FROST Trusted Dealer

.github/workflows/ci.yml

@@ -41,6 +41,7 @@ env:
   ECDSAADAPTOR: 'no'
   BPPP: 'no'
   SCHNORRSIG_HALFAGG: 'no'
+  FROST: 'no'
   ### test options
   SECP256K1_TEST_ITERS:
   BENCH: 'yes'
@@ -79,14 +80,14 @@ jobs:
       matrix:
         configuration:
           - env_vars: { WIDEMUL: 'int64',  RECOVERY: 'yes' }
-          - env_vars: { WIDEMUL: 'int64',                   ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes',  RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes'}
+          - env_vars: { WIDEMUL: 'int64',                   ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes',  RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes' }
           - env_vars: { WIDEMUL: 'int128' }
           - env_vars: { WIDEMUL: 'int128_struct',                                           ELLSWIFT: 'yes' }
           - env_vars: { WIDEMUL: 'int128', RECOVERY: 'yes',              SCHNORRSIG: 'yes', ELLSWIFT: 'yes' }
-          - env_vars: { WIDEMUL: 'int128',                  ECDH: 'yes', SCHNORRSIG: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes'}
+          - env_vars: { WIDEMUL: 'int128',                  ECDH: 'yes', SCHNORRSIG: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes' }
           - env_vars: { WIDEMUL: 'int128', ASM: 'x86_64',                                   ELLSWIFT: 'yes' }
-          - env_vars: {                    RECOVERY: 'yes',              SCHNORRSIG: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes'}
-          - env_vars: { CTIMETESTS: 'no',  RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', CPPFLAGS: '-DVERIFY' }
+          - env_vars: {                    RECOVERY: 'yes',              SCHNORRSIG: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes' }
+          - env_vars: { CTIMETESTS: 'no',  RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes', CPPFLAGS: '-DVERIFY' }
           - env_vars: { BUILD: 'distcheck', WITH_VALGRIND: 'no', CTIMETESTS: 'no', BENCH: 'no' }
           - env_vars: { CPPFLAGS: '-DDETERMINISTIC' }
           - env_vars: { CFLAGS: '-O0', CTIMETESTS: 'no' }
@@ -158,6 +159,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CC: ${{ matrix.cc }}
 
     steps:
@@ -211,6 +213,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
 
     steps:
@@ -271,6 +274,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
 
     steps:
@@ -325,6 +329,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
 
     strategy:
@@ -389,6 +394,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
 
     steps:
@@ -450,6 +456,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
       SECP256K1_TEST_ITERS: 2
 
@@ -510,6 +517,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
       CFLAGS: '-fsanitize=undefined,address -g'
       UBSAN_OPTIONS: 'print_stacktrace=1:halt_on_error=1'
@@ -576,6 +584,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'yes'
       CC: 'clang'
       SECP256K1_TEST_ITERS: 32
@@ -632,6 +641,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
       CTIMETESTS: 'no'
 
     strategy:
@@ -688,15 +698,15 @@ jobs:
       fail-fast: false
       matrix:
         env_vars:
-          - { WIDEMUL: 'int64',  RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes' }
+          - { WIDEMUL: 'int64',  RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes' }
           - { WIDEMUL: 'int128_struct', ECMULTGENPRECISION: 2, ECMULTWINDOW: 4 }
-          - { WIDEMUL: 'int128',                  ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes' }
+          - { WIDEMUL: 'int128',                  ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes' }
           - { WIDEMUL: 'int128', RECOVERY: 'yes' }
-          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes' }
-          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', CC: 'gcc' }
-          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes',           WRAPPER_CMD: 'valgrind --error-exitcode=42', SECP256K1_TEST_ITERS: 2 }
-          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', CC: 'gcc', WRAPPER_CMD: 'valgrind --error-exitcode=42', SECP256K1_TEST_ITERS: 2 }
-          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', CPPFLAGS: '-DVERIFY', CTIMETESTS: 'no' }
+          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes' }
+          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', CC: 'gcc', FROST: 'yes' }
+          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes', WRAPPER_CMD: 'valgrind --error-exitcode=42', SECP256K1_TEST_ITERS: 2 }
+          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes', CC: 'gcc', WRAPPER_CMD: 'valgrind --error-exitcode=42', SECP256K1_TEST_ITERS: 2 }
+          - { WIDEMUL: 'int128', RECOVERY: 'yes', ECDH: 'yes', SCHNORRSIG: 'yes', ELLSWIFT: 'yes', EXPERIMENTAL: 'yes', ECDSA_S2C: 'yes', RANGEPROOF: 'yes', WHITELIST: 'yes', GENERATOR: 'yes', MUSIG: 'yes', ECDSAADAPTOR: 'yes', BPPP: 'yes', SCHNORRSIG_HALFAGG: 'yes', FROST: 'yes', CPPFLAGS: '-DVERIFY', CTIMETESTS: 'no' }
           - BUILD: 'distcheck'
 
     steps:
@@ -816,6 +826,7 @@ jobs:
       ECDSAADAPTOR: 'yes'
       BPPP: 'yes'
       SCHNORRSIG_HALFAGG: 'yes'
+      FROST: 'yes'
 
     steps:
       - name: Checkout

.gitignore

@@ -66,6 +66,7 @@ libsecp256k1.pc
 contrib/gh-pr-create.sh
 
 musig_example
+frost_example
 
 ### CMake
 /CMakeUserPresets.json

CMakeLists.txt

@@ -69,9 +69,18 @@ option(SECP256K1_ENABLE_MODULE_MUSIG "Enable MuSig module." ON)
 option(SECP256K1_ENABLE_MODULE_ECDSA_ADAPTOR "Enable ecdsa adaptor signatures module." ON)
 option(SECP256K1_ENABLE_MODULE_ECDSA_S2C "Enable ECDSA sign-to-contract module." ON)
 option(SECP256K1_ENABLE_MODULE_BPPP "Enable Bulletproofs++ module." ON)
+option(SECP256K1_ENABLE_MODULE_FROST "Enable FROST module." ON)
 
 # Processing must be done in a topological sorting of the dependency graph
 # (dependent module first).
+if(SECP256K1_ENABLE_MODULE_FROST)
+  if(DEFINED SECP256K1_ENABLE_MODULE_SCHNORRSIG AND NOT SECP256K1_ENABLE_MODULE_SCHNORRSIG)
+    message(FATAL_ERROR "Module dependency error: You have disabled the schnorrsig module explicitly, but it is required by the frost module.")
+  endif()
+  set(SECP256K1_ENABLE_MODULE_SCHNORRSIG ON)
+  add_compile_definitions(ENABLE_MODULE_FROST=1)
+endif()
+
 if(SECP256K1_ENABLE_MODULE_BPPP)
   if(DEFINED SECP256K1_ENABLE_MODULE_GENERATOR AND NOT SECP256K1_ENABLE_MODULE_GENERATOR)
     message(FATAL_ERROR "Module dependency error: You have disabled the generator module explicitly, but it is required by the bppp module.")
@@ -362,6 +371,7 @@ message("  musig ............................... ${SECP256K1_ENABLE_MODULE_MUSIG
 message("  ecdsa-s2c ........................... ${SECP256K1_ENABLE_MODULE_ECDSA_S2C}")
 message("  ecdsa-adaptor ....................... ${SECP256K1_ENABLE_MODULE_ECDSA_ADAPTOR}")
 message("  bppp ................................ ${SECP256K1_ENABLE_MODULE_BPPP}")
+message("  frost ............................... ${SECP256K1_ENABLE_MODULE_FROST}")
 message("Parameters:")
 message("  ecmult window size .................. ${SECP256K1_ECMULT_WINDOW_SIZE}")
 message("  ecmult gen precision bits ........... ${SECP256K1_ECMULT_GEN_PREC_BITS}")

Makefile.am

@@ -195,6 +195,17 @@ musig_example_LDFLAGS += -lbcrypt
 endif
 TESTS += musig_example
 endif
+if ENABLE_MODULE_FROST
+noinst_PROGRAMS += frost_example
+frost_example_SOURCES = examples/frost.c
+frost_example_CPPFLAGS = -I$(top_srcdir)/include -DSECP256K1_STATIC
+frost_example_LDADD = libsecp256k1.la
+frost_example_LDFLAGS = -static
+if BUILD_WINDOWS
+frost_example_LDFLAGS += -lbcrypt
+endif
+TESTS += frost_example
+endif
 endif
 
 ### Precomputed tables
@@ -320,3 +331,7 @@ endif
 if ENABLE_MODULE_ECDSA_ADAPTOR
 include src/modules/ecdsa_adaptor/Makefile.am.include
 endif
+
+if ENABLE_MODULE_FROST
+include src/modules/frost/Makefile.am.include
+endif

README.md

@@ -12,6 +12,7 @@ Added features:
 * Experimental module for Confidential Assets (Pedersen commitments, range proofs, and [surjection proofs](src/modules/surjection/surjection.md)).
 * Experimental module for Bulletproofs++ range proofs.
 * Experimental module for [address whitelisting](src/modules/whitelist/whitelist.md).
+* Experimental module for [FROST](src/modules/frost/frost.md).
 
 Experimental features are made available for testing and review by the community. The APIs of these features should not be considered stable.
 

ci/ci.sh

@@ -15,7 +15,7 @@ print_environment() {
             ECMULTWINDOW ECMULTGENPRECISION ASM WIDEMUL WITH_VALGRIND EXTRAFLAGS \
             EXPERIMENTAL ECDH RECOVERY SCHNORRSIG SCHNORRSIG_HALFAGG ELLSWIFT \
             ECDSA_S2C GENERATOR RANGEPROOF WHITELIST MUSIG ECDSAADAPTOR BPPP \
-            SECP256K1_TEST_ITERS BENCH SECP256K1_BENCH_ITERS CTIMETESTS\
+            FROST SECP256K1_TEST_ITERS BENCH SECP256K1_BENCH_ITERS CTIMETESTS\
             EXAMPLES \
             HOST WRAPPER_CMD \
             CC CFLAGS CPPFLAGS AR NM
@@ -83,6 +83,7 @@ esac
     --enable-module-schnorrsig="$SCHNORRSIG"  --enable-module-musig="$MUSIG" --enable-module-ecdsa-adaptor="$ECDSAADAPTOR" \
     --enable-module-schnorrsig="$SCHNORRSIG" \
     --enable-module-schnorrsig-halfagg="$SCHNORRSIG_HALFAGG" \
+    --enable-module-frost="$FROST" \
     --enable-examples="$EXAMPLES" \
     --enable-ctime-tests="$CTIMETESTS" \
     --with-valgrind="$WITH_VALGRIND" \

configure.ac

@@ -240,6 +240,11 @@ AC_ARG_ENABLE(external_default_callbacks,
     AS_HELP_STRING([--enable-external-default-callbacks],[enable external default callback functions [default=no]]), [],
     [SECP_SET_DEFAULT([enable_external_default_callbacks], [no], [no])])
 
+AC_ARG_ENABLE(module_frost,
+    AS_HELP_STRING([--enable-module-frost],[enable FROST module (experimental)]),
+    [],
+    [SECP_SET_DEFAULT([enable_module_frost], [no], [yes])])
+
 # Test-only override of the (autodetected by the C code) "widemul" setting.
 # Legal values are:
 #  * int64 (for [u]int64_t),
@@ -530,6 +535,14 @@ if test x"$enable_module_ecdh" = x"yes"; then
   SECP_CONFIG_DEFINES="$SECP_CONFIG_DEFINES -DENABLE_MODULE_ECDH=1"
 fi
 
+if test x"$enable_module_frost" = x"yes"; then
+  if test x"$enable_module_schnorrsig" = x"no"; then
+    AC_MSG_ERROR([Module dependency error: You have disabled the schnorrsig module explicitly, but it is required by the frost module.])
+  fi
+  SECP_CONFIG_DEFINES="$SECP_CONFIG_DEFINES -DENABLE_MODULE_FROST=1"
+  enable_module_schnorrsig=yes
+fi
+
 if test x"$enable_external_default_callbacks" = x"yes"; then
   SECP_CONFIG_DEFINES="$SECP_CONFIG_DEFINES -DUSE_EXTERNAL_DEFAULT_CALLBACKS=1"
 fi
@@ -582,6 +595,9 @@ else
   if test x"$set_asm" = x"arm32"; then
     AC_MSG_ERROR([ARM32 assembly is experimental. Use --enable-experimental to allow.])
   fi
+  if test x"$enable_module_frost" = x"yes"; then
+    AC_MSG_ERROR([FROST module is experimental. Use --enable-experimental to allow.])
+  fi
 fi
 
 ###
@@ -611,6 +627,7 @@ AM_CONDITIONAL([ENABLE_MODULE_ECDSA_S2C], [test x"$enable_module_ecdsa_s2c" = x"
 AM_CONDITIONAL([ENABLE_MODULE_ECDSA_ADAPTOR], [test x"$enable_module_ecdsa_adaptor" = x"yes"])
 AM_CONDITIONAL([ENABLE_MODULE_BPPP], [test x"$enable_module_bppp" = x"yes"])
 AM_CONDITIONAL([ENABLE_MODULE_SCHNORRSIG_HALFAGG], [test x"$enable_module_schnorrsig_halfagg" = x"yes"])
+AM_CONDITIONAL([ENABLE_MODULE_FROST], [test x"$enable_module_frost" = x"yes"])
 AM_CONDITIONAL([USE_REDUCED_SURJECTION_PROOF_SIZE], [test x"$use_reduced_surjection_proof_size" = x"yes"])
 AM_CONDITIONAL([USE_EXTERNAL_ASM], [test x"$enable_external_asm" = x"yes"])
 AM_CONDITIONAL([USE_ASM_ARM], [test x"$set_asm" = x"arm32"])
@@ -651,6 +668,7 @@ echo "  module ecdsa-s2c        = $enable_module_ecdsa_s2c"
 echo "  module ecdsa-adaptor    = $enable_module_ecdsa_adaptor"
 echo "  module bppp             = $enable_module_bppp"
 echo "  module schnorrsig-halfagg = $enable_module_schnorrsig_halfagg"
+echo "  module frost            = $enable_module_frost"
 echo
 echo "  asm                       = $set_asm"
 echo "  ecmult window size        = $set_ecmult_window"