make([]id.EventID, numVoters)
+	hashSeeds := make([]string, numVoters)
+	for i, joinID := range *el.FinalJoinIDs {
 		voter := el.Joins[joinID]
-		evalsIDs = append(evalsIDs, *voter.EvalsID)
+		evalsIDs[i] = *voter.EvalsID
+		hashSeeds[i] = voter.HashSeed
 	}
 
+	var r1cs r1cs.R1CS
+	{
+		var circuit math.SumCircuit
+		circuit.HashSeeds = hashSeeds
+		circuit.HashSelects = make([]frontend.Variable, numVoters)
+		circuit.OutputHashes = make([]frontend.Variable, numVoters)
+		circuit.Outputs = make([]frontend.Variable, numVoters)
+		var err error
+		r1cs, err = frontend.Compile(gurvy.BLS381, &circuit)
+		if err != nil {
+			el.RUnlock()
+			return fmt.Errorf("couldn't compile sum circuit: %s", err)
+		}
+	}
+
+	proofs := make([]string, numVoters)
+	{
+		hashSelects := make([]frontend.Variable, numVoters)
+		outputHashes := make([]frontend.Variable, numVoters)
+		outputs := make([]frontend.Variable, numVoters)
+
+		var witness math.SumCircuit
+		witness.HashSeeds = hashSeeds
+		witness.HashSelects = hashSelects
+		witness.OutputHashes = outputHashes
+		witness.Sum.Assign(el.LocalVoter.Sum)
+		witness.Outputs = outputs
+
+		ourJoinIDIndex := *el.LocalVoter.JoinIDIndex
+		for i, joinID := range *el.FinalJoinIDs {
+			if i == ourJoinIDIndex {
+				hashSelects[i].Assign(1)
+			} else {
+				hashSelects[i].Assign(0)
+			}
+			evaler := el.Joins[joinID]
+			outputHashForUs := (*evaler.OutputHashes)[ourJoinIDIndex]
+			outputHashes[i].Assign(outputHashForUs)
+			outputs[i].Assign(evaler.Output)
+		}
+
+		for i, joinID := range *el.FinalJoinIDs {
+			voter := *el.Joins[joinID]
+			proof, err := groth16.Prove(r1cs, *voter.SumProvingKey, &witness)
+			if err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't prove sum: %s", err)
+			}
+			var nonce [24]byte
+			if _, err := io.ReadFull(rand.Reader, nonce[:]); err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't read random bytes for sum proof nonce: %s", err)
+			}
+			var buf bytes.Buffer
+			_, err = proof.WriteTo(&buf)
+			if err != nil {
+				el.RUnlock()
+				return fmt.Errorf("couldn't write sum proof to buffer: %s", err)
+			}
+			encryptedProof := box.Seal(nonce[:], buf.Bytes(), &nonce, &voter.PubKey, &el.LocalVoter.PrivKey)
+			proofs[i] = base64.StdEncoding.EncodeToString(encryptedProof)
+		}
+	}
+
+	sumBytes := el.LocalVoter.Sum.Bytes()
 	resp, err := client.SendMessageEvent(el.RoomID, SumMessage, SumMessageContent{
 		Version:  Version,
+
 		EvalsIDs: evalsIDs,
 		JoinID:   el.LocalVoter.JoinEvt.ID,
-		Sum:      base64.StdEncoding.EncodeToString(el.LocalVoter.Sum.Bytes()),
+		Sum:      base64.StdEncoding.EncodeToString(sumBytes[:]),
+		Proofs:   proofs,
 	})
 	el.RUnlock()
 	if err != nil {

@@ 4,8 4,9 @@ go 1.13
 
 require (
 	github.com/cbergoon/merkletree v0.2.0
+	github.com/consensys/gnark v0.3.8
+	github.com/consensys/gurvy v0.3.8
 	github.com/gdamore/tcell/v2 v2.1.0
-	github.com/kr/pretty v0.1.0 // indirect
 	github.com/kyoh86/xdg v1.2.0
 	github.com/lucasb-eyer/go-colorful v1.2.0 // indirect
 	github.com/magefile/mage v1.11.0 // indirect

@@ 1,4 1,48 @@
+cloud.google.com/go v0.26.0/go.mod h1:aQUYkXzVsufM+DwF1aE+0xfcU+56JwCaLick0ClmMTw=
+cloud.google.com/go v0.34.0/go.mod h1:aQUYkXzVsufM+DwF1aE+0xfcU+56JwCaLick0ClmMTw=
+cloud.google.com/go v0.38.0/go.mod h1:990N+gfupTy94rShfmMCWGDn0LpTmnzTp2qbd1dvSRU=
+cloud.google.com/go v0.44.1/go.mod h1:iSa0KzasP4Uvy3f1mN/7PiObzGgflwredwwASm/v6AU=
+cloud.google.com/go v0.44.2/go.mod h1:60680Gw3Yr4ikxnPRS/oxxkBccT6SA1yMk63TGekxKY=
+cloud.google.com/go v0.45.1/go.mod h1:RpBamKRgapWJb87xiFSdk4g1CME7QZg3uwTez+TSTjc=
+cloud.google.com/go v0.46.3/go.mod h1:a6bKKbmY7er1mI7TEI4lsAkts/mkhTSZK8w33B4RAg0=
+cloud.google.com/go/bigquery v1.0.1/go.mod h1:i/xbL2UlR5RvWAURpBYZTtm/cXjCha9lbfbpx4poX+o=
+cloud.google.com/go/datastore v1.0.0/go.mod h1:LXYbyblFSglQ5pkeyhO+Qmw7ukd3C+pD7TKLgZqpHYE=
+cloud.google.com/go/firestore v1.1.0/go.mod h1:ulACoGHTpvq5r8rxGJ4ddJZBZqakUQqClKRT5SZwBmk=
+cloud.google.com/go/pubsub v1.0.1/go.mod h1:R0Gpsv3s54REJCy4fxDixWD93lHJMoZTyQ2kNxGRt3I=
+cloud.google.com/go/storage v1.0.0/go.mod h1:IhtSnM/ZTZV8YYJWCY8RULGVqBDmpoyjwiyrjsg+URw=
+dmitri.shuralyov.com/gpu/mtl v0.0.0-20190408044501-666a987793e9/go.mod h1:H6x//7gZCb22OMCxBHrMx7a5I7Hp++hsVxbQ4BYO7hU=
+github.com/Azure/azure-pipeline-go v0.2.1/go.mod h1:UGSo8XybXnIGZ3epmeBw7Jdz+HiUVpqIlpz/HKHylF4=
+github.com/Azure/azure-pipeline-go v0.2.2/go.mod h1:4rQ/NZncSvGqNkkOsNpOU1tgoNuIlp9AfUH5G1tvCHc=
+github.com/Azure/azure-storage-blob-go v0.7.0/go.mod h1:f9YQKtsG1nMisotuTPpO0tjNuEjKRYAcJU8/ydDI++4=
+github.com/Azure/go-autorest/autorest v0.9.0/go.mod h1:xyHB1BMZT0cuDHU7I0+g046+BFDTQ8rEZB0s4Yfa6bI=
+github.com/Azure/go-autorest/autorest/adal v0.5.0/go.mod h1:8Z9fGy2MpX0PvDjB1pEgQTmVqjGhiHBW7RJJEciWzS0=
+github.com/Azure/go-autorest/autorest/adal v0.8.0/go.mod h1:Z6vX6WXXuyieHAXwMj0S6HY6e6wcHn37qQMBQlvY3lc=
+github.com/Azure/go-autorest/autorest/date v0.1.0/go.mod h1:plvfp3oPSKwf2DNjlBjWF/7vwR+cUD/ELuzDCXwHUVA=
+github.com/Azure/go-autorest/autorest/date v0.2.0/go.mod h1:vcORJHLJEh643/Ioh9+vPmf1Ij9AEBM5FuBIXLmIy0g=
+github.com/Azure/go-autorest/autorest/mocks v0.1.0/go.mod h1:OTyCOPRA2IgIlWxVYxBee2F5Gr4kF2zd2J5cFRaIDN0=
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+github.com/BurntSushi/toml v0.3.1/go.mod h1:xHWCNGjB5oqiDr8zfno3MHue2Ht5sIBksp03qcyfWMU=
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+github.com/OneOfOne/xxhash v1.2.2/go.mod h1:HSdplMjZKSmBqAxg5vPj2TmRDmfkzw+cTzAElWljhcU=
+github.com/StackExchange/wmi v0.0.0-20180116203802-5d049714c4a6/go.mod h1:3eOhrUMpNV+6aFIbp5/iudMxNCF27Vw2OZgy4xEx0Fg=
+github.com/VictoriaMetrics/fastcache v1.5.7/go.mod h1:ptDBkNMQI4RtmVo8VS/XwRY6RoTu1dAWCbrk+6WsEM8=
 github.com/aead/siphash v1.0.1/go.mod h1:Nywa3cDsYNNK3gaciGTWPwHt0wlpNV15vwmswBAUSII=
+github.com/alecthomas/template v0.0.0-20160405071501-a0175ee3bccc/go.mod h1:LOuyumcjzFXgccqObfd/Ljyb9UuFJ6TxHnclSeseNhc=
+github.com/alecthomas/units v0.0.0-20151022065526-2efee857e7cf/go.mod h1:ybxpYRFXyAe+OPACYpWeL0wqObRcbAqCMya13uyzqw0=
+github.com/allegro/bigcache v1.2.1-0.20190218064605-e24eb225f156/go.mod h1:Cb/ax3seSYIx7SuZdm2G2xzfwmv3TPSk2ucNfQESPXM=
+github.com/aristanetworks/goarista v0.0.0-20170210015632-ea17b1a17847/go.mod h1:D/tb0zPVXnP7fmsLZjtdUhSsumbK/ij54UXjjVgMGxQ=
+github.com/armon/circbuf v0.0.0-20150827004946-bbbad097214e/go.mod h1:3U/XgcO3hCbHZ8TKRvWD2dDTCfh9M9ya+I9JpbB7O8o=
+github.com/armon/go-metrics v0.0.0-20180917152333-f0300d1749da/go.mod h1:Q73ZrmVTwzkszR9V5SSuryQ31EELlFMUz1kKyl939pY=
+github.com/armon/go-radix v0.0.0-20180808171621-7fddfc383310/go.mod h1:ufUuZ+zHj4x4TnLV4JWEpy2hxWSpsRywHrMgIH9cCH8=
+github.com/aws/aws-sdk-go v1.25.48/go.mod h1:KmX6BPdI08NWTb3/sm4ZGu5ShLoqVDhKgpiN924inxo=
+github.com/beorn7/perks v0.0.0-20180321164747-3a771d992973/go.mod h1:Dwedo/Wpr24TaqPxmxbtue+5NUziq4I4S80YR8gNf3Q=
+github.com/beorn7/perks v1.0.0/go.mod h1:KWe93zE9D1o94FZ5RNwFwVgaQK1VOXiVxmqh+CedLV8=
+github.com/bgentry/speakeasy v0.1.0/go.mod h1:+zsyZBPWlz7T6j88CTgSN5bM796AkVf0kBD4zp0CCIs=
+github.com/bketelsen/crypt v0.0.3-0.20200106085610-5cbc8cc4026c/go.mod h1:MKsuJmJgSg28kpZDP6UIiPt0e0Oz0kqKNGyRaWEPv84=
+github.com/btcsuite/btcd v0.0.0-20171128150713-2e60448ffcc6/go.mod h1:Dmm/EzmjnCiweXmzRIAiUWCInVmPgjkzgv5k4tVyXiQ=
 github.com/btcsuite/btcd v0.20.1-beta/go.mod h1:wVuoA8VJLEcwgqHBwHmzLRazpKxTv13Px/pDuV7OomQ=
 github.com/btcsuite/btclog v0.0.0-20170628155309-84c8d2346e9f/go.mod h1:TdznJufoqS23FtqVCzL0ZqgP5MqXbb4fg/WgDys70nA=
 github.com/btcsuite/btcutil v0.0.0-20190425235716-9e5f4b9a998d/go.mod h1:+5NJ2+qvTyV9exUAL/rxXi3DcLg2Ts+ymUAY5y4NvMg=


@@ 11,115 55,473 @@ github.com/btcsuite/websocket v0.0.0-20150119174127-31079b680792/go.mod h1:ghJtE
 github.com/btcsuite/winsvc v1.0.0/go.mod h1:jsenWakMcC0zFBFurPLEAyrnc/teJEM1O46fmI40EZs=
 github.com/cbergoon/merkletree v0.2.0 h1:Bttqr3OuoiZEo4ed1L7fTasHka9II+BF9fhBfbNEEoQ=
 github.com/cbergoon/merkletree v0.2.0/go.mod h1:5c15eckUgiucMGDOCanvalj/yJnD+KAZj1qyJtRW5aM=
+github.com/cespare/cp v0.1.0/go.mod h1:SOGHArjBr4JWaSDEVpWpo/hNg6RoKrls6Oh40hiwW+s=
+github.com/cespare/xxhash v1.1.0/go.mod h1:XrSqR1VqqWfGrhpAt58auRo0WTKS1nRRg3ghfAqPWnc=
+github.com/cespare/xxhash/v2 v2.1.1/go.mod h1:VGX0DQ3Q6kWi7AoAeZDth3/j3BFtOZR5XLFGgcrjCOs=
+github.com/client9/misspell v0.3.4/go.mod h1:qj6jICC3Q7zFZvVWo7KLAzC3yx5G7kyvSDkc90ppPyw=
+github.com/cloudflare/cloudflare-go v0.10.2-0.20190916151808-a80f83b9add9/go.mod h1:1MxXX1Ux4x6mqPmjkUgTP1CdXIBXKX7T+Jk9Gxrmx+U=
+github.com/consensys/bavard v0.1.7/go.mod h1:Bpd0/3mZuaj6Sj+PqrmIquiOKy397AKGThQPaGzNXAQ=
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+github.com/consensys/gnark v0.3.8 h1:4t8tdqDLOAWRRycw4dsmI29v+ESii0up+vHAekn/Sgw=
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 github.com/davecgh/go-spew v1.1.1 h1:vj9j/u1bqnvCEfJOwUhtlOARqs3+rkHYY13jYWTU97c=
 github.com/davecgh/go-spew v1.1.1/go.mod h1:J7Y8YcW2NihsgmVo/mv3lAwl/skON4iLHjSsI+c5H38=
+github.com/deckarep/golang-set v0.0.0-20180603214616-504e848d77ea/go.mod h1:93vsz/8Wt4joVM7c2AVqh+YRMiUSc14yDtF28KmMOgQ=
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@@ 0,0 1,31 @@
+package math
+
+import "github.com/consensys/gurvy/bls381/fr"
+
+// p[0] is x coordinate, p[1] is y coordinate
+type Point [2]*fr.Element
+
+// uses https://en.wikipedia.org/wiki/Lagrange_polynomial#Definition but returns
+// the evaluation at zero (the only one we care about)
+func LagrangeEvalAtZero(points []Point) *fr.Element {
+	zero := new(fr.Element).SetZero()
+
+	sum := new(fr.Element).SetZero()
+	tmp := new(fr.Element)
+
+	for j, pointj := range points {
+		term := new(fr.Element).Set(pointj[1])
+		for k, pointk := range points {
+			if k == j {
+				continue
+			}
+			// numerator
+			term.Mul(term, tmp.Sub(zero, pointk[0]))
+			// denominator
+			term.Div(term, tmp.Sub(pointj[0], pointk[0]))
+		}
+		sum.Add(sum, term)
+	}
+
+	return sum
+}

@@ 0,0 1,81 @@
+package math
+
+import (
+	"testing"
+
+	"github.com/consensys/gurvy/bls381/fr"
+)
+
+func TestWikipedia(t *testing.T) {
+	// https://en.wikipedia.org/wiki/Lagrange_polynomial#Example_1
+	points := make([]Point, 3)
+	points[0][0] = new(fr.Element).SetUint64(1)
+	points[0][1] = new(fr.Element).SetUint64(1)
+	points[1][0] = new(fr.Element).SetUint64(2)
+	points[1][1] = new(fr.Element).SetUint64(4)
+	points[2][0] = new(fr.Element).SetUint64(3)
+	points[2][1] = new(fr.Element).SetUint64(9)
+	r := LagrangeEvalAtZero(points)
+	if !r.Equal(new(fr.Element).SetZero()) {
+		t.Fail()
+	}
+
+	// https://en.wikipedia.org/wiki/Lagrange_polynomial#Example_2
+	points = make([]Point, 4)
+	points[0][0] = new(fr.Element).SetUint64(1)
+	points[0][1] = new(fr.Element).SetUint64(1)
+	points[1][0] = new(fr.Element).SetUint64(2)
+	points[1][1] = new(fr.Element).SetUint64(8)
+	points[2][0] = new(fr.Element).SetUint64(3)
+	points[2][1] = new(fr.Element).SetUint64(27)
+	points[3][0] = new(fr.Element).SetUint64(4)
+	points[3][1] = new(fr.Element).SetUint64(64)
+	if !r.Equal(new(fr.Element).SetZero()) {
+		t.Fail()
+	}
+}
+
+func TestRandomPoly(t *testing.T) {
+	// a simple ballot where the first candidate is ranked above the second
+	ballot := make([][]byte, 2)
+	ballot[0] = make([]byte, 2)
+	ballot[1] = make([]byte, 2)
+	ballot[0][1] = 1
+
+	// pinput doesn't matter
+genpinput:
+	pinput, err := new(fr.Element).SetRandom()
+	if err != nil {
+		panic(err)
+	}
+	if pinput.IsZero() {
+		goto genpinput
+	}
+
+	zero := new(fr.Element).SetZero()
+	// vary the number of points
+	for n := 2; n <= 20; n++ {
+		poly := NewRandomPoly(ballot, n, pinput)
+		points := make([]Point, n)
+		for i := 0; i < n; i++ {
+		genx:
+			x, err := new(fr.Element).SetRandom()
+			// TODO: check that x doesn't repeat (very unlikely)
+			if err != nil {
+				panic(err)
+			}
+			if x.IsZero() {
+				goto genx
+			}
+			y := poly.eval(x)
+			points[i][0] = x
+			points[i][1] = y
+		}
+		lagrangeZeroOutput := LagrangeEvalAtZero(points)
+		actualZeroOutput := poly.eval(zero)
+		if lagrangeZeroOutput.Cmp(actualZeroOutput) != 0 {
+			t.Fatalf("lagrange zero output %s does not equal %s",
+				lagrangeZeroOutput, actualZeroOutput)
+		}
+	}
+}

@@ 1,73 0,0 @@
-package math
-
-import (
-	"math/big"
-)
-
-type Matrix [][]big.Rat
-
-func (M Matrix) Rows() int {
-	return len(M)
-}
-
-func (M Matrix) Cols() int {
-	if len(M) == 0 {
-		return 0
-	}
-	return len(M[0])
-}
-
-func (M Matrix) RREF() {
-	lead := 0
-	zero := big.NewRat(0, 1)
-
-	for r := 0; r < M.Rows(); r++ {
-		if lead >= M.Cols() {
-			return
-		}
-		i := r
-		for M[i][lead].Cmp(zero) == 0 {
-			i++
-			if M.Rows() == i {
-				i = r
-				lead++
-				if M.Cols() == lead {
-					return
-				}
-			}
-		}
-		M[i], M[r] = M[r], M[i]
-		f := &big.Rat{}
-		f.Inv(&M[r][lead])
-		for j := range M[r] {
-			M[r][j].Mul(&M[r][j], f)
-		}
-		for i = 0; i < M.Rows(); i++ {
-			if i != r {
-				f.Set(&M[i][lead])
-				for j, e := range M[r] {
-					M[i][j].Sub(&M[i][j], new(big.Rat).Mul(&e, f))
-				}
-			}
-		}
-		lead++
-	}
-}
-
-// assumes `M' and `other' are valid matrices
-func (M Matrix) Equals(other Matrix) bool {
-	if len(M) != len(other) {
-		return false
-	}
-	if len(M) > 0 && len(M[0]) != len(other[0]) {
-		return false
-	}
-	for r := range M {
-		for c := range M[r] {
-			if M[r][c].Cmp(&other[r][c]) != 0 {
-				return false
-			}
-		}
-	}
-	return true
-}

@@ 1,36 0,0 @@
-package math
-
-import (
-	"math/big"
-	"testing"
-)
-
-func TestRREF(t *testing.T) {
-	mat := Matrix{
-		{*big.NewRat( 1, 1), *big.NewRat(2, 1), *big.NewRat(-1, 1), *big.NewRat( -4, 1)},
-		{*big.NewRat( 2, 1), *big.NewRat(3, 1), *big.NewRat(-1, 1), *big.NewRat(-11, 1)},
-		{*big.NewRat(-2, 1), *big.NewRat(0, 1), *big.NewRat(-3, 1), *big.NewRat( 22, 1)},
-	}
-	mat.RREF()
-	result := Matrix{
-		{*big.NewRat(1, 1), *big.NewRat(0, 1), *big.NewRat(0, 1), *big.NewRat(-8, 1)},
-		{*big.NewRat(0, 1), *big.NewRat(1, 1), *big.NewRat(0, 1), *big.NewRat( 1, 1)},
-		{*big.NewRat(0, 1), *big.NewRat(0, 1), *big.NewRat(1, 1), *big.NewRat(-2, 1)},
-	}
-	if !mat.Equals(result) {
-		t.Fail()
-	}
-
-	mat = Matrix{
-		{*big.NewRat(1, 1), *big.NewRat(3, 1), *big.NewRat(-1, 1)},
-		{*big.NewRat(0, 1), *big.NewRat(1, 1),  *big.NewRat(7, 1)},
-	}
-	mat.RREF()
-	result = Matrix{
-		{*big.NewRat(1, 1), *big.NewRat(0, 1), *big.NewRat(-22, 1)},
-		{*big.NewRat(0, 1), *big.NewRat(1, 1), *big.NewRat(  7, 1)},
-	}
-	if !mat.Equals(result) {
-		t.Fail()
-	}
-}

@@ 1,69 1,166 @@
 package math
 
 import (
-	"crypto/rand"
-	"math"
-	"math/big"
+	"fmt"
+
+	"github.com/consensys/gnark/backend/groth16"
+	"github.com/consensys/gnark/backend/r1cs"
+	"github.com/consensys/gnark/crypto/hash/mimc/bls381"
+	"github.com/consensys/gnark/frontend"
+	"github.com/consensys/gurvy"
+	"github.com/consensys/gurvy/bls381/fr"
 )
 
 type Poly struct {
-	Constant *big.Int   `json:"constant"`
-	Coefs    []*big.Int `json:"coefs"`
+	Ballot [][]byte      `json:"ballot"`
+	Coeffs []*fr.Element `json:"coeffs"`
+	PInput *fr.Element   `json:"pinput"`
+
+	cache *polyCache `json:"-"`
 }
 
-func RandomBigInt(numBytes uint, allowAllZeros bool) (*big.Int, error) {
-	randBytes := make([]byte, numBytes)
-	_, err := rand.Read(randBytes)
-	if err != nil {
-		return nil, err
-	}
-	bi := new(big.Int).SetBytes(randBytes)
-	if !allowAllZeros && bi.Cmp(big.NewInt(0)) == 0 {
-		return RandomBigInt(numBytes, allowAllZeros)
-	}
-	return bi, nil
+type polyCache struct {
+	ballotBits []*fr.Element
+	constant   *fr.Element
+	outputs    map[*fr.Element]outputCache
+	poutput    *fr.Element
 }
 
-func NewPoly(coefficients []*big.Int, ballot []byte) *Poly {
-	constant := new(big.Int).SetBytes(ballot)
-	return &Poly{constant, coefficients}
+type outputCache struct{
+	output      *fr.Element
+	poutputHash []byte
+	proof       groth16.Proof
 }
 
-// NewRandomPoly generates a random `degree' degree polynomial, using at least
-// `entropy' bits of entropy for the random coefficients and inserting `ballot'
-// as the constant term.
-func NewRandomPoly(degree uint, entropy uint, ballot []byte) *Poly {
-	coefficients := make([]*big.Int, degree)
+// NewRandomPoly generates a random polynomial of degree numVoters-1, using
+// `ballot' as the constant term
+func NewRandomPoly(ballot [][]byte, numVoters int, pinput *fr.Element) *Poly {
+	coeffs := make([]*fr.Element, numVoters - 1)
+
+	var err error
+	for i := range coeffs {
+		coeffs[i], err = new(fr.Element).SetRandom()
+		if err != nil {
+			panic(err)
+		}
+	}
+
+	return &Poly{ballot, coeffs, pinput, nil}
+}
 
-	// number of bits per coefficient
-	numBits := uint(math.Ceil(float64(entropy) / float64(degree)))
+func (p *Poly) setupCache() {
+	p.cache = &polyCache{}
 
-	// number of bytes per coefficient
-	numBytes := numBits / 8
-	if numBits%8 > 0 {
-		numBytes += 1
+	// lay out ballot bits slice and compute constant term
+	n := len(p.Ballot)
+	ballotBits := make([]*fr.Element, n*n)
+	ballot1DBytes := make([]byte, 0, n*n)
+	for i, row := range p.Ballot {
+		ballot1DBytes = append(ballot1DBytes, row...)
+		for j, bool := range row {
+			if bool == 0 {
+				ballotBits[i * n + j] = new(fr.Element).SetZero()
+			} else if bool == 1 {
+				ballotBits[i * n + j] = new(fr.Element).SetOne()
+			} else {
+				panic(fmt.Sprintf("invalid ballot byte value: %d", bool))
+			}
+		}
 	}
+	p.cache.ballotBits = ballotBits
+	p.cache.constant = new(fr.Element).SetBytes(ballot1DBytes)
 
-	var err error
-	for i := range coefficients {
-		coefficients[i], err = RandomBigInt(numBytes, false)
+	// malloc outputs map
+	p.cache.outputs = make(map[*fr.Element]outputCache)
+
+	// calculate verifier's output
+	poutput := new(fr.Element).Set(p.cache.constant)
+	term := new(fr.Element)
+	for i, coeff := range p.Coeffs {
+		term.Set(coeff)
+		// <= because we want there to be at least one multiplication
+		for j := 0; j <= i; j++ {
+			term.Mul(term, p.PInput)
+		}
+		poutput.Add(poutput, term)
+	}
+	p.cache.poutput = poutput
+
+}
+
+func (p *Poly) EvalAndProve(input *fr.Element, provingKey groth16.ProvingKey, hashSeed string) (*fr.Element, []byte, groth16.Proof) {
+	if p.cache == nil {
+		p.setupCache()
+	} else if outputCache, exists := p.cache.outputs[input]; exists {
+		return outputCache.output, outputCache.poutputHash, outputCache.proof
+	}
+
+	// compile R1CS
+	var r1cs r1cs.R1CS
+	{
+		var circuit EvalCircuit
+		circuit.Coeffs = make([]frontend.Variable, len(p.Coeffs))
+		circuit.BallotBits = make([]frontend.Variable, len(p.Ballot)*len(p.Ballot[0]))
+		circuit.HashSeed = hashSeed
+		var err error
+		r1cs, err = frontend.Compile(gurvy.BLS381, &circuit)
 		if err != nil {
 			panic(err)
 		}
 	}
 
-	return NewPoly(coefficients, ballot)
-}
+	output := p.eval(input)
 
-func (p *Poly) Eval(input *big.Int) *big.Int {
-	res := new(big.Int).Set(p.Constant)
+	var witness EvalCircuit
+	// public
+	witness.Input.Assign(input)
+	witness.Output.Assign(output)
+	witness.PInput.Assign(p.PInput)
+	var poutputHash []byte
+	{
+		poutputBytes := p.cache.poutput.Bytes()
+		var err error
+		poutputHash, err = bls381.Sum(hashSeed, poutputBytes[:])
+		if err != nil {
+			panic(err)
+		}
+		witness.POutputHash.Assign(poutputHash)
+	}
+	witness.HashSeed = hashSeed
+	// private
+	witness.BallotBits = make([]frontend.Variable, len(p.Ballot)*len(p.Ballot[0]))
+	for i, bit := range p.cache.ballotBits {
+		witness.BallotBits[i].Assign(bit)
+	}
+	witness.Coeffs = make([]frontend.Variable, len(p.Coeffs))
+	for i, coeff := range p.Coeffs {
+		witness.Coeffs[i].Assign(coeff)
+	}
+	witness.Constant.Assign(p.cache.constant)
+	witness.POutput.Assign(p.cache.poutput)
 
-	for i, coef := range p.Coefs {
-		degree := big.NewInt(int64(i + 1))
-		term := new(big.Int).Exp(input, degree, nil)
-		term.Mul(term, coef)
-		res.Add(res, term)
+	proof, err := groth16.Prove(r1cs, provingKey, &witness)
+	if err != nil {
+		panic(err)
 	}
 
-	return res
+	p.cache.outputs[input] = outputCache{output, poutputHash, proof}
+	return output, poutputHash, proof
+}
+
+func (p *Poly) eval(input *fr.Element) *fr.Element {
+	if p.cache == nil {
+		p.setupCache()
+	}
+	output := new(fr.Element).Set(p.cache.constant)
+	term := new(fr.Element)
+	for i, coeff := range p.Coeffs {
+		term.Set(coeff)
+		// <= because we want there to be at least one multiplication
+		for j := 0; j <= i; j++ {
+			term.Mul(term, input)
+		}
+		output.Add(output, term)
+	}
+	return output
 }

@@ 0,0 1,117 @@
+package math
+
+import (
+	"errors"
+
+	"github.com/consensys/gnark/frontend"
+	"github.com/consensys/gnark/std/hash/mimc"
+	"github.com/consensys/gurvy"
+)
+
+type EvalCircuit struct {
+	Input       frontend.Variable `gnark:",public"`
+	Output      frontend.Variable `gnark:",public"`
+	// prover's input
+	PInput      frontend.Variable `gnark:",public"`
+	POutputHash frontend.Variable `gnark:",public"`
+	HashSeed    string
+
+	// must be created with length numCandidates^2
+	BallotBits []frontend.Variable `gnark:",private"`
+	// must be created with length numVoters - 1
+	Coeffs     []frontend.Variable `gnark:",private"`
+	Constant   frontend.Variable   `gnark:",private"`
+	// prover's hidden output for their own polynomial
+	POutput    frontend.Variable   `gnark:",private"`
+}
+
+func (circuit *EvalCircuit) Define(curveID gurvy.ID, cs *frontend.ConstraintSystem) error {
+	// prove the input yields the given output
+	output := circuit.Constant
+	poutput := circuit.Constant
+	for i, coeff := range circuit.Coeffs {
+		term := coeff
+		vterm := coeff
+		// <= because we want there to be at least one multiplication
+		for j := 0; j <= i; j++ {
+			term = cs.Mul(term, circuit.Input)
+			vterm = cs.Mul(vterm, circuit.PInput)
+		}
+		output = cs.Add(output, term)
+		poutput = cs.Add(poutput, vterm)
+	}
+	cs.AssertIsEqual(output, circuit.Output)
+	cs.AssertIsEqual(poutput, circuit.POutput)
+
+	// prove the constant is valid (i.e. a string of bits separated by 7
+	// zeros)
+	// TODO: support more than 255 voters (limit from usage of byte)
+	zero := cs.Constant(0)
+	shift := cs.Constant(1 << 8)
+	slot := cs.Constant(1)
+	constructedConstant := zero
+	for i := len(circuit.BallotBits)-1; i >= 0; i-- {
+		bit := circuit.BallotBits[i]
+		cs.AssertIsBoolean(bit)
+		constructedConstant = cs.Add(constructedConstant, cs.Select(bit, slot, zero))
+		slot = cs.Mul(slot, shift)
+	}
+	cs.AssertIsEqual(constructedConstant, circuit.Constant)
+
+	mimc, err := mimc.NewMiMC(circuit.HashSeed, gurvy.BLS381)
+	if err != nil {
+		return err
+	}
+
+	cs.AssertIsEqual(mimc.Hash(cs, circuit.POutput), circuit.POutputHash)
+
+	return nil
+}
+
+type SumCircuit struct {
+	HashSeeds    []string
+	HashSelects  []frontend.Variable `gnark:",public"`
+	OutputHashes []frontend.Variable `gnark:",public"`
+	Sum          frontend.Variable   `gnark:",public"`
+
+	Outputs      []frontend.Variable `gnark:",private"`
+}
+
+func (circuit *SumCircuit) Define(curveID gurvy.ID, cs *frontend.ConstraintSystem) error {
+	if len(circuit.HashSeeds) != len(circuit.HashSelects) {
+		return errors.New("len(circuit.HashSeeds) != len(circuit.HashSelects)")
+	}
+	if len(circuit.HashSelects) != len(circuit.OutputHashes) {
+		return errors.New("len(circuit.HashSelects) != len(circuit.OutputHashes)")
+	}
+	if len(circuit.OutputHashes) != len(circuit.Outputs) {
+		return errors.New("len(circuit.OutputHashes) != len(circuit.Outputs)")
+	}
+
+	// prove the secret outputs hash to public hashes
+	zero := cs.Constant(0)
+	// TODO: this seems verify inefficient
+	for i, hashSeed := range circuit.HashSeeds {
+		mimc, err := mimc.NewMiMC(hashSeed, gurvy.BLS381)
+		if err != nil {
+			return err
+		}
+		bit := circuit.HashSelects[i]
+		for j, output := range circuit.Outputs {
+			cs.AssertIsEqual(
+				cs.Select(bit, mimc.Hash(cs, output), zero),
+				cs.Select(bit, circuit.OutputHashes[j], zero),
+			)
+		}
+	}
+
+	// prove that sum of outputs yields Sum
+	sum := zero
+	for _, output := range circuit.Outputs {
+		sum = cs.Add(sum, output)
+	}
+
+	cs.AssertIsEqual(sum, circuit.Sum)
+
+	return nil
+}

@@ 232,8 232,10 @@ func CreateElectionTUI() (title string, candidates []election.Candidate) {
 	n := 2
 	app := newTallyardApplication()
 	plus := func() {
-		form.AddInputField(fmt.Sprintf("%d.", n+1), "", 50, nil, nil)
-		n++
+		if n < 5 {
+			form.AddInputField(fmt.Sprintf("%d.", n+1), "", 50, nil, nil)
+			n++
+		}
 	}
 	minus := func() {
 		// TODO: ensure from joiner that there are at least two


@@ 274,7 276,7 @@ func CreateElectionTUI() (title string, candidates []election.Candidate) {
 	return title, candidates
 }
 
-func ElectionWaitTUI(client *mautrix.Client, el *election.Election, eventStore *election.EventStore) error {
+func ElectionWaitTUI(client *mautrix.Client, el *election.Election, eventStore *election.EventStore) (err error) {
 	votersTextView := tview.NewTextView()
 	frame := tview.NewFrame(votersTextView)
 	frame.SetTitle(el.Title).SetBorder(true)


@@ 286,20 288,21 @@ func ElectionWaitTUI(client *mautrix.Client, el *election.Election, eventStore *
 			if event.Key() == tcell.KeyEnter {
 				frame.Clear()
 				frame.AddText("Starting election...", false, tview.AlignCenter, tcell.ColorWhite)
-				err := el.StartElection(client, eventStore)
-				if err != nil {
-					panic(err)
-				}
+				err = el.StartElection(client, eventStore)
 			}
 			return event
 		})
 	} else {
 		frame.AddText("Waiting for election to start...", false, tview.AlignCenter, tcell.ColorWhite)
 	}
+	if err != nil {
+		return
+	}
 	el.RUnlock()
 	update := func() {
 		el.RLock()
 		// TODO: handle duplicate joins from one UserID
+		// TODO: handle users who joined after cutoff
 		voters := make([]string, 0, len(el.Joins))
 		for _, voter := range el.Joins {
 			voters = append(voters, voter.JoinEvt.Sender.String())


@@ 311,28 314,59 @@ func ElectionWaitTUI(client *mautrix.Client, el *election.Election, eventStore *
 		votersTextView.SetText(text)
 	}
 	go func() {
+		var electionStarted bool
 		for app.alive {
-			app.QueueUpdateDraw(update)
-
-			// has the election started?
 			el.RLock()
-			started := el.StartID != nil
-			el.RUnlock()
-			if started {
-				app.Stop()
-				break
+			if electionStarted {
+				// have we received everyone's keys?
+				receivedAllKeys := true
+				for _, joinID := range *el.FinalJoinIDs {
+					if el.Joins[joinID].KeysID == nil {
+						receivedAllKeys = false
+						break
+					}
+				}
+				el.RUnlock()
+				if receivedAllKeys {
+					app.Stop()
+					return
+				}
+			} else if el.FinalJoinIDs != nil {
+				electionStarted = true
+				el.RUnlock()
+				if el.LocalVoter.KeysID == nil {
+					err = el.SendProvingKeys(client, eventStore)
+					if err != nil {
+						app.Stop()
+						return
+					}
+				}
+				app.QueueUpdateDraw(func() {
+					frame.Clear()
+					frame.AddText("Waiting for everyone's keys...", false, tview.AlignCenter, tcell.ColorWhite)
+					update()
+				})
+			} else {
+				el.RUnlock()
+				app.QueueUpdateDraw(update)
 			}
 
 			time.Sleep(1 * time.Second)
 		}
 	}()
 	app.SetRoot(frame, true)
-	err := app.Run()
-	return err
+	err2 := app.Run()
+	if err != nil {
+		return err
+	}
+	if err2 != nil {
+		return err2
+	}
+	return nil
 }
 
 // displays a voting UI to the user and returns the encoded ballot
-func VoteTUI(candidates []election.Candidate) (ballot *[]byte, err error) {
+func VoteTUI(candidates []election.Candidate) (ballot *[][]byte, err error) {
 	app := newTallyardApplication()
 	form := tview.NewForm()
 	form.SetTitle("Vote").SetBorder(true)


@@ 398,17 432,15 @@ func (t *tallyardApplication) Run() error {
 	return err
 }
 
-func getBallotFromRankings(ranks []int) []byte {
+func getBallotFromRankings(ranks []int) [][]byte {
 	n := len(ranks)
 	candidates := make([]int, n)
-
 	for i := 0; i < n; i++ {
 		candidates[i] = i
 	}
 
 	// sort candidates by ranking
-	cr := candidateRanking{candidates, ranks}
-	sort.Sort(&cr)
+	sort.Sort(&candidateRanking{candidates, ranks})
 
 	// TODO: support more than 255 voters (limit from usage of byte)
 	prefMatrix := make([][]byte, n)


@@ 431,13 463,7 @@ func getBallotFromRankings(ranks []int) []byte {
 		candidates = candidates[i:]
 	}
 
-	// convert 2D array into 1D array
-	barray := make([]byte, 0, n*n)
-	for _, row := range prefMatrix {
-		barray = append(barray, row...)
-	}
-
-	return barray
+	return prefMatrix
 }
 
 type candidateRanking struct {