htlcswitch: accept failure reason for intercepted htlcs

This commit is contained in:
Joost Jager 2022-01-18 15:17:05 +01:00
parent 60625b6c1a
commit e2b56f212f
No known key found for this signature in database
GPG Key ID: A61B9D4C393C59C7
8 changed files with 456 additions and 221 deletions

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@ -64,6 +64,11 @@
* [Add dev only RPC subserver and the devrpc.ImportGraph
call](https://github.com/lightningnetwork/lnd/pull/6149)
* [Extend](https://github.com/lightningnetwork/lnd/pull/6177) the HTLC
interceptor API to provide more control over failure messages. With this
change, it allows encrypted failure messages to be returned to the sender.
Additionally it is possible to signal a malformed htlc.
## Documentation

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@ -1,6 +1,7 @@
package htlcswitch
import (
"crypto/sha256"
"fmt"
"sync"
@ -15,6 +16,10 @@ var (
// ErrFwdNotExists is an error returned when the caller tries to resolve
// a forward that doesn't exist anymore.
ErrFwdNotExists = errors.New("forward does not exist")
// ErrUnsupportedFailureCode when processing of an unsupported failure
// code is attempted.
ErrUnsupportedFailureCode = errors.New("unsupported failure code")
)
// InterceptableSwitch is an implementation of ForwardingSwitch interface.
@ -137,21 +142,63 @@ func (f *interceptedForward) Resume() error {
return f.htlcSwitch.ForwardPackets(f.linkQuit, f.packet)
}
// Fail forward a failed packet to the switch.
func (f *interceptedForward) Fail() error {
update, err := f.htlcSwitch.cfg.FetchLastChannelUpdate(
f.packet.incomingChanID,
)
if err != nil {
return err
// Fail notifies the intention to fail an existing hold forward with an
// encrypted failure reason.
func (f *interceptedForward) Fail(reason []byte) error {
obfuscatedReason := f.packet.obfuscator.IntermediateEncrypt(reason)
return f.resolve(&lnwire.UpdateFailHTLC{
Reason: obfuscatedReason,
})
}
// FailWithCode notifies the intention to fail an existing hold forward with the
// specified failure code.
func (f *interceptedForward) FailWithCode(code lnwire.FailCode) error {
shaOnionBlob := func() [32]byte {
return sha256.Sum256(f.htlc.OnionBlob[:])
}
reason, err := f.packet.obfuscator.EncryptFirstHop(
lnwire.NewTemporaryChannelFailure(update),
)
// Create a local failure.
var failureMsg lnwire.FailureMessage
switch code {
case lnwire.CodeInvalidOnionVersion:
failureMsg = &lnwire.FailInvalidOnionVersion{
OnionSHA256: shaOnionBlob(),
}
case lnwire.CodeInvalidOnionHmac:
failureMsg = &lnwire.FailInvalidOnionHmac{
OnionSHA256: shaOnionBlob(),
}
case lnwire.CodeInvalidOnionKey:
failureMsg = &lnwire.FailInvalidOnionKey{
OnionSHA256: shaOnionBlob(),
}
case lnwire.CodeTemporaryChannelFailure:
update, err := f.htlcSwitch.cfg.FetchLastChannelUpdate(
f.packet.incomingChanID,
)
if err != nil {
return err
}
failureMsg = lnwire.NewTemporaryChannelFailure(update)
default:
return ErrUnsupportedFailureCode
}
// Encrypt the failure for the first hop. This node will be the origin
// of the failure.
reason, err := f.packet.obfuscator.EncryptFirstHop(failureMsg)
if err != nil {
return fmt.Errorf("failed to encrypt failure reason %v", err)
}
return f.resolve(&lnwire.UpdateFailHTLC{
Reason: reason,
})

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@ -297,8 +297,13 @@ type InterceptedForward interface {
// forward with a given preimage.
Settle(lntypes.Preimage) error
// Fails notifies the intention to fail an existing hold forward
Fail() error
// Fail notifies the intention to fail an existing hold forward with an
// encrypted failure reason.
Fail(reason []byte) error
// FailWithCode notifies the intention to fail an existing hold forward
// with the specified failure code.
FailWithCode(code lnwire.FailCode) error
}
// htlcNotifier is an interface which represents the input side of the

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@ -1,6 +1,7 @@
package htlcswitch
import (
"bytes"
"crypto/rand"
"crypto/sha256"
"fmt"
@ -18,6 +19,7 @@ import (
"github.com/lightningnetwork/lnd/lntypes"
"github.com/lightningnetwork/lnd/lnwire"
"github.com/lightningnetwork/lnd/ticker"
"github.com/stretchr/testify/assert"
"github.com/stretchr/testify/require"
)
@ -3142,7 +3144,8 @@ type mockForwardInterceptor struct {
intercepted InterceptedForward
}
func (m *mockForwardInterceptor) InterceptForwardHtlc(intercepted InterceptedForward) bool {
func (m *mockForwardInterceptor) InterceptForwardHtlc(
intercepted InterceptedForward) bool {
m.intercepted = intercepted
return true
@ -3152,8 +3155,14 @@ func (m *mockForwardInterceptor) settle(preimage lntypes.Preimage) error {
return m.intercepted.Settle(preimage)
}
func (m *mockForwardInterceptor) fail() error {
return m.intercepted.Fail()
func (m *mockForwardInterceptor) fail(reason []byte) error {
return m.intercepted.Fail(reason)
}
func (m *mockForwardInterceptor) failWithCode(
code lnwire.FailCode) error {
return m.intercepted.FailWithCode(code)
}
func (m *mockForwardInterceptor) resume() error {
@ -3170,7 +3179,7 @@ func assertNumCircuits(t *testing.T, s *Switch, pending, opened int) {
}
func assertOutgoingLinkReceive(t *testing.T, targetLink *mockChannelLink,
expectReceive bool) {
expectReceive bool) *htlcPacket {
// Pull packet from targetLink link.
select {
@ -3181,11 +3190,15 @@ func assertOutgoingLinkReceive(t *testing.T, targetLink *mockChannelLink,
t.Fatalf("unable to complete payment circuit: %v", err)
}
return packet
case <-time.After(time.Second):
if expectReceive {
t.Fatal("request was not propagated to destination")
}
}
return nil
}
func TestSwitchHoldForward(t *testing.T) {
@ -3247,6 +3260,7 @@ func TestSwitchHoldForward(t *testing.T) {
// bob channel link.
preimage := [sha256.Size]byte{1}
rhash := sha256.Sum256(preimage[:])
onionBlob := [1366]byte{4, 5, 6}
ogPacket := &htlcPacket{
incomingChanID: aliceChannelLink.ShortChanID(),
incomingHTLCID: 0,
@ -3255,6 +3269,7 @@ func TestSwitchHoldForward(t *testing.T) {
htlc: &lnwire.UpdateAddHTLC{
PaymentHash: rhash,
Amount: 1,
OnionBlob: onionBlob,
},
}
@ -3299,13 +3314,55 @@ func TestSwitchHoldForward(t *testing.T) {
assertNumCircuits(t, s, 0, 0)
assertOutgoingLinkReceive(t, bobChannelLink, false)
if err := forwardInterceptor.fail(); err != nil {
if err := forwardInterceptor.fail(nil); err != nil {
t.Fatalf("failed to cancel forward %v", err)
}
assertOutgoingLinkReceive(t, bobChannelLink, false)
assertOutgoingLinkReceive(t, aliceChannelLink, true)
assertNumCircuits(t, s, 0, 0)
// Test failing a hold forward with a failure message.
require.NoError(t,
switchForwardInterceptor.ForwardPackets(linkQuit, ogPacket),
)
assertNumCircuits(t, s, 0, 0)
assertOutgoingLinkReceive(t, bobChannelLink, false)
reason := lnwire.OpaqueReason([]byte{1, 2, 3})
require.NoError(t, forwardInterceptor.fail(reason))
assertOutgoingLinkReceive(t, bobChannelLink, false)
packet := assertOutgoingLinkReceive(t, aliceChannelLink, true)
require.Equal(t, reason, packet.htlc.(*lnwire.UpdateFailHTLC).Reason)
assertNumCircuits(t, s, 0, 0)
// Test failing a hold forward with a malformed htlc failure.
err = switchForwardInterceptor.ForwardPackets(linkQuit, ogPacket)
require.NoError(t, err)
assertNumCircuits(t, s, 0, 0)
assertOutgoingLinkReceive(t, bobChannelLink, false)
code := lnwire.CodeInvalidOnionKey
require.NoError(t, forwardInterceptor.failWithCode(code))
assertOutgoingLinkReceive(t, bobChannelLink, false)
packet = assertOutgoingLinkReceive(t, aliceChannelLink, true)
failPacket := packet.htlc.(*lnwire.UpdateFailHTLC)
shaOnionBlob := sha256.Sum256(onionBlob[:])
expectedFailure := &lnwire.FailInvalidOnionKey{
OnionSHA256: shaOnionBlob,
}
var b bytes.Buffer
require.NoError(t, lnwire.EncodeFailure(&b, expectedFailure, 0))
assert.Equal(t, lnwire.OpaqueReason(b.Bytes()), failPacket.Reason)
assertNumCircuits(t, s, 0, 0)
// Test settling a hold forward
if err := switchForwardInterceptor.ForwardPackets(linkQuit, ogPacket); err != nil {
t.Fatalf("can't forward htlc packet: %v", err)

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@ -2,13 +2,15 @@ package routerrpc
import (
"errors"
"fmt"
"sync"
"github.com/lightningnetwork/lnd/channeldb"
"github.com/lightningnetwork/lnd/htlcswitch"
"github.com/lightningnetwork/lnd/lnrpc"
"github.com/lightningnetwork/lnd/lntypes"
"github.com/lightningnetwork/lnd/lnwire"
"google.golang.org/grpc/codes"
"google.golang.org/grpc/status"
)
var (
@ -193,8 +195,64 @@ func (r *forwardInterceptor) resolveFromClient(
switch in.Action {
case ResolveHoldForwardAction_RESUME:
return interceptedForward.Resume()
case ResolveHoldForwardAction_FAIL:
return interceptedForward.Fail()
// Fail with an encrypted reason.
if in.FailureMessage != nil {
if in.FailureCode != 0 {
return status.Errorf(
codes.InvalidArgument,
"failure message and failure code "+
"are mutually exclusive",
)
}
// Verify that the size is equal to the fixed failure
// message size + hmac + two uint16 lengths. See BOLT
// #4.
if len(in.FailureMessage) !=
lnwire.FailureMessageLength+32+2+2 {
return status.Errorf(
codes.InvalidArgument,
"failure message length invalid",
)
}
return interceptedForward.Fail(in.FailureMessage)
}
var code lnwire.FailCode
switch in.FailureCode {
case lnrpc.Failure_INVALID_ONION_HMAC:
code = lnwire.CodeInvalidOnionHmac
case lnrpc.Failure_INVALID_ONION_KEY:
code = lnwire.CodeInvalidOnionKey
case lnrpc.Failure_INVALID_ONION_VERSION:
code = lnwire.CodeInvalidOnionVersion
// Default to TemporaryChannelFailure.
case 0, lnrpc.Failure_TEMPORARY_CHANNEL_FAILURE:
code = lnwire.CodeTemporaryChannelFailure
default:
return status.Errorf(
codes.InvalidArgument,
"unsupported failure code: %v", in.FailureCode,
)
}
err := interceptedForward.FailWithCode(code)
if err == htlcswitch.ErrUnsupportedFailureCode {
return status.Errorf(
codes.InvalidArgument, err.Error(),
)
}
return err
case ResolveHoldForwardAction_SETTLE:
if in.Preimage == nil {
return ErrMissingPreimage
@ -204,8 +262,12 @@ func (r *forwardInterceptor) resolveFromClient(
return err
}
return interceptedForward.Settle(preimage)
default:
return fmt.Errorf("unrecognized resolve action %v", in.Action)
return status.Errorf(
codes.InvalidArgument,
"unrecognized resolve action %v", in.Action,
)
}
}

View File

@ -2690,6 +2690,19 @@ type ForwardHtlcInterceptResponse struct {
Action ResolveHoldForwardAction `protobuf:"varint,2,opt,name=action,proto3,enum=routerrpc.ResolveHoldForwardAction" json:"action,omitempty"`
// The preimage in case the resolve action is Settle.
Preimage []byte `protobuf:"bytes,3,opt,name=preimage,proto3" json:"preimage,omitempty"`
// Encrypted failure message in case the resolve action is Fail.
//
// If failure_message is specified, the failure_code field must be set
// to zero.
FailureMessage []byte `protobuf:"bytes,4,opt,name=failure_message,json=failureMessage,proto3" json:"failure_message,omitempty"`
// Return the specified failure code in case the resolve action is Fail. The
// message data fields are populated automatically.
//
// If a non-zero failure_code is specified, failure_message must not be set.
//
// For backwards-compatibility reasons, TEMPORARY_CHANNEL_FAILURE is the
// default value for this field.
FailureCode lnrpc.Failure_FailureCode `protobuf:"varint,5,opt,name=failure_code,json=failureCode,proto3,enum=lnrpc.Failure_FailureCode" json:"failure_code,omitempty"`
}
func (x *ForwardHtlcInterceptResponse) Reset() {
@ -2745,6 +2758,20 @@ func (x *ForwardHtlcInterceptResponse) GetPreimage() []byte {
return nil
}
func (x *ForwardHtlcInterceptResponse) GetFailureMessage() []byte {
if x != nil {
return x.FailureMessage
}
return nil
}
func (x *ForwardHtlcInterceptResponse) GetFailureCode() lnrpc.Failure_FailureCode {
if x != nil {
return x.FailureCode
}
return lnrpc.Failure_RESERVED
}
type UpdateChanStatusRequest struct {
state protoimpl.MessageState
sizeCache protoimpl.SizeCache
@ -3163,7 +3190,7 @@ var file_routerrpc_router_proto_rawDesc = []byte{
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@ -3175,166 +3202,173 @@ var file_routerrpc_router_proto_rawDesc = []byte{
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}
var (
@ -3432,47 +3466,48 @@ var file_routerrpc_router_proto_depIdxs = []int32{
42, // 24: routerrpc.ForwardHtlcInterceptRequest.custom_records:type_name -> routerrpc.ForwardHtlcInterceptRequest.CustomRecordsEntry
36, // 25: routerrpc.ForwardHtlcInterceptResponse.incoming_circuit_key:type_name -> routerrpc.CircuitKey
2, // 26: routerrpc.ForwardHtlcInterceptResponse.action:type_name -> routerrpc.ResolveHoldForwardAction
49, // 27: routerrpc.UpdateChanStatusRequest.chan_point:type_name -> lnrpc.ChannelPoint
3, // 28: routerrpc.UpdateChanStatusRequest.action:type_name -> routerrpc.ChanStatusAction
5, // 29: routerrpc.Router.SendPaymentV2:input_type -> routerrpc.SendPaymentRequest
6, // 30: routerrpc.Router.TrackPaymentV2:input_type -> routerrpc.TrackPaymentRequest
7, // 31: routerrpc.Router.EstimateRouteFee:input_type -> routerrpc.RouteFeeRequest
9, // 32: routerrpc.Router.SendToRoute:input_type -> routerrpc.SendToRouteRequest
9, // 33: routerrpc.Router.SendToRouteV2:input_type -> routerrpc.SendToRouteRequest
11, // 34: routerrpc.Router.ResetMissionControl:input_type -> routerrpc.ResetMissionControlRequest
13, // 35: routerrpc.Router.QueryMissionControl:input_type -> routerrpc.QueryMissionControlRequest
15, // 36: routerrpc.Router.XImportMissionControl:input_type -> routerrpc.XImportMissionControlRequest
19, // 37: routerrpc.Router.GetMissionControlConfig:input_type -> routerrpc.GetMissionControlConfigRequest
21, // 38: routerrpc.Router.SetMissionControlConfig:input_type -> routerrpc.SetMissionControlConfigRequest
24, // 39: routerrpc.Router.QueryProbability:input_type -> routerrpc.QueryProbabilityRequest
26, // 40: routerrpc.Router.BuildRoute:input_type -> routerrpc.BuildRouteRequest
28, // 41: routerrpc.Router.SubscribeHtlcEvents:input_type -> routerrpc.SubscribeHtlcEventsRequest
5, // 42: routerrpc.Router.SendPayment:input_type -> routerrpc.SendPaymentRequest
6, // 43: routerrpc.Router.TrackPayment:input_type -> routerrpc.TrackPaymentRequest
38, // 44: routerrpc.Router.HtlcInterceptor:input_type -> routerrpc.ForwardHtlcInterceptResponse
39, // 45: routerrpc.Router.UpdateChanStatus:input_type -> routerrpc.UpdateChanStatusRequest
50, // 46: routerrpc.Router.SendPaymentV2:output_type -> lnrpc.Payment
50, // 47: routerrpc.Router.TrackPaymentV2:output_type -> lnrpc.Payment
8, // 48: routerrpc.Router.EstimateRouteFee:output_type -> routerrpc.RouteFeeResponse
10, // 49: routerrpc.Router.SendToRoute:output_type -> routerrpc.SendToRouteResponse
48, // 50: routerrpc.Router.SendToRouteV2:output_type -> lnrpc.HTLCAttempt
12, // 51: routerrpc.Router.ResetMissionControl:output_type -> routerrpc.ResetMissionControlResponse
14, // 52: routerrpc.Router.QueryMissionControl:output_type -> routerrpc.QueryMissionControlResponse
16, // 53: routerrpc.Router.XImportMissionControl:output_type -> routerrpc.XImportMissionControlResponse
20, // 54: routerrpc.Router.GetMissionControlConfig:output_type -> routerrpc.GetMissionControlConfigResponse
22, // 55: routerrpc.Router.SetMissionControlConfig:output_type -> routerrpc.SetMissionControlConfigResponse
25, // 56: routerrpc.Router.QueryProbability:output_type -> routerrpc.QueryProbabilityResponse
27, // 57: routerrpc.Router.BuildRoute:output_type -> routerrpc.BuildRouteResponse
29, // 58: routerrpc.Router.SubscribeHtlcEvents:output_type -> routerrpc.HtlcEvent
35, // 59: routerrpc.Router.SendPayment:output_type -> routerrpc.PaymentStatus
35, // 60: routerrpc.Router.TrackPayment:output_type -> routerrpc.PaymentStatus
37, // 61: routerrpc.Router.HtlcInterceptor:output_type -> routerrpc.ForwardHtlcInterceptRequest
40, // 62: routerrpc.Router.UpdateChanStatus:output_type -> routerrpc.UpdateChanStatusResponse
46, // [46:63] is the sub-list for method output_type
29, // [29:46] is the sub-list for method input_type
29, // [29:29] is the sub-list for extension type_name
29, // [29:29] is the sub-list for extension extendee
0, // [0:29] is the sub-list for field type_name
47, // 27: routerrpc.ForwardHtlcInterceptResponse.failure_code:type_name -> lnrpc.Failure.FailureCode
49, // 28: routerrpc.UpdateChanStatusRequest.chan_point:type_name -> lnrpc.ChannelPoint
3, // 29: routerrpc.UpdateChanStatusRequest.action:type_name -> routerrpc.ChanStatusAction
5, // 30: routerrpc.Router.SendPaymentV2:input_type -> routerrpc.SendPaymentRequest
6, // 31: routerrpc.Router.TrackPaymentV2:input_type -> routerrpc.TrackPaymentRequest
7, // 32: routerrpc.Router.EstimateRouteFee:input_type -> routerrpc.RouteFeeRequest
9, // 33: routerrpc.Router.SendToRoute:input_type -> routerrpc.SendToRouteRequest
9, // 34: routerrpc.Router.SendToRouteV2:input_type -> routerrpc.SendToRouteRequest
11, // 35: routerrpc.Router.ResetMissionControl:input_type -> routerrpc.ResetMissionControlRequest
13, // 36: routerrpc.Router.QueryMissionControl:input_type -> routerrpc.QueryMissionControlRequest
15, // 37: routerrpc.Router.XImportMissionControl:input_type -> routerrpc.XImportMissionControlRequest
19, // 38: routerrpc.Router.GetMissionControlConfig:input_type -> routerrpc.GetMissionControlConfigRequest
21, // 39: routerrpc.Router.SetMissionControlConfig:input_type -> routerrpc.SetMissionControlConfigRequest
24, // 40: routerrpc.Router.QueryProbability:input_type -> routerrpc.QueryProbabilityRequest
26, // 41: routerrpc.Router.BuildRoute:input_type -> routerrpc.BuildRouteRequest
28, // 42: routerrpc.Router.SubscribeHtlcEvents:input_type -> routerrpc.SubscribeHtlcEventsRequest
5, // 43: routerrpc.Router.SendPayment:input_type -> routerrpc.SendPaymentRequest
6, // 44: routerrpc.Router.TrackPayment:input_type -> routerrpc.TrackPaymentRequest
38, // 45: routerrpc.Router.HtlcInterceptor:input_type -> routerrpc.ForwardHtlcInterceptResponse
39, // 46: routerrpc.Router.UpdateChanStatus:input_type -> routerrpc.UpdateChanStatusRequest
50, // 47: routerrpc.Router.SendPaymentV2:output_type -> lnrpc.Payment
50, // 48: routerrpc.Router.TrackPaymentV2:output_type -> lnrpc.Payment
8, // 49: routerrpc.Router.EstimateRouteFee:output_type -> routerrpc.RouteFeeResponse
10, // 50: routerrpc.Router.SendToRoute:output_type -> routerrpc.SendToRouteResponse
48, // 51: routerrpc.Router.SendToRouteV2:output_type -> lnrpc.HTLCAttempt
12, // 52: routerrpc.Router.ResetMissionControl:output_type -> routerrpc.ResetMissionControlResponse
14, // 53: routerrpc.Router.QueryMissionControl:output_type -> routerrpc.QueryMissionControlResponse
16, // 54: routerrpc.Router.XImportMissionControl:output_type -> routerrpc.XImportMissionControlResponse
20, // 55: routerrpc.Router.GetMissionControlConfig:output_type -> routerrpc.GetMissionControlConfigResponse
22, // 56: routerrpc.Router.SetMissionControlConfig:output_type -> routerrpc.SetMissionControlConfigResponse
25, // 57: routerrpc.Router.QueryProbability:output_type -> routerrpc.QueryProbabilityResponse
27, // 58: routerrpc.Router.BuildRoute:output_type -> routerrpc.BuildRouteResponse
29, // 59: routerrpc.Router.SubscribeHtlcEvents:output_type -> routerrpc.HtlcEvent
35, // 60: routerrpc.Router.SendPayment:output_type -> routerrpc.PaymentStatus
35, // 61: routerrpc.Router.TrackPayment:output_type -> routerrpc.PaymentStatus
37, // 62: routerrpc.Router.HtlcInterceptor:output_type -> routerrpc.ForwardHtlcInterceptRequest
40, // 63: routerrpc.Router.UpdateChanStatus:output_type -> routerrpc.UpdateChanStatusResponse
47, // [47:64] is the sub-list for method output_type
30, // [30:47] is the sub-list for method input_type
30, // [30:30] is the sub-list for extension type_name
30, // [30:30] is the sub-list for extension extendee
0, // [0:30] is the sub-list for field type_name
}
func init() { file_routerrpc_router_proto_init() }

View File

@ -781,6 +781,21 @@ message ForwardHtlcInterceptResponse {
// The preimage in case the resolve action is Settle.
bytes preimage = 3;
// Encrypted failure message in case the resolve action is Fail.
//
// If failure_message is specified, the failure_code field must be set
// to zero.
bytes failure_message = 4;
// Return the specified failure code in case the resolve action is Fail. The
// message data fields are populated automatically.
//
// If a non-zero failure_code is specified, failure_message must not be set.
//
// For backwards-compatibility reasons, TEMPORARY_CHANNEL_FAILURE is the
// default value for this field.
lnrpc.Failure.FailureCode failure_code = 5;
}
enum ResolveHoldForwardAction {

View File

@ -1201,6 +1201,15 @@
"type": "string",
"format": "byte",
"description": "The preimage in case the resolve action is Settle."
},
"failure_message": {
"type": "string",
"format": "byte",
"description": "Encrypted failure message in case the resolve action is Fail.\n\nIf failure_message is specified, the failure_code field must be set\nto zero."
},
"failure_code": {
"$ref": "#/definitions/FailureFailureCode",
"description": "Return the specified failure code in case the resolve action is Fail. The\nmessage data fields are populated automatically.\n\nIf a non-zero failure_code is specified, failure_message must not be set.\n\nFor backwards-compatibility reasons, TEMPORARY_CHANNEL_FAILURE is the\ndefault value for this field."
}
},
"description": "*\nForwardHtlcInterceptResponse enables the caller to resolve a previously hold\nforward. The caller can choose either to:\n- `Resume`: Execute the default behavior (usually forward).\n- `Reject`: Fail the htlc backwards.\n- `Settle`: Settle this htlc with a given preimage."