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This project is a C++ implementation of {t,n}-Threshold Signature Scheme

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Safeheron/multi-party-sig-cpp

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multi-party-sig-cpp

This code repository has been renamed to "multi-party-sig-cpp" from "multi-party-ecdsa-cpp".

This project is a C++ implementation of {t,n}-Threshold Signature Scheme. It supports ECDSA (elliptic curve digital signature algorithm) now. Ed25519, Schnorr and BLS will be supported soon.

Introduction

The library implements three different mpc protocols for threshold ECDSA now.

  • GG18[1]: The GG18 protocol implements a {t,n}-threshold signature scheme according to Gennaro & Goldfeder's paper in 2018.
    • Key Generation: Create a new ECDSA private key shared among all the given participants.
    • Key Refresh: Refresh the secret shares while keeping the public key the same.
    • Sign: Generates an ECDSA signature for specified message.
  • GG20[2]: The GG20 protocol is an optimization of GG18, which reduced the MPC protocol of signature generation to 7 rounds.
    • Key Generation: the same as GG18.
    • Key Refresh: the same as GG18.
    • Sign: Signing for using the secret shares to generate a signature.
  • CMP[3]: A optimized threshold ECDSA protocol proposed built on the Gennaro & Goldfeder and Lindell's protocols (GG18).
    • Minimal Key Generation: Create a new ECDSA private key shared among all the given participants. Note that only the key relevant to the asset is generated.
    • Auxiliary info and key Refresh: Refresh of the auxiliary info and the secret shares while keeping the public key the same.
    • Sign: Generates an ECDSA signature for specified message.

More details about the protocols can be found in the papers.

We define structured data for each round via Protobuf for efficient message transfer. The cryptographic primitives used in the mpc protocols are defined in our library safeheron-crypto-suites-cpp . You can follow the Prerequisites to acquire them.

Comparison of the protocols

Protocol Paper Scheme Number of Rounds
Gennaro, Goldfeder 18 clicked here {t - n} 9
Gennaro, Goldfeder 20 clicked here {t - n} 7
Canetti, Makriyannis, Peled clicked here {n - n} / {t - n} 4
  • {t - n} means that the threshold is t and the number of participants is n. At least t participants are required to recover the private key and to sign a message.
  • {n - n} means that the threshold is n and the number of participants is n.

Hint: In order to make the description more concise, the CMP paper describes the entire algorithm in the form of n-n signatures, and mentions that it can be adapted to t-n threshold signatures, but has not elaborated yet. We have supplemented the detailed description of how to modify CMP to the t-n threshold signature algorithm. Refer to "Modify MPC-CMP as a Threshold Signature Scheme".

Supported Curves

  • Secp256k1
  • Secp256r1 (P256)
  • STARK Curve

Hint: The STARK Curve is a new curve used by STARK Ware. It is a 256-bit curve with a 128-bit security level. It is designed to be used in the STARK extension protocol. If you wish to use Stark curve, please follow the instructions below:

  • Install the OpenSSL extended by Safeheron (refer to the "stark_curve" branch). The StarkCurve is not supported by OpenSSL. We have extended OpenSSL to support the StarkCurve. Otherwise, you will get an error when you use the Stark curve.
  • Set -DENABLE_STARK=ON and complile the library safeheron-crypto-suites-cpp.
  • Set -DTEST_STARK_CURVE=ON and complile current library.

Prerequisites

Build and Install

Linux and Mac are supported now. After obtaining the Source, have a look at the installation script.

git clone https://github.com/safeheron/multi-party-sig-cpp.git
cd multi-party-sig-cpp
git submodule update --recursive --init
mkdir build && cd build
# Run "cmake .. -DOPENSSL_ROOT_DIR=Your-Root-Directory-of-OPENSSL  -DENABLE_TESTS=ON" instead of the command below on Mac OS.
# Use cmake .. -DENABLE_TESTS=ON -DTEST_STARK_CURVE=ON for stark curve.
cmake ..  -DENABLE_TESTS=ON
# Add the path to the LD_LIBRARY_PATH environment variable on Mac OS; Ignore it on Linux
export LIBRARY_PATH=$LIBRARY_PATH:/usr/local/lib/
make
make test
sudo make install

You can use the MACRO to exclude the specified mpc protocol like this:

# Run "cmake .. -DOPENSSL_ROOT_DIR=Your-Root-Directory-of-OPENSSL  -DENABLE_TESTS=ON -DNO_MPC_GG18=ON" instead of the command below on Mac OS.
cmake .. -DENABLE_TESTS=ON -DNO_MPC_GG18=ON

The macros are listed as follows:

option(NO_MPC_GG18 "NO_MPC_GG18" OFF)
option(NO_MPC_GG20 "NO_MPC_GG20" OFF)
option(NO_MPC_MPC_CMP "NO_MPC_CMP" OFF)

More platforms such as Windows would be supported soon.

Link the library with CMake

CMake is your best option. It supports building on Linux, MacOS and Windows (soon) but also has a good chance of working on other platforms (no promises!). cmake has good support for crosscompiling and can be used for targeting the Android platform.

To build multi-party-sig-cpp from source, follow the BUILDING guide.

The canonical way to discover dependencies in CMake is the find_package command.

project(XXXX)

set(CMAKE_CXX_STANDARD 11)
set(CMAKE_BUILD_TYPE "Release")

find_package(PkgConfig REQUIRED)
pkg_search_module(PROTOBUF REQUIRED protobuf)  # this looks for *.pc file
#set(OPENSSL_USE_STATIC_LIBS TRUE)
find_package(OpenSSL REQUIRED)
find_package(MultiPartySig REQUIRED)

add_executable(${PROJECT_NAME} XXXX.cpp)
target_include_directories(${PROJECT_NAME} PUBLIC
        ${MultiPartySig_INCLUDE_DIRS}
        ${PROTOBUF_INCLUDE_DIRS}
        )
          
target_link_libraries(${PROJECT_NAME} PUBLIC
        MultiPartySig
        OpenSSL::Crypto
        ${PROTOBUF_LINK_LIBRARIES}
        pthread )

Usage

GG18

Key Generation

Refer to the test case for more details.

At the end of the MPC Procedure, each signer will get the secret share:

  • The secret share in base64 format:
"Cgt3b3Jrc3BhY2UgMBADGAUqrUYKCmNvX3NpZ25lcjIaAjAyMqckCoAEQTRCRDMxQkExN0EyMUYyOEExRjhBRkZEOEU0MUFBOTZEMjg0NDgzQkUwNzE0OEQ2MEUzN0I4RjI5RDQ3ODQ1N0I0MEY1RkZEQ0Y3MjBEMkUyQzExNEE1QjM3OUY5RUEwN0IxRkNEMjczMzgzMjU3RUM2Qzc2M0VEMzdENjRGMTZDQjgwNDk2NDJEQTdDNzI2RDMxOUFENTQ1RjkxQ0Q0RDNFOENGMjY2MUQyNTFBNUY4MjlBRkFCNUI0NERCQjZFNzZDRkY2N0E1NTE1NzRDN0ZDMDJCRDUwQUEzMDY3QUI4RUQ1RDhFOTRCOTgxNkU2REE2QzAzNDRCNDVERTRDMzFEMjNDMDc5Nzg2Q0VGOUJBOTJGQUQ3OUJFQTVGODdBMTYwRDFFOTJFNjNBQTM4MUU4QzEyMUVCOEUzM0VERThEMEYxNEY5MzNBOTI4QjBBQUE4OEM5RENDRjAwQ0QzMTgzMUY0RjUwNTVBQ0ZBNDQwMjAzNkI2MzcxMUUwNUJGMDlDQkI4MDUwMzU4NEE4NEYxRDYwRTJFQTM3RkJCOUVCQTBDMDg5RTQ1REE1NzExMTgyMTE4RjREQkMyOTRFMTBFNzZGNjVGMTdGNzE1NzMyMDY3QjI5OEE4NEZEQTZCRTA3MjVFQTdENUZFRkUwQzE5OEU1MDdEMTU5OTMyQzESgARBNEJEMzFCQTE3QTIxRjI4QTFGOEFGRkQ4RTQxQUE5NkQyODQ0ODNCRTA3MTQ4RDYwRTM3QjhGMjlENDc4NDU3QjQwRjVGRkRDRjcyMEQyRTJDMTE0QTVCMzc5RjlFQTA3QjFGQ0QyNzMzODMyNTdFQzZDNzYzRUQzN0Q2NEYxNkNCODA0OTY0MkRBN0M3MjZEMzE5QUQ1NDVGOTFDRDREM0U4Q0YyNjYxRDI1MUE1RjgyOUFGQUI1QjQ0REJCNkU3NkNGRjY3QTU1MTU3NEM3RkMwMkJENTBBQTMwNjdBQjhFRDVEOEU5NEI5ODE2RTZEQTZDMDM0NEI0NURFNEMxODIyRjQ0ODVDQzJCMkM5MzE2MTNGQkY3Mzk3MzhBMTE2MDAzQTM4RkUxQTFCMTJEMDczMzU5MzM0N0JDOTZFNkE3NUNCN0Y3QzU5MEUyODRFOURDOTMzRUYzQTIzNTVFNUQ2MDg2OEFGOTVFREQwMzkyOTFEM0M1QjY2NzM0NUU4NTE0MDdGMkI4NjMwODAyMzM0RkY4NTAzN0ZFNkE5MTI5RTE3RUE0MkQzNjBFNzhGRjI3NkUwM0E5QzI3RTNFREI5QjEyM0ExMERCOTI5OEQyNzA2RjAwNThBMjhDMDdDMjM0QzMwNjFFMTUxN0JBNkQ4RTNFMkRDM0Q1MDE4NBqABDlFNEEzRTlEQjJCOTc2RjFCN0UyNkYzRDI2MjI2QkUxRUFGRDQ2QUVGMUZDOTE4NzE5OTM3NjQxMjgzNDU4RjQ3MDM2MzlGNzAwNDlFRUUxMUVCNjU4NjZFQTU3MTU3N0FFRkQwREYzMjk4MjY2OEYzN0U5RjE5NjQxMjA3MUNDN0VBNjA4MUJCOUIyQjUyQzkxQjE2QzcyOURCRDA2NUFCOTAwOUJCMjk1QUQ2RDJGN0JGRDg4MEQ4QTRCNjk5NzUyNjkwM0QyRTFCMThGNTBDQTlFQ0YxMUE0QzgyQkQ0MzcxMkRFNjM5RDhCMDJGRDM1RTNGMTVBNzE2MkE2MDQ2MDc3Q0YzRkUwNTc3RDhCREQ0RTIxODg1ODlDQjZCRUY2OThFQjQwNzRCOERFQzcxQjc3NzlCRDgyOTlDOTMwMkQ1QTJEQTEyRDVDQzZFRUYxQzE5RTlEOTIwNEMzQUNDMzFGN0QyMjg2NDY1M0UxQzcwNzU0NDMzMDMwRDEyMUFDNTY1Q0MwNDRENUYwREI4ODg4RDMxRkU3REE0QkFFQjkzMDA3NEZCNTczRkY1M0EyRkM0NjJEQzQ3QUM2RUYzQzk2MDRCNDJGNEY5OTNBMTlFRTM3MzNFQjMwNDE4Q0I5NzE2NzRGMDhBNjk4OUM4N0IwM0NFQzBBREI0Q0I5IoACRDQ2NDYzM0FCNDhGQjQ4QURBMjZFRkRERjcyOTI2MUY4Mzc5N0IzRTZDMUY2QzE3M0QxRjFEODFEMjI1NTEzRTIwOEI0QkU5MzdDRTgxQjQ5MkNFMUJENDRGREMwQTEzMUEwNDcyN0Y4MDZCMjBCMDFFRjRFOTU0OEM0RDA1MzYzREYzMTM5RjI2MjFERTk4ODcwREFFMzI3OUZFMDAxN0Q5M0I2RDBFQzU1QUZERTJBNEZCMEZCQTg2RUEyQzZCRERGRDM0ODUxMDUwMUNBQ0IyNzlGRTIxRjRDN0QyQjc1Rjg3RTY5NURDNUZBM0EwQTc2NTU4MTc0MjFGNTc5RiqAAkM2OTAxOEI4RjdCMjBFN0RCOEY0QzFBNDhFMDk0ODQ5MzI4RkZGQzQ5ODc5ODYzREE0NkVBQjM2NzQ1MjA1QzQwOTA5NEJCMjNEMzMyNkQxMkRERTFBQzk4QjgyOERDMDBCQkE1NjQ1REJFMkZDOTA1MDdDQUU0OTJFNjlDQzJBNDZDNDlDNzMyNEQzNjNFQTM3Nzg2N0FCRjE4MzUwRjVCNkVGMUNFQjUzNDk0QUI1NzNGRTlCMzZBQjE1RUEzNjU0REVBRjlGRjZDQjY2MkQ5QjdENDU3NjVBRTU3QkFDQkVCNUI1MEJEQjhBNDJCMTA0OUFCQTM4MEZBNEQ5OUYygARCMDM2NkJCNkU3NjE4ODY4N0JFNjg1RTY3MzI5QTNFRTk3M0VBM0U2OTFERjZFNzhDNTZDRTU3QUNDMzVFMTBGQzEwNEZBQUQ1MTc3MDAzNjRFQkMxMTYzRDhCNUY5NTQzQjY1NjVEN0VCNzRDMzc0MEJEQkI5MTk2MTM2NzQ3QzgwQTMyRTA3RUU2NkQwNDdEQzBBQjhERjdBRjQ4ODYxNzU1NDE1RDc3RDNERDM2MzQ4REIxNjU4MTQzOUUwNTkxRDg3REE0MDA1NzhBRjdCOEY4NDBDMTlFOURBMUI1MTFFRTcxRUU5QzMwNkY1N0M3MTA0NDhCM0FFQkY3ODJDNURFN0IzMTIwMUI5QjZCNEEwNTI3NkRGMEVGNUZGMTgwQUZFQUE2N0ZCRjdGMjQwMzM0MDgxNUJCQ0Q5ODkzMzM0QTEzRjk1MDJDQjAyMERDNkM5MDUzRTUzRkNGMkYzOEVEQzM5QTZBRTU1RjM0RkI2RTRDQ0NBOTU4NURFNDAwNTBDREI3OUZGOTlERjc3MkE2Nzk1NDQwNTk0MDk5MzRFMTZEN0U0QzRFMzc3NUFDNkVCQzEwNEY5MzkxNkQ1QzAwNTE5MEYxMEUzNzQyOUZFOUMyRjgzNjEyN0RERTdDQUQ5QzE0NUQ5QjMxMTVGMTQ2OTNCOTY1MjdGNzRDMTqABDlBMDMzNzU5RjFBMTRBRDEyOUUwMkExQUVCOEFERDcyMzQzNDhENTc0NkE2NzVFQkQxQjEzMDM1M0E4MjI5NjgzQjcwRDA2NDU4Qzk4NkQxRDI5ODQwRjUxRUNFRTM0MTJCQTJCNUMzN0Y1NUVCNTlGRjRGQzU4MkFDRkU2RUQ1NDg0N0NERDExMDc4N0Y4RjFGMDFEOEFCQzE4MzUzRUM1Nzk2REQzM0YwMjlFNTE2Mzg2ODc0QTJDNzkzNEIwRTk5QjdFQkVDRTc3QzYwQzlENUM2MDBDNjQ2MkJEM0IwMTgxQTIxMEEzRTlFN0EwNUUwMjUzRDBERTQ0NTlFNkExMUNGN0IyRjYwQkEyNTBERUJEMDIxQzlCODVFMzc3RTI2NDkyQUY3RTY1MzI4NjFBRkIxODE1MUJCOTg2NzlGNjM4MjQwNzM1OUM3OEFCNDk2RUU5MEY5RjNFMkE4QTIzRTQ0OUJBNzY1QzIyMDg4OTIwQjAyMjZFNkJFNUYxNzQ5RTgzMzMyQzBBNTdEQkY3MEQyRjE4MkJFMjNGNkUxNTQzNUY4NzRGMTM1OEEzMDE2NUYyM0QxNTI4ODc1NDZBOTdCMjI3RkJBQkUzMjEwRjg1MjY4OTkyMjUwQzJGMDA4MDAxMEE3OUVGQzY2MDMzMDlBRkQzODJFQjZGMEMxQoACRDQ2NDYzM0FCNDhGQjQ4QURBMjZFRkRERjcyOTI2MUY4Mzc5N0IzRTZDMUY2QzE3M0QxRjFEODFEMjI1NTEzRTIwOEI0QkU5MzdDRTgxQjQ5MkNFMUJENDRGREMwQTEzMUEwNDcyN0Y4MDZCMjBCMDFFRjRFOTU0OEM0RDA1MzYzREYzMTM5RjI2MjFERTk4ODcwREFFMzI3OUZFMDAxN0Q5M0I2RDBFQzU1QUZERTJBNEZCMEZCQTg2RUEyQzZCRERGRDM0ODUxMDUwMUNBQ0IyNzlGRTIxRjRDN0QyQjc1Rjg3RTY5NURDNUZBM0EwQTc2NTU4MTc0MjFGNTc5RUqAAkM2OTAxOEI4RjdCMjBFN0RCOEY0QzFBNDhFMDk0ODQ5MzI4RkZGQzQ5ODc5ODYzREE0NkVBQjM2NzQ1MjA1QzQwOTA5NEJCMjNEMzMyNkQxMkRERTFBQzk4QjgyOERDMDBCQkE1NjQ1REJFMkZDOTA1MDdDQUU0OTJFNjlDQzJBNDZDNDlDNzMyNEQzNjNFQTM3Nzg2N0FCRjE4MzUwRjVCNkVGMUNFQjUzNDk0QUI1NzNGRTlCMzZBQjE1RUEzNjU0REVBRjlGRjZDQjY2MkQ5QjdENDU3NjVBRTU3QkFDQkVCNUI1MEJEQjhBNDJCMTA0OUFCQTM4MEZBNEQ5OUVSgAIyQ0YxQkQ2OTZDRkNGRkZBN0UzQzY0N0E3MjA5QUY3N0Y0ODczMTMzRjg3Qjk4OUJBM0Y0QTc3MUExODQ1MTUxQThGREMxMjhDMDVCN0RGQTEyREI4MDQ4MjM0MEU0MTM2RUJERUYyODU4QjMyREE5OUNCNDBENkE3NkQ4MUNBQUJGQkVBOEM0QjdCQTRBQjNGQTIyQjJDMDU0MkZDNDYxQTk5RjIzQkQ4RUQ0QkI0N0JENkY0QUIxMjUyRDRCNEZBOUQ5MUY1NjYxMDlEM0MyMDk1NDdDNEE2QzRDQjdCQzMzNUQzRDA2Q0JENEU0MzQxRDEyRDAzMUZGMjY4MUIyWoACOUM4Qjg0ODE4NTBDQkYwQjQ4RjJENkNENzM0QzJBQ0IyRjhGN0RGODIwNEE5MTFFQzk3QjI2MUIyNTVGNzVGQTU4RENEMDAzQkQ3MEI1NDM4MzIzRDlFRUQyMDhFQUZDNkUzM0QyM0NDMzIxMEY2M0QxQTIxM0FERDQ5RkYzRkMyRTM4MEJFRDgxOEIwM0VFRUNBOEIzQTgyMkY1RTRGQTYzMjEyMTYxQTQzRTA5QTk5NjEzMkE2RDg4NjJGMUQ4OUYzMkZDNDFBNjdGNTQxREFFQkE3REExRTQ3QTI4MUEyNzlGRDkxOUIwOENGQjQ3NUJBRUZENUE0OERGMzc4RWKAAkE3NzJBNUQxNDc5MkI0OTA1QkVBOEI2Mzg1MUY3NkE3OEVGMjRBMEE3M0EzRDM3Qjk5MkE3NjEwMzBBMEZGRUM3NzhEOEFDMDc3NzMwM0JBN0ZGMjlCOEMyQzlCMjVGRkFCNDY4MzU3MjdCN0YzMDY4MjQwREJFQTE1NzRFODhCN0UzNDZBREE2RTY3OTNFNDhDRUFGQjcyMjVDRTNCQjYyRjlDNDk1MTM2ODY0MjlBRTc4QkM1MDk2MUJDRTExQzM0MjQxNTJFQUY0NjQ4RUFBOTI1ODFENzg4N0IxQUZCMkMyQUE5OEYxMDhBQkY2QzhBNTI4N0U1NDJGOEQ1RURqgAIyQTA0OTQzNzcyQTU0RjcyNzAwMUVBRDcxQUJEMUQ3RTAzMDA4MUNDNzgyRUY1MUVEQUYzODUxQjRFRjI4RkM5QjAyQzdCQUU3RkMyNzE4REFBQkE0MERBQjk3OUEyQzM5RDg2ODQwOTE4QzFFRDJDN0VEQTlBOUI1OUM5RDgyRTE4OEM5MDg1QTM0ODVGRkI0QUNGQjQwM0NFOEQ2QkZCNTNDREZCODlBRjBCNDEwQkRERUI3MEM5MjJCMkY4NURCNUFCQjM1RTUwNEMxMjBGRUNDMkM3RDQ3NjZCNTM5Mjk3MTVEQkYyMkFGRDQ3NjlBOEVCQkNEREM2QzVBMjExOoYICoAEQTRCRDMxQkExN0EyMUYyOEExRjhBRkZEOEU0MUFBOTZEMjg0NDgzQkUwNzE0OEQ2MEUzN0I4RjI5RDQ3ODQ1N0I0MEY1RkZEQ0Y3MjBEMkUyQzExNEE1QjM3OUY5RUEwN0IxRkNEMjczMzgzMjU3RUM2Qzc2M0VEMzdENjRGMTZDQjgwNDk2NDJEQTdDNzI2RDMxOUFENTQ1RjkxQ0Q0RDNFOENGMjY2MUQyNTFBNUY4MjlBRkFCNUI0NERCQjZFNzZDRkY2N0E1NTE1NzRDN0ZDMDJCRDUwQUEzMDY3QUI4RUQ1RDhFOTRCOTgxNkU2REE2QzAzNDRCNDVERTRDMzFEMjNDMDc5Nzg2Q0VGOUJBOTJGQUQ3OUJFQTVGODdBMTYwRDFFOTJFNjNBQTM4MUU4QzEyMUVCOEUzM0VERThEMEYxNEY5MzNBOTI4QjBBQUE4OEM5RENDRjAwQ0QzMTgzMUY0RjUwNTVBQ0ZBNDQwMjAzNkI2MzcxMUUwNUJGMDlDQkI4MDUwMzU4NEE4NEYxRDYwRTJFQTM3RkJCOUVCQTBDMDg5RTQ1REE1NzExMTgyMTE4RjREQkMyOTRFMTBFNzZGNjVGMTdGNzE1NzMyMDY3QjI5OEE4NEZEQTZCRTA3MjVFQTdENUZFRkUwQzE5OEU1MDdEMTU5OTMyQzESgARBNEJEMzFCQTE3QTIxRjI4QTFGOEFGRkQ4RTQxQUE5NkQyODQ0ODNCRTA3MTQ4RDYwRTM3QjhGMjlENDc4NDU3QjQwRjVGRkRDRjcyMEQyRTJDMTE0QTVCMzc5RjlFQTA3QjFGQ0QyNzMzODMyNTdFQzZDNzYzRUQzN0Q2NEYxNkNCODA0OTY0MkRBN0M3MjZEMzE5QUQ1NDVGOTFDRDREM0U4Q0YyNjYxRDI1MUE1RjgyOUFGQUI1QjQ0REJCNkU3NkNGRjY3QTU1MTU3NEM3RkMwMkJENTBBQTMwNjdBQjhFRDVEOEU5NEI5ODE2RTZEQTZDMDM0NEI0NURFNEMzMUQyM0MwNzk3ODZDRUY5QkE5MkZBRDc5QkVBNUY4N0ExNjBEMUU5MkU2M0FBMzgxRThDMTIxRUI4RTMzRURFOEQwRjE0RjkzM0E5MjhCMEFBQTg4QzlEQ0NGMDBDRDMxODMxRjRGNTA1NUFDRkE0NDAyMDM2QjYzNzExRTA1QkYwOUNCQjgwNTAzNTg0QTg0RjFENjBFMkVBMzdGQkI5RUJBMEMwODlFNDVEQTU3MTExODIxMThGNERCQzI5NEUxMEU3NkY2NUYxN0Y3MTU3MzIwNjdCMjk4QTg0RkRBNkJFMDcyNUVBN0Q1RkVGRTBDMTk4RTUwN0QxNTk5MzJDMkJARDE5ODYxQjRFNEZERjhBMUZGQzUwMzAwM0ZGMDJGOTE0QTUwQjRCNENDN0QxODg4RkIxNUNERUVDODc4NUQwM0qPAQpANDdEODMwOEZEQzI3QzUyNEE0NDVBQTM2RDFFN0RGRTQzNDVCRDdFNzYyNEQ1NzBEQjkzM0U4MzE0MUZBMTM0OBJANzAxNTI4NDAwODEyRTU1MzEwRDNERjM5QzVEMkY2NDMxODFFRUQ4OTU3REQ4OTYyMzdGM0RFQjMyQUUyOUFGRBoJc2VjcDI1NmsxUoAEQkE2RkI3MTNGM0E4MEU4QkZCRkM0NEE4RkE4MEYzMDA3NTQ3NkJDODk3NTI4MTM5MkRDOEM2OEFFMEFFRUE0NkM4NDZFRDg1MUUwRUE2NTY5RjU0MjYwOTIxODQ4Q0ZFMkZBODRGNjhCNDg4QzZFRkEzMjczMzZGN0RDQ0MyMTI1OUJDMThGQkU0MjNDODE1QkFDRTA2OTA1MkQ1MUMwRkRGMTg2MDAwQ0EwQkYyRjE2QjlFQkUyMUI5MjYwMzhFRjZCNTk4OUYwQjU5Q0IwNDZDMDFGOUJCNEQ5OTZGQzg5RDhCQTc0NjAxQjY5QzlBNEZGM0REMjMzNUI4MDI1NUM3OTYzNEVCMzU0OEUwNkZBREJGOTJGQzNCMDgxMjYwM0REMUY1QjM3NjA5MzYwQjVDQ0I1MDM4RUEwNjQ0NzUyRDYxRDU3QjFDMjhGOTdFOTg2RjNGRjQ4QjVFMEJBMkVCNEQwMDlFREFENzc4NjQ0OEQ5Mjc0MzhFNEMyMTgwMjdFRkY4Q0I4M0YyMzg5RDdGOTUyQkY5M0I2QzE5RTk1NTk5QkUzNUMyRTRCQUE5MjQyMzhDN0E0NTFENDlEN0M5MDZCRkJEQzMzQkI1MzBGRTgwNUI3MkRGN0VFMEZGNTM1QTgwM0IzRjAwNDU3RUIyQ0ZEREVBN0UzNDRDOTFagAI2RjIxNjJCNDQ5RjM3MTY1Qzg2NEI5NDNFMzMxRjE4NTUwRDI1NDI2NDZGRUQ1N0EwODdGN0FFQzVDMzE2RjBCMkE1NUUzNjE2QjIyQTlBQUM0MDE0QzcxOEZFNkZERUI4RkVDQ0Q0OEFEQTg5RjI5MzVCOTBGQzhBQ0VDMUU0MTRGQTUwM0Q5QjdFQzNCQTNDRUM2MjQ4M0JGNkRDQTg4QTEzM0QxNjEyQUQ0QkRDQjRCRTcyMzY2RUYzRDhDNzY4MTE2MEY1RUE4NEQ5QTdFQzVDMjEzRTcyMTNCM0EyNjUwM0U0REQ1NThFOTE3MDYxMzhGQ0Q2QUYzRUFCOTU3YoACNkI1RTYzQTRBOEJCMjg4NzI0NjZFRjhDODU4QjUwOTczOTQzNzFEMTE1QTRDQzc5MzBBODEyMTQyNkVBNUE5QTdFQkFEQzczMzdCRkQwNkIxRTg4NjA1NkJGNDFBQUUyQzhCQkYyRkIwNzNBMDk2RUM0N0Y3MzZBQzc3QTgwMENGNTYxOEZENjhFNEM4NTQzODk0MjZCRjU5ODkwOTQwRjM3Njc1MjZBMzVBMzQzNjlGQzFBNDBGNzc5RkQ4MEUzM0NEQTEwNDhDMDY0QThGMTU0RUMyRERGQzVDRDdEREY3NENGREJDQkJFRjY0NEFDNTFDMDYwREM1MDUzODI1RmqABEE3QjY0M0JDRTkyNzJBNjFGQkM0M0NDN0I4MkQ5NkI3Q0FEMkI3REIyODE1RUI4RkQ0NjMyODlFRTQ4OUYyMUIzQkQ1N0Q2NkI3QzVEMDI5NzE4MEQ3NkM0NjczNDc1RDRGRURCRTVDNTM1NzI4REYwMUI0QjgyMDk2RjRGQzU3QTM2RUQyNDc0QjIyNjREQjQ4RjdBQzI4NTEwNEM3MjhGRkE3MEM0Q0M4NkE1RDM0NDkzRDU3MjQ3RTcwRjY2RUFEODVEQTVBNjFGRERGMUQ0Mzc4ODQ0QkJFOUY3MjdGQzc0MzM1NDAwMjQxMzZERDZCNDdBMzkxQTkzOThBODEzNTY1NzE3Mjc4NjE3MEI3RURFMTcxNTkwQjM3Q0ZERjU2Q0Q5Q0E3REQ3NjRCOEI5NTg4QjFGMDJEMkM2OTFGMzg1MkY4NkVFNzZDNDI0REM5ODIwRUI1MTdEOTc2NUIxRkIyQ0EwQjFCMjk3QjRBMEMyNjM5OTNDMTNENjBERUUxOEYxRkEyMkRBMTEyN0YyQURDNEMzMzE5OTEyNjc5QTM3M0FCRjdFOTY4MTJEQjEyQzUxMEM4QjhBQzMzQ0FCMTE1OTRBQUJENkFBQ0RDQkREQ0I3NDdDNUIxRDRBRTM5MEMzQkIxMDQ3RjQ2QTY0N0NBNEFFRjE0RTMyRDRCcoAEMUUzM0UyNjY5N0MyMjNCRjI3RTUyQkVFREM4MTFCNUU3Qjk1ODQ4OUNBNUM4OEE1QTg1MEY1OUZGMjIwRjFDQkFFMURFMjczRkQ4RkFEOUFERTEwQjBDNDU4MTc3RDEwQjk4RDdCREE2MDFDRDIzQTUwMERENjRBMzg4MjUyNUY2M0ZEMEI4MjA3M0M5QjEwMDVGNTMzQ0VERkU2RDk1QjBENTg1QkZCOTEwNEE0QjdGM0ZCMUI1RTE0NEEzRTAyMjcxMzc5MkQzQ0YyNzNBODU1NkFFRjk5RDQ2NDFFMkY4NTEzMDE1MDJBMzI1NkEzQjMxNEU4NTVBRkIwRDg5NEUxNTA1QjE2NUZGODU0MzY4RDYxNjg0NzhDNURFN0QwMjBGOTIwMUM2OTg2RjM1RDA0MDFDQ0I5QjE5OUIzOUZDNTgxNTY0M0UwQzZGODQ4QzVFODFGNTFGMTk0QzBFMTc2N0Y4NTFBNUEwMUMwNTVBMTI5REIxRDZGOTMwRDdBMTcxMTg2MTY5Qzk2NzNCREY0NzFEREEzNzFCNDJDQTIzQTMxNDVFQjYwNEI1QUU3QTc4NjRDMDQ1M0Q3QTMwQTRBOTZCN0RDQkFENzU5RDFCNUQ4MDAzRTFCOTRGRTVGMjAyQzAyQjExNjFBNDA1MTY5OENEMERBOUYyOTAyMDN6gAQyM0MyNjkxREE2ODU3ODc1MkI5NDhENDk2NTY4QzIwOUQwMjhCQzkwRTMxNjEzQkU3NTMzQzkwODg2NjM0M0Y3QUQxNDhDRDM3MUU2MjQwNTQyMEFBMTcxMzRGMDJFOUM1QkE2MTMzNjY0M0Y0RjYyOTc0OEZEOTNFQ0Q0NDVBQzg0Mjk5NUQ5MDI4NEEzQUIyMTUxQTgxMzE2MzU2RUY1ODcwOTYwNjFEMzA3QzQ5MzdFNUE0NjFCMDlEOTA3RDdFMDY2RjRFREEzRTBCN0YzREE5NTUwOEQ3QjgzQUU1MTcwMDQyNkRERTI2QUJENEU0MUZBNTBBNENCMzQ4NUVFRkM2RjE0NUQwQkMzNzM3REFGRTExQTBBODJENTRGQTZCQTM0NTc5OUEyNDJFRUI4NEYwQzQ5Qjg4NDMwQjE5NjYwQTFEMzVBQjI0NDIwMEMzREZEREFGRUUxQzhBM0M2NTRDRDY0RTUwMTc5M0MyQjFFRTVGOTEwRDk1OEY2RjdGMzIyOTAzRDEwRTc1MTUxQjZFMzcxRjVDMDM1MUVEQTNBMUIzQzJCQTAzMjkyMjlFRDI1MEMxRjE2N0I5OUY1ODRCNzdGRjhGNUMzM0Q0RTU3MzNDRTE3QTJBMzNGQTQ4NUM3Mzg5ODlCODlBNUNFQkU3NDk4Q0M5QkNGRkJFRYIBgAQyNDkxRDczQjkwQTRBQkIzQzc0QkU1QzY5OUJGQjZGQ0Y0QUVGRDE4NkFBMTM0Q0MwRkZCQjk4NjEwMDcxOEY3ODVDMTREOTRDQjBFOTlBMjhCRThCQzI3N0QyOENBMjZCMDY5MUZFNDVGNjBFNjUwOTcxQzkwMTg1QzI1NEMyOUI1OUY2QjRCNzQwQkNBNzhCMTZBOERENzNGOTg5NENERkE1QUMwODg0RUVFQkM2MzIzODQwRTJCMTg0MUJFMkJFNDc2OEJFQUQ4MjBCREZGMTc2NUEzQkY2NjE4MUVGQ0M5MjgyRjA5QjI1Q0EwRUUxQzcxN0ExQzc3RTlEOEIzQUU5MjkzQTJCNTJFOTQwMTZGNkU2MTVFRjY5OTY5MDQ3QzlFNTJDNjlDNUYxM0I0RjFFNTlDRkI4MjMyQ0M1OEI4QTJDRjI4M0RBODRDNzhGQUE5ODY2QkU0MUUzQTNCQkQ2MTI4ODE4ODBCNkZERTYzREREQkEwNzc5NjBFRkMwRjE0MUI2RDczMEYwOENDRjU5RUJGQjMyRkVEN0FEMTI2RkU5NkREQTlBRjQ2REY2REY2MzgzOUUyNDRDMjczODM0NzZDRENFMDg3MjY2Q0Q5RDQ5MTBCNUREMkYzRUU0QzhDRjkxMEVERkY3QTM1RTdCMDM3QUZEMDZCN0ZGNDK0FQoKY29fc2lnbmVyMRoCMDE6hggKgAREQzNGRTgxRDREN0Q5MkVFMDFFNjhDNDBDMEIyMTJBQjgxRTgwMEMzNjMyM0RENjY3QjA5ODQ2NjkwREYyQTg5MUU0RUY3RDhDMzYzRkI1ODJDQTU0QzFENzdCNzQzNjI1M0I1RjhERTMzMzk0OEYxNkYxREY1MTlENjhGNTE4QzU0ODcxMkU4QTk0QTBCNUU0QUQzMjREMjA2RjVBQUJEQ0FENkE4RUI5QTVGRTQxMjY5MzVDRjZEODcwMERGMzZDRDU0OTYwOERBNjU0NjlGMzFDRUZFRTI5QUY2MTEyRTI2QjM0QUJCMENBOTZDQ0JDNDFDODQ4NTU0MUNCRkEyMkExMjhCQUM2MDBFMThCN0I0OUEyMjVFQTk5NkFCOTMzMkU3MkE2NjAxRTIzRjFBRTJFQzQyMkEwRDRFMUUyMTAyMUQ3RDEwQzkzQzE1REY4MzRGRDIwMDVFMzc2MUU3NjAyNkQyQzg5MDA4REI2RDk0RjI4NDM0OTdDQUM0M0NBQ0U0QkM1NzZGREEyN0ZEOTNGRTZFRkREOTE3NzA2MjUyNDlDNTJCRTM4OTk3MEM3QjdDRUQyNUI4OTZFMjVCQ0I4N0VGMTU5MUY3Qzg4QzU0QjUwMTMzOTNCMTM4ODE0QUVFMDY4MjlGNzcwOUI5MTI0RTIwNTE3RTM3QUQ4NRKABERDM0ZFODFENEQ3RDkyRUUwMUU2OEM0MEMwQjIxMkFCODFFODAwQzM2MzIzREQ2NjdCMDk4NDY2OTBERjJBODkxRTRFRjdEOEMzNjNGQjU4MkNBNTRDMUQ3N0I3NDM2MjUzQjVGOERFMzMzOTQ4RjE2RjFERjUxOUQ2OEY1MThDNTQ4NzEyRThBOTRBMEI1RTRBRDMyNEQyMDZGNUFBQkRDQUQ2QThFQjlBNUZFNDEyNjkzNUNGNkQ4NzAwREYzNkNENTQ5NjA4REE2NTQ2OUYzMUNFRkVFMjlBRjYxMTJFMjZCMzRBQkIwQ0E5NkNDQkM0MUM4NDg1NTQxQ0JGQTIyQTEyOEJBQzYwMEUxOEI3QjQ5QTIyNUVBOTk2QUI5MzMyRTcyQTY2MDFFMjNGMUFFMkVDNDIyQTBENEUxRTIxMDIxRDdEMTBDOTNDMTVERjgzNEZEMjAwNUUzNzYxRTc2MDI2RDJDODkwMDhEQjZEOTRGMjg0MzQ5N0NBQzQzQ0FDRTRCQzU3NkZEQTI3RkQ5M0ZFNkVGREQ5MTc3MDYyNTI0OUM1MkJFMzg5OTcwQzdCN0NFRDI1Qjg5NkUyNUJDQjg3RUYxNTkxRjdDODhDNTRCNTAxMzM5M0IxMzg4MTRBRUUwNjgyOUY3NzA5QjkxMjRFMjA1MTdFMzdBRDg2So8BCkA4QkZEMDlGQjc3MjMxMjFGNkEyMEJFQjk3RDI3QTE5MjczOUE4MDY3NUI4NjdENEY0Qjk2RDdDRUFCQTIzNEQwEkBFQzAyNkIyQUM1NTQxNUJERjMyQ0JFNzM5NEMwMUY3QjNGREE3REVBQUIwM0Q2RTZDMDNBNzdGRTBGRDA5MDhFGglzZWNwMjU2azFSgARCNzcxQTJBQ0QyNUU4OTdENzdDNkY3MjI2NjBEQUZGMjI5NDNGQzRFQjE3RDZEQTYyODc5REVEOEJBNTZCOTY1NkE3Q0RGOUVGRjNFRUUwNTYyREE4NzZENDhEODEwOEU5MUY3MzFDNTM5RTFGNTc4MERDMzUzRDY2QjdCNjg1MDM2OTc2ODBCN0Y1OEIwQTVDODZERkE5RUI2QkQ5NkQyMTNBOTlEN0E0Q0ZERjYyRkI3RDBFNzMxMEVERTFEODBDQjc1RkI4Qjg1MDNCMThFODg2REI1NkVGNzMyMkJDNTI5MTFBMjc1MDVFRkYxNzI3MEJEQjI5MzRGNTZDN0M5RkFFNTI0QjA2QzExRjgzQTkwNDczNUJFRUU0RDE3NUVDNDQzQjVFOUY0QTNDMUZDNTcxMzBEOEFENzIxOEJBMjkyODdCNEVDRDZFQzNBOTlEQjEzRjlDNEU0QjM3QzMwMENGQkM3MjFEMTNBMUM3MDU2RDRERjNCNEE2NTdDRTA1MjZGREUzODk1OTM0M0NCNDdGMUNCQTlDMUI2NDhFQjE3ODk5QTRBMDFDOURGMzZGNDg1QUVGMEI3NjRDOUFCODMxNkJCNzVGQ0Q5MUI3QjIzN0Q1NTBFNThFQzFFM0Y4QTIyMUU5RDUwQjI3NkY5M0I2REJCNTE1RDNFQzVENWqABDRFOUIwRUFCQ0JENUQzQ0YxM0ZCQUIxMUExRTAxRUQ2NzJEQTVFNDE5MDgyOUNGRTdFMzRCQUFERTE3MUY0NkNCMzk1NzBCMUQ5QjJGMEIwRjU2NEQ1NjVFNTA3QkIzRTkwMDZDMEYxRTY1ODYzOTg5NUJCRjU0NkNFMjJGRkM2MkJCNEUwQUY0QUI5NDY5MDlEN0EzNEVCMjM0RTEzQUEzMTI2OUQ5RDMxQjY5RkFDMkQ4OUYxNkRGRkQyMEIxRjI4QkE0ODhBRTJDNzg3QUIxNTREQjg2MEY3M0YzOTc2M0NFMjhBRjI1MTRDNzVDRjA5MDE3NzE2REY5QTcxNzFBODUyMDdDRUE2MTQxODQ3M0E3QkM4ODU0REI0RjNGNEIyMjgwRjM2QTA0RDVDRDlGNzAyMzI1RDI1QTAyMThGQTA4NDFEMTQwREU3MUZBNUY3MkRFNjZCMzVBMTQ4OTU4QzM1RkVCODM4MTZENEJCMkZDOTg2MEU3M0YxMDFBMDA3RDdDODI4NzI1RDUyRjhBQjczMTIzNzNCQTA0Qjc4NUNDRkVFNjNDRjMzODRBNzcwNEJCQkZGOTY1NkY5OTI1M0Y4QkIzODJCQkM5NkUzRjRFNDY5N0Q5RkQ4MTlFOEQ5RDIwQTBDNTQ0ODhBQjFFNDZEMEIxMjI2MjdCMjEycoAENTM3QjQ5RjgzRjU2QTUzN0JBM0UxNzlBRjk2NTgwOTM4NkFBMEZDQzFCOUFDMzMyOEZCMDBGMTE5QTZENzVBQjQ0NzBBOTQyQzA0ODEzRDUzM0E4NTA5MkFDRDE4RURFNDczMzlFMjYwNDM3QTJFMDc1MEQzRkM5QjcwRkVGMDE4RTA5RENENDI3OEFCMjY0MkQxNzhBMkE5RjRCNDFGMjQxRTI3QTRBM0FEQjhERjcyQ0ZCMUYzOTk1RUZEQjU0NjE0NzlGNEE3QTIzQTc2MzExODJBRUFCNEFCMzk0NzEwMkRCRUFGRjFCRTRBRjFDRDM5NTY0NDhDNzQ5NjcxRDk1MUMyNTU5QjI2QUY0RkZEMTQyRDUxODRGRDI0RjI2QjlEQjVBRjgxRTE2RTJCNDQ4RDMxMTc5NTdFNUNCNDZFMjEzRTlGMEM2NzhGQTJGQ0YwMTc4RTNFQjZGN0ZCNkEwOTkyM0NDRTE2QUVGOThGNjkxQkMwQ0I1MEUwODhCODI4MEFEQzI0NDdFRUYwOUQ2MjJERERCNUUwNTEzQzFCRjhCREYxQUMzMjA3RUI0NDUwRUJGRUNEMUJEODUzOTcwMzNCQkRDMjQyNTU4NDkyNjU0RUJBQTlEQjA1NjU1MkUzRTAwNjE3NTYzMEEyNDg0RUI3MjAyRjI4RTFBRTYytBUKCmNvX3NpZ25lcjMaAjAzOoYICoAEQUQ0MjUwODRCMDI4QzIyNTYxQTJDQkVEMTQ0OUNCOTQ5NEI1QUY0NDBGNjg3RkU3NkM0RUUwMzJCOTM5Q0Q4NzY5MzBBOEJBNTI1N0Y1MjEwREQ4RUVBOTUwODI3RTIxQzg3NUY4MzU1NEUxMDhCNTUwQjc2NTdERTA2MzlDMDhGMjdFMzlDMzExRUQ4NkZBQjc4QkY4RTQ3N0M1MjU5MzQwREY3NEVCRDA3MEY2ODQxMTc0MkMxRDIyMzFFNUFBRkRFQUY4RTM2RkU1MEJEODAyOTlFQTQ2MzNFMjg0NDdGQjU1RUM5MDk1QkNBMDkzOUMzOEZDMDQwQUE5QzNGMTQ5NkNDREUxMDM4NjlGNjlDOThBMkEyOUZBRUU4ODYwQjIyNTEzNTU3RjRCMTVDMDBCMTZBQURGRDc5MERDN0JGNkE0MDZDRUUxQjVEQjhERDE1QUNEMThBMEJGQUJCQjk1MDY4QjJEMjIwODgxNDI4RjkyMjYzMTY4M0NFOEExNUM1RTE1RjNDMkYyRDY0NDQxNzIwRkMzOTA0NkE1NEZGMjdENjlCMjIxOUI1MDUzQzU5MUFGN0M2ODk0OTkxNDA5QzM4MjFBQUE4OTlBN0QzODg5ODMyQ0I0RUQ4NjY5MzU3RTg3Mjk0NjNBODI2MjlDNjVEQUZDQzM1NjQwNEQSgARBRDQyNTA4NEIwMjhDMjI1NjFBMkNCRUQxNDQ5Q0I5NDk0QjVBRjQ0MEY2ODdGRTc2QzRFRTAzMkI5MzlDRDg3NjkzMEE4QkE1MjU3RjUyMTBERDhFRUE5NTA4MjdFMjFDODc1RjgzNTU0RTEwOEI1NTBCNzY1N0RFMDYzOUMwOEYyN0UzOUMzMTFFRDg2RkFCNzhCRjhFNDc3QzUyNTkzNDBERjc0RUJEMDcwRjY4NDExNzQyQzFEMjIzMUU1QUFGREVBRjhFMzZGRTUwQkQ4MDI5OUVBNDYzM0UyODQ0N0ZCNTVFQzkwOTVCQ0EwOTM5QzM4RkMwNDBBQTlDM0YxNDk2Q0NERTEwMzg2OUY2OUM5OEEyQTI5RkFFRTg4NjBCMjI1MTM1NTdGNEIxNUMwMEIxNkFBREZENzkwREM3QkY2QTQwNkNFRTFCNURCOEREMTVBQ0QxOEEwQkZBQkJCOTUwNjhCMkQyMjA4ODE0MjhGOTIyNjMxNjgzQ0U4QTE1QzVFMTVGM0MyRjJENjQ0NDE3MjBGQzM5MDQ2QTU0RkYyN0Q2OUIyMjE5QjUwNTNDNTkxQUY3QzY4OTQ5OTE0MDlDMzgyMUFBQTg5OUE3RDM4ODk4MzJDQjRFRDg2NjkzNTdFODcyOTQ2M0E4MjYyOUM2NURBRkNDMzU2NDA0RUqPAQpANkZGRjQ0NTgwREI2MUJBMDJEQzg2REY4ODU4NDhCREIyNTkyMDBENzUyNEU4QjhGREQxQjhGQkFCMUZBQzczORJAM0ExOUM0QzhFNEE3Q0I4Q0VCOTUwNTA1RDREQTg5OEM0RjlEQzU4NjVEQ0RFMEUyOEMzMTY0NjkzNUE1NzQ3RRoJc2VjcDI1NmsxUoAEQjA3MEU5ODA5MjA2Q0REM0NCOUQyOEEwQjhCRUFCMDJCMjc4RThEQjk4M0JGM0I1M0ExQjdEM0FDMTIxREQ4MDgzNkM5M0NDMUZDRkQyNkZCRUI4MTI5QzA1NDRDMUEwMjU0OTA0MjZENTVFRDE5ODA3RDAwQzBEQTczMEI0M0I1MzU1MDA1MjQyQkFEQzI4MkNCN0QyRjE5OUY3NkI0QzMzMUVCRDA2NTExMkJFQkM2M0ZEQUYxQzcyNzkyNkEyNjIxM0IyQUYzMjY3QjQ2QUFBNTFGRTFDNTE2MEFGNjc0QkFEMkJCM0FEQTJBMDdCNjc4MTcwQTJDMTM4MDdFNDA3NkRDOEFDQzVBM0Q3NUFDNDBENkQ1NEZCMTE5NjNBNDYzOTdGMjQ1NUNFRkYwMjU3RDBGQTMxOTkxNUVEQzczNTM0MUIyMEVCNDhDMUJGODM2RUMyQ0UwQjI4MEFEQUYxRUY4NjIzMkE0Rjg5QkE2MTUxMzE4REM4RjgwODg2Q0UzMEYyRDFBQzVGMTM0RTRBOTBBQjc3MjVEODc0OTQxRDJDRkNBNTcxRjk0Q0MwOTdDMzZBRjdBREZGMUQ1NjM0Mzk2MThCNjE5M0ExMkUwMUYyNjJBOERFOEYyRTAxRDNBRTk3MUEwOUVGMzVEMkZCMTcwRDE3QkU5QjM4MzFqgAQ5RTJENDE3Rjk1M0RBQzE5QkIzNUNGN0Q4MDM1NzcyMDlFRDg5ODc4QzI5MzlFMTRFREZCQjQ2RDMxMUU0OURDRTE2QjE2REI5N0E2RTgxQTQyOEI0M0VCQkI3NUJEMjNFNkJDNTNFM0I4MzkwNjUzRkFCN0NBQUE0MjRFQ0Y4MTQ4RDIyNTc4NUZBNTBCQkNBRjNBREE1MDQ1QTBERTc0Q0JDOUEyQjU3MkIyMkRGNkU1QzFGOTIzQjJCRkNBRjgwNjhCMjE2RUMzRUMyOERBNDJCMTUxQzc2OTdEMDQ0NzYyRkJDOTAzQ0QyNDQ1NUU1NjBDQzA3MDZFRUVCMUU4MDZBRDFGN0E3QzI2MUJEQjExRTA2NkMxOEUyNTREMjE0Q0MxNkNCNDNCQTkwMjY0MUE5NUI1OEI0RkQ1QkMwOTlGOEVDOTQ5MDVFQjk5Q0Y1QTUxNTE3ODQxMjg1OUJBMjRCNEUyNDAwQkZBNDczQTEwNDMwMzZDMTQzQ0RDQTlBMjk1NzExQjA1NzIyNkM2NTIwRjQ1MDFERDkzMkFCNDBFNjIxRDg2MUZGMThEOUVCMEVFRkIyNzY1RTZEMUIzMzEyNDIzNTFDOEM0RkUzRjk4MDc2NDYyRTJGMTRGQUY1MjZFQTY2OTAwMzI1RTEyNjY5OTkxOEQ4NzFBMDE0OHKABDVDQTlDQzRCMTJEQThGNzc4MDUzRDNGQTJGMTY1NTBERUU1NDFGNjBCRDJDMkM0RDRFQTQxQUY0NjZBMDMyOTgwNTU4RUVFQzBFQTVCMkQyQkZCQjE0QUE1QUMyQTg1Qzc0RkU2MDQ5MDFCM0I1OUREMzE4OTNGODE2OTMzMEU2MzJDQUQxQzIzQzhCODFEMDE0QzE2NTIyRjMyMTJGMEZCNERFQkY1RUJGNzEzQ0NEMzBDQjQ3RkEyRTBDMERGRUNEOEI5RDE4OEU2RDA4RDVEMkFFNjZFMTM2QzAwREY5RDM4RjE2MTlCNzNDQzI4NDYzNzhGQTgxMEU1NkQzRjUxQ0M5Q0Q5QzA0N0RENzU4QUExQ0Q5NDE0RjRENTMyODkzMkYwQkNCRTQ5N0QyRjEyM0U3QjI0OUE5MzA3OTkwRTg0MUVFOTkzRjMzRTUzQTFFOEM3ODE2RUNDODRBQkYzRDJFNEI4RDM4RTUzOTdGMzU3MEJFNjYxNEFCQjQxOEFCRjg1MjhBNkVBRDQ5Mjc0QTA5NzA0NzBCNjY1NjFBNUMyQjhEOTVENUY2OENFNzVCQUQ1ODBCODI2RTU2RkY4Mzk2NzVERTdEREFBNUU1RjMyQjgzMjAyRDhCMTE4RDlCOTczRDEwMDc2RjU0OTU4QjkxRjMzQTYyOTBFMjQ5MrQVCgpjb19zaWduZXI0GgIwNDqGCAqABEZCMTY4RTNENTRERDk4RDFDN0YyMkRCNDBBQUEzNkY3N0JBMENCNEQ1Mjk5QjVDRTNCODYzQTIwQjRDODhBMDNCRjYyOUU1REZGMDAxRURBOTQzMzg5NjhEREEyQTY1MzMxODc2MTRCQjcyNTg4NUY2RkM3RTYyQjI4NzhBNDk1OUU0OTgxOTZDOEQ5MTEyREM4NUU0NkUzM0E3RERCMzRDNjlCQzk2NDcyOUQ2RTBCOURDMkYzRDgwNEQ1QTY2QjVBQUI4MkJBQUZFRTREN0E5NkY2QUIzMUEwN0U2NUU1NDZERUUyMUFGMTVFMTlEMEY5NDJDRTQ2MThDQUFENThCRDYwRUExRTM4NTAzNkU4QTQyNkIwNTY4QTU3NkI1Q0NFMTczNEMxMTcwRDg1NjQ3QjIzMTYxN0NFMDdCODc0RTk5MTYxODFDMEZBNTZDREQ5ODg1QjYwNDcyMEFDRkE4QzBDNzM0ODVERTcxQ0RBMkEzNzZDMjJCQkQ3QTRENjRGQzcxQkVCNjE4NEM5QTk2Mzc3RUQ3NEUzN0NCRTFFOENBNkY1NkE2OTBGMjhFNjU3RTBDMUQ2OTYxQkMyNDQ3M0QzODIyRjNCMkMwMTA1REExODk3OEUwQzQ3MUUxRTk3Mjk3MEIxQjg3NkE2Qjc1MjI0M0I5NEMyQTc4MDU1EoAERkIxNjhFM0Q1NEREOThEMUM3RjIyREI0MEFBQTM2Rjc3QkEwQ0I0RDUyOTlCNUNFM0I4NjNBMjBCNEM4OEEwM0JGNjI5RTVERkYwMDFFREE5NDMzODk2OEREQTJBNjUzMzE4NzYxNEJCNzI1ODg1RjZGQzdFNjJCMjg3OEE0OTU5RTQ5ODE5NkM4RDkxMTJEQzg1RTQ2RTMzQTdEREIzNEM2OUJDOTY0NzI5RDZFMEI5REMyRjNEODA0RDVBNjZCNUFBQjgyQkFBRkVFNEQ3QTk2RjZBQjMxQTA3RTY1RTU0NkRFRTIxQUYxNUUxOUQwRjk0MkNFNDYxOENBQUQ1OEJENjBFQTFFMzg1MDM2RThBNDI2QjA1NjhBNTc2QjVDQ0UxNzM0QzExNzBEODU2NDdCMjMxNjE3Q0UwN0I4NzRFOTkxNjE4MUMwRkE1NkNERDk4ODVCNjA0NzIwQUNGQThDMEM3MzQ4NURFNzFDREEyQTM3NkMyMkJCRDdBNEQ2NEZDNzFCRUI2MTg0QzlBOTYzNzdFRDc0RTM3Q0JFMUU4Q0E2RjU2QTY5MEYyOEU2NTdFMEMxRDY5NjFCQzI0NDczRDM4MjJGM0IyQzAxMDVEQTE4OTc4RTBDNDcxRTFFOTcyOTcwQjFCODc2QTZCNzUyMjQzQjk0QzJBNzgwNTZKjwEKQDZBMEM5RDc1RUVDOUFBRUQxNzUxNzhBQjRFMzk2RjlCQUU4MzEzOUYxOTIzMkU5NzZEOTdGNDAyNUQ5MjFFMEISQDU5QUFGMjM2Q0I2NzRDQUIyQkNBMDQzNkEzRjI2QTNERkUxNzVGQTY3MTFGOEYzOUY3RTYzMUQ1QUQwOEY2MDEaCXNlY3AyNTZrMVKABEIxRDQ3NUREMTA4MjRCODY3Q0FFNzUyQjQyMDk0OUM4NkMyQkVBQzhCQjFEQTg4Nzk5Q0UzNTA3RjM1RThBODk1QzJEREYyNTI3RDA4NkQ2NzBCRUYxOUZGODgxMDg3RjBENDQ3MEE1MTZFM0Q0OTNBNDRERDJGQ0YzMTdFMUExOTNGRTc5ODc5MzJENjk2N0ExNzc2NTNCQTgxNEQwNkYyNzZBODkzRUZBNjAyOENEODlBNzVCQTY3NjNERTRGNkZGQTQxOURFQ0VBMkRENThFN0Y1M0VGRkQzQjYyMzBEN0IxRkQ1MEU4N0IxODMxNjFCMTU4REE4QUUzMjI2REU1MUY0RTBDQzMzM0E2NjM4RTNGRDk3M0Q3REIyQTgzRDFGNTQxQUM1RDQyQ0MzNDQ0ODhEMTE5OUYxMzdFRURCMjJCQ0RGQkNFMkE5NDVFODBBOTc2RTQxMTkyOEU0MDJBOTNENTNEOTAzQ0Y3RjJBMUFBNEU5RDg4NjZENjlDRDdERDBFMTNCOTA2M0YzQzQxQUE5QjQyOTE0NzE2OUQxQTRDNDgzMjc1MTJENEFDQjJFQkZDMzhENzgwRTg4RTcxNTYxQUMyQUY2NDhENjVBN0ZEMDlGOUREQTE5NUEzREEzOTUwQkNCOUJEMjhBMjhDQTM5Q0Y2QkQyMkJDMEVEaoAENEU3OTZBQTlCRkQwQTU1RjcyRDc1MUM3RUE5M0ZFRTE4Rjc3NTdEMThEMjlBNzM0NDkwNDlGNjBEQzY5RkQxQjI4MDA0Nzg3MEZENTZCN0ZFNTJBREM3OUFGQ0QzRjlFMjE0NERBQ0FCRTUwMjcxMUY2RDVEMkNBRDUyMThDQjYzQzhGNkRBMjE3MDdEOTUzQzk1MTFFREFDODRCREFDNEJDNzExRUM4NkZBMjVCOUMxMkZDMzZGQTk0RDgzRjg4MjEyNEMyNEQ0MzRGRkM4NjJGNTVGQjRFMkMzRUJBNTRFNTYyMzdGNDRENzM5NDM5NEFDOUU3RDVBMjIzRkM5MUEwQ0VCQkIwRkE4ODM3MzAwRUJBOTEwMzkxNDNDMTZEMkI0QUM3QzA5RDQ4REE1OTdEN0Y1OEM4QjI4NTVDMEQ4MDhGRjdBM0M5QzEwMjczNDk2MDAxRTEzQkEyNzkzRDdFNzM3NjY5MEI0Q0RFMDE1Mzk5OTBDNDMyNDQzOEM2NzBBQzE5NzU5QUYyMzNFMTEzRTAwQzgzRThFRDUwRjVGN0Y3MDIxQzc2NkE2RUFGNDVGODc4Mzc3OTkyN0UxRjAyRDg5MDA0NTJDREI1MkM5OUREOUM3Qjg3RTBDM0MzNDgxQTZDMDg0NTZFQkM4OEE2OTgyMzYyRUNGODNCMEZygAQ2MkI2ODA2MkZBOEQxOEJFOUFBODEzN0E2MjYwN0MxNjRFMUYzNzg3NEQwODBCODU4NzZEMDI5NzE5RDAyMDFCODY1RTY5QzM3QUM4RDc3NDU0RTFFMkFDMUVFNjg4MEY0RTYxMTI2NkM4MDIyMTNDQkY1MDVBOEFBRkRCRTBCMzYxOUIzM0I2OTU3N0VDOTUzQzdBRkNEMjFFNDhCNzdCQjNDMzgxODk0NjU1MjcxNTgwQjE4RTcxMkQ5Qzk1QzkyRURBRUY3NDcwQzc2QzgzNkY5ODg5RDExNzM5QzYxNDMxQ0Y4RDVDOTFEOTZDNzgzNTA1MUUzQTg1NDU3NzQ5OUI0NkE2MDEyMDNCRTA5NTVEODAxMzRFQjcxQzhBRjkwNjdBRjI1OUM4Q0EyMkQ0NTRCQzFCQjU1RTNEQjUwNjFDMDUxREQwRkI4MjA4M0U1MjQyQzBEMTkzNjAyNTMyQjY5MEY1NTBBQkVCRDhGQTBDQjQ2NjAzNzBFNTA5RDdFNTA2OEI4NTlEOTQzMDY3Q0QwNzIzQzEyRTA5MDcyRUMzRDgyMzMxRDRCNzI5NDdFRTBDQTlEREQzMUI4OEQ4Q0RDQzgxRUQzMzhDQkEzMzhCMjdCMDg0RTEyMjI1RUMxRjAwOUM2RDBEM0M3RDBCRUZDNzY5NUZCMjMwNUQwMDK0FQoKY29fc2lnbmVyNRoCMDU6hggKgARCQUEwOUJEREMyRkJEMzlCMzAzMzNCMERCNEYyNDRDNERGNzMwMDI1M0Y5MEJDQ0U4MTM0MTFENDkzQUE5MDgwRjRGQkVCOURDMERCM0UxM0RENjQwM0UwMjA0REVCRkQ4QTU2NDc5NDE0OEY4RkExRDE5MzkwMEIzN0UyQkZEQUY1Mjk3QzFGNDJGQTdGNjQxN0E2N0I1M0ZCRTNEMUYyNUNENDc4RDI1MkVFNjQwREY3QjMwQUU0NTA4MkFDQzZCQjA1QjE3ODcxRDhFOEQ3QjhFQjE4RTI5MjAyQjEwRkE2MDc1MEFEQzk3QkJFMDk3RTg4NUJDMzI4NTI2OEM2OTVDQzNDODZFNEZFN0I4QTZEMjE4NEUzRTRCMERDNTc5QUU4MDVCNkFCMUE2RDRFMzI0MkM0RjNGRTVGQ0Y0OTUzNEVCMkNFQzM0N0YzMUYzQzQyQjMwMkU5N0UyRUYyQ0EzQzQ0MUIwRjJGMTk2RjFBOTAzQTBEQjcyNzJGODExMEVCN0ZEQUI0OUIwNEQwRkZCQ0VFNURBQTkxRTJENzczMUQ5Mjg1QjFCMUUxMkY3NTM1QUEyMjVBMUEyOTAxOUIzNDhDNTBEOTYzQ0U0NjA4QUVDMjNERkEyM0RCNUJDRkNFNDI1ODk1QTEwRkJCNkQzMzBCN0JBRTk3NDM1RBKABEJBQTA5QkREQzJGQkQzOUIzMDMzM0IwREI0RjI0NEM0REY3MzAwMjUzRjkwQkNDRTgxMzQxMUQ0OTNBQTkwODBGNEZCRUI5REMwREIzRTEzREQ2NDAzRTAyMDRERUJGRDhBNTY0Nzk0MTQ4RjhGQTFEMTkzOTAwQjM3RTJCRkRBRjUyOTdDMUY0MkZBN0Y2NDE3QTY3QjUzRkJFM0QxRjI1Q0Q0NzhEMjUyRUU2NDBERjdCMzBBRTQ1MDgyQUNDNkJCMDVCMTc4NzFEOEU4RDdCOEVCMThFMjkyMDJCMTBGQTYwNzUwQURDOTdCQkUwOTdFODg1QkMzMjg1MjY4QzY5NUNDM0M4NkU0RkU3QjhBNkQyMTg0RTNFNEIwREM1NzlBRTgwNUI2QUIxQTZENEUzMjQyQzRGM0ZFNUZDRjQ5NTM0RUIyQ0VDMzQ3RjMxRjNDNDJCMzAyRTk3RTJFRjJDQTNDNDQxQjBGMkYxOTZGMUE5MDNBMERCNzI3MkY4MTEwRUI3RkRBQjQ5QjA0RDBGRkJDRUU1REFBOTFFMkQ3NzMxRDkyODVCMUIxRTEyRjc1MzVBQTIyNUExQTI5MDE5QjM0OEM1MEQ5NjNDRTQ2MDhBRUMyM0RGQTIzREI1QkNGQ0U0MjU4OTVBMTBGQkI2RDMzMEI3QkFFOTc0MzVFSo8BCkAzMDdCRTEyNEE2MjI4NkRBRDlCMDAzQURBREMwQUYyRUI3NTE2NzU3Q0UxNzA5OEYwRjMxNDBBMTkzRUQ1MDNEEkAxMDdCRkIxQUI1MkU2RjFDN0UyRjUzNDIwQUNGMjBBMDUwMzE1MTNFNDVGNTY1M0I3MTUzMTUxNzAyNzZFRjJCGglzZWNwMjU2azFSgARDODMxMkM5MUNDQzM5MTI0REY5RDkxQUZCRjk5QkMxQ0EyNDY5RTUzQjNCRjlGQTE3REZBQjEwNDUxQkE1OTNEREZDOUU0RTlFQzA1MzAwREQ5QjREOTgzOUU0QUU4NzgyQ0E1RkYzQUY5RDY3QjE3RDlGMEQ5MjY4MEI3RDcxMDBDMzFFOTE3QjQxREIxNUMwRDY0MTYyRkZGODlGMzg0NkFGNThBQjI1MTk2RTNGNjIxNkE3MjFEQTc0NERCRjczNjBDOTFFQUJCNUZDMzMwODYyMUZBMzQ2RkU3RDE1MDJDOTU2MDk4QTAxOTNDRTYxNzNCQzNDNTA5ODA5QzUxMkJENzYyMkUyNDAwRjFDNDlDNkNCOTUxQ0MwMTgyMzM5NkNFNkVGMDMyNDFDQTBBNURCOEU0NzU4QTJDOEMwQUQzNDE4N0Y1MjRGM0ZGRTlEMEMzNkE1RDQwNTU2QTBBNDBBRjk2OEE5MEVCNkQyQ0JBQzNGQTEwRTk1NkIzNjk3NjlDM0QxRUM4ODBBMzAwQkQ5RTIyRTEzMzlCNTQwMEM0RUU4NTJGMTYwODYyMjI3MjFCMkZCNEFGNjRFQUJFQ0U5Rjc1NDgxQTg2QzEyREMwRkYxRkFDOUFGQUI3NTY5RTVFQUE3QTlEOTdDMjg0MDNGMUQ3RUUyRTg4NjFCNWqABDg5NjBCOEJEOUNFN0MzRTY5QzA4M0ZEOEVCRkI0RDU1NDhGMzlBRjlFMkM3MDI3Q0EzRjJBQ0UxREI5MzY1NENCODBBQjVEQTM4MjUwODU0RjUyMDAxN0MxMDQ4Q0U1MzI4MEUxRDlBNTRCOEUzMjY3NzJEODZFMUU1NzlBNTY2MjJCNURBMTRDQTQxRTdGNUE3RjQ3OUQyQUMxNTI0MDk5REQzQUM5RUIzQjE2MjY0OTI3MTVGMkRFOUY5NDhFNDNFNkM1NTJBNjhFNzY0RjRBQ0U1MEY1OUI4MDA5RDMyMDg3MDAzNjlGNkUxMkUwM0RDOTY1N0VDOTZBMDlBMDQ2QkM1QjNFQ0NBQzVEQzY4MUVDRDk4QkZEMEY2QTY0RTkyQjRBMUE5ODk2NzcwRTYwMUI3MkFFRkVBQjQ5MzVGMTc4RkE3OThEODhGQzA3OUVCQzVEOTA2RjI3NEYxNEFDMkM1RkVDRjQ3MjRDNzYwNkMxMUU1QjcwOENDREJGNDI4QUU0NUVGQTVBMzNDMUZGQ0VCNDY3M0JCRkEwREVBOEI5OEJFMDY1NTc3MTAxRjg0NkExNTVFN0Y0Mjk1NURBNUFENzg3N0EyMkI2MjFBOTQxQ0I1QjRBQUYzQjRGRkY3MjRFRkNFRkYxMzlCMjhCMTI0NjgyRURFRkEwNkRBcoAEMzYyMkUzRjY2QjVGMDhDMThCNzU3MjdENDZFNEQ5NkRFRTY3NTg2RUNENjdFRkRGRTQzQTc2NzA4MDU1OUI1Q0FDNDE5NEIxMzg0MzFDOEIyNEFCODY4RUREMDBDNTM1MDIwODNGQ0Q2ODQ0ODYyMUU1OThBOEYxODUwQUU1ODg4NTJBMTMxODlBMzcwNDkwRjZGQjExNEZENUM3RjdBOUU5QjUyNUExMDIyQTA5MjYxQjE5QjY2NDYxQ0FGREQ0QURBNEMwNTRGOUI4MjMyRDc0NkZFRkI3NDExREU3RjVCM0U3NUU2NTcxMDQ5NkVCM0M5RUI1NDU0RkQyNzY5NjQwQkExNzU5MzE2OUM5NUI0RTAzMEExODBDNDFDRjU4NDgxMDU0NkRCRURDNzI3OEY1RDdFNzg2Q0EzREYwQjIzMEE3OUM3RDY3NkVCOUQ1RTNFMDZCQUJEODhGODE3QkQ0NjYzMDczNTlBOUVCRDkyMkIyQ0U4NDkzNkVENEIxMDMxNTkzNzJGRUZCQUU1MDRENDBDNERCNEU4REJGRTVFNTA4QzI5RjA0OTE2RTZDODM0REVEM0JFNDMwRjczQjg2NzU5REI5MjJBNzgxQzk0MUZFQUZBNDNFRTQ2MjJEMUREMjZDN0QxMjQ2MTM5M0QxOENCOTc2QUZCRDg2NDQ6jwEKQDY2MDgwRTAyRjgzNjI1OUU0M0UzREE3MkYyOTFENjY5ODBERkQ0NTc3Q0JEQjVFQ0UzMUQ3M0M5ODU1REYyMDkSQDU5NDNBRUYzMkM5MjYwRUE0Q0U2ODc0NjM0QzY3NkI1Q0Y3RUEzOUEwRUNEMEUxRUNBMzNGQUQxNTg0N0JGOUQaCXNlY3AyNTZrMQ.."
  • The secret share in json format:
{
 "workspaceId": "workspace 0",
 "threshold": 3,
 "nParties": 5,
 "localParty": {
  "partyId": "co_signer2",
  "index": "02",
  "pailPriv": {
   "n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
   "lambda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
   "mu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
   "p": "D464633AB48FB48ADA26EFDDF729261F83797B3E6C1F6C173D1F1D81D225513E208B4BE937CE81B492CE1BD44FDC0A131A04727F806B20B01EF4E9548C4D05363DF3139F2621DE98870DAE3279FE0017D93B6D0EC55AFDE2A4FB0FBA86EA2C6BDDFD348510501CACB279FE21F4C7D2B75F87E695DC5FA3A0A7655817421F579F",
   "q": "C69018B8F7B20E7DB8F4C1A48E094849328FFFC49879863DA46EAB36745205C409094BB23D3326D12DDE1AC98B828DC00BBA5645DBE2FC90507CAE492E69CC2A46C49C7324D363EA377867ABF18350F5B6EF1CEB53494AB573FE9B36AB15EA3654DEAF9FF6CB662D9B7D45765AE57BACBEB5B50BDB8A42B1049ABA380FA4D99F",
   "pSqr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
   "qSqr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
   "pMinus1": "D464633AB48FB48ADA26EFDDF729261F83797B3E6C1F6C173D1F1D81D225513E208B4BE937CE81B492CE1BD44FDC0A131A04727F806B20B01EF4E9548C4D05363DF3139F2621DE98870DAE3279FE0017D93B6D0EC55AFDE2A4FB0FBA86EA2C6BDDFD348510501CACB279FE21F4C7D2B75F87E695DC5FA3A0A7655817421F579E",
   "qMinus1": "C69018B8F7B20E7DB8F4C1A48E094849328FFFC49879863DA46EAB36745205C409094BB23D3326D12DDE1AC98B828DC00BBA5645DBE2FC90507CAE492E69CC2A46C49C7324D363EA377867ABF18350F5B6EF1CEB53494AB573FE9B36AB15EA3654DEAF9FF6CB662D9B7D45765AE57BACBEB5B50BDB8A42B1049ABA380FA4D99E",
   "hp": "2CF1BD696CFCFFFA7E3C647A7209AF77F4873133F87B989BA3F4A771A1845151A8FDC128C05B7DFA12DB80482340E4136EBDEF2858B32DA99CB40D6A76D81CAABFBEA8C4B7BA4AB3FA22B2C0542FC461A99F23BD8ED4BB47BD6F4AB1252D4B4FA9D91F566109D3C209547C4A6C4CB7BC335D3D06CBD4E4341D12D031FF2681B2",
   "hq": "9C8B8481850CBF0B48F2D6CD734C2ACB2F8F7DF8204A911EC97B261B255F75FA58DCD003BD70B5438323D9EED208EAFC6E33D23CC3210F63D1A213ADD49FF3FC2E380BED818B03EEECA8B3A822F5E4FA63212161A43E09A996132A6D8862F1D89F32FC41A67F541DAEBA7DA1E47A281A279FD919B08CFB475BAEFD5A48DF378E",
   "qInvP": "A772A5D14792B4905BEA8B63851F76A78EF24A0A73A3D37B992A761030A0FFEC778D8AC0777303BA7FF29B8C2C9B25FFAB46835727B7F3068240DBEA1574E88B7E346ADA6E6793E48CEAFB7225CE3BB62F9C49513686429AE78BC50961BCE11C3424152EAF4648EAA92581D7887B1AFB2C2AA98F108ABF6C8A5287E542F8D5ED",
   "pInvQ": "2A04943772A54F727001EAD71ABD1D7E030081CC782EF51EDAF3851B4EF28FC9B02C7BAE7FC2718DAABA40DAB979A2C39D86840918C1ED2C7EDA9A9B59C9D82E188C9085A3485FFB4ACFB403CE8D6BFB53CDFB89AF0B410BDDEB70C922B2F85DB5ABB35E504C120FECC2C7D4766B53929715DBF22AFD4769A8EBBCDDC6C5A211"
  },
  "pailPub": {
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   "index": "05",
   "pailPub": {
    "n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
    "g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
   },
   "gX": {
    "x": "307BE124A62286DAD9B003ADADC0AF2EB7516757CE17098F0F3140A193ED503D",
    "y": "107BFB1AB52E6F1C7E2F53420ACF20A05031513E45F5653B715315170276EF2B",
    "curve": "secp256k1"
   },
   "NTilde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
   "h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
   "h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
  }
 ],
 "gX": {
  "x": "66080E02F836259E43E3DA72F291D66980DFD4577CBDB5ECE31D73C9855DF209",
  "y": "5943AEF32C9260EA4CE6874634C676B5CF7EA39A0ECD0E1ECA33FAD15847BF9D",
  "curve": "secp256k1"
 }
}

Key Refresh

Refer to the test case for more details.

Sign

Refer to the test case for more details.

GG20

Sign

Refer to the test case for more details.

CMP

Minimal Key Genaeration

Refer to the test case for more details.

At the end of the MPC Procedure, each signer will get the secret share:

  • The secret share in base64 format:
"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"
  • The secret share in json format:
{
 "threshold": 3,
 "nParties": 3,
 "localParty": {
  "partyId": "co_signer1",
  "index": "01",
  "x": "4FD09F2AE63BFB995ABE1BA9163D83623BBE91629D9BE0204A58A51811141E80",
  "gX": {
   "x": "1DCD11C5D9099B70700CFCCFC6DAF4823964E6772A9A98D443E9AD762DAA483D",
   "y": "29BAC2A1961B1A6A4863B7F155FFB39AFAE127C2CFA9989D025DB4B6A6C46FAE",
   "curve": "secp256k1"
  }
 },
 "remoteParties": [
  {
   "partyId": "co_signer2",
   "index": "02",
   "gX": {
    "x": "DE13E92D582AD635F45EA9D4AC45C8BBA589166F6E6FEB02A26447EEEC2EB03A",
    "y": "69FFE7777D28E28A4229DD3EE0EA7D249CAE856D8B5D258533CFD501906978A3",
    "curve": "secp256k1"
   }
  },
  {
   "partyId": "co_signer3",
   "index": "03",
   "gX": {
    "x": "09E05B38E939A3894DBD0D37B1DC91BDE7CA35064965580E669894564BE4548F",
    "y": "EAC9E6C4602D658EC1FC08CCA24BCEBAE0107B4A56AE4894D4130CABBFE0A824",
    "curve": "secp256k1"
   }
  }
 ],
 "gX": {
  "x": "DFEA26A0B1398DF52D802913DB12C1D0692F050047E7B3BCA2D34DCFC3669F14",
  "y": "B27F26A0C38E8E9B04B2D7E6D602D4C0B3C5DCEFCBAA50ABE7CFB0191ADC2BFD",
  "curve": "secp256k1"
 }
}

Note that the minimal sign key couldn't be used as the input of protocol "Sign".

Auxiliary Info and Key Refresh

Refer to the test case for more details.

Note that the minimal sign key, the output of protocol "Minimal Key Generation" should be used as the input of protocol "Auxiliary Info and Key Refresh".

At the end of the MPC Procedure, each signer will get the secret share which could be used to execute the mpc protocol "Sign".:

  • The secret share in base64 format:
"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."
  • The secret share in json format:
{
 "threshold": 2,
 "nParties": 3,
 "localParty": {
  "partyId": "co_signer1",
  "index": "01",
  "x": "D427D9FFE3A948EBA5AE2ED5FBCD3FADF0C39D8377F4DA118E5D349D070CBCC5",
  "gX": {
   "x": "963EC791F8AA2E19DB4B032A8A6E3581CA75A2E0556835E1C5B2D23D9713CD9C",
   "y": "B347E42471A81B067C3BFA78CBFE276F7AE381ADB742769E1EA41818EE9D83A4",
   "curve": "secp256k1"
  },
  "gY": {
   "x": "35295FB1B4DC3FBFA0C5FADEE37CE88706B009CA4C44C269139A6FDF0AA3519C",
   "y": "4A195A2C1DFD5E3CF0238F6DFA9E0EAE88098608D37B59BB1181198078BAF1B8",
   "curve": "secp256k1"
  },
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
  "s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
  "t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
  "p": "DF1C1C7FE555507BC514413D78AFAC1214C86FB0395664EFBC2E44DD11EA38928772F24D792FD10B126441EFD5FD1AB0AD0D7F13AC806DFC6A4F342F13B0219251D0D4AD7AD43AAC0555586587EE57186F1C436383B20C2FE8517F5E49448B2E02EACA19A55CB4FE250799E9AD1DA2BFDE8D32FBE3610034DFC941CA9489852B",
  "q": "EA5D97AA23C1243E577F6107EFADD101377C2E52E9F88B3E28344FE84F098562646B58123103C5B56A02CB4067FC63E371FB235E194C8437D2D9359D8EC5025D82970A4063004BC2313CC33DEAB4989F87F12F21A6DFC2107335BDC5E3856B3458946E35499B04D9378F3FE62BB693A2DBDF62561DB08A1E644716B373538683"
 },
 "remoteParties": [
  {
   "partyId": "co_signer2",
   "index": "02",
   "gX": {
    "x": "105BC27C54023A5143CD96AE11E41DB68B83024B95F8E3A9FF8A15BCE5C547DC",
    "y": "020C31C3A6FB859C399B21E3220C638CC56618E15391DA298619479CD3FE7966",
    "curve": "secp256k1"
   },
   "gY": {
    "x": "174B7A368F939A991AB89E1343ACE41A53B336B7209FB874DF98A49E760D9C10",
    "y": "DC0126367F9F55D7DF11EF6524D34C87BCB65976ADA28CEDAA00A4DC52354050",
    "curve": "secp256k1"
   },
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
   "s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
   "t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
  },
  {
   "partyId": "co_signer3",
   "index": "03",
   "gX": {
    "x": "A528F8605F5BDCA6B92CC5AD64649F171F004D69527364F52B043BAF2D0EDC19",
    "y": "2F5B8759E1A01771EB31D349B16E59071363172ED2B2D1B308A56117B793C8C1",
    "curve": "secp256k1"
   },
   "gY": {
    "x": "0E2A7B6C75BF04DDA6206216A9107226BBC4C55BDB98E5163B8D990F3A59D1FB",
    "y": "3036E4B8283E10776209EB9DB900C1D93ECE550DC0C58B807A2E55FC2F33FB71",
    "curve": "secp256k1"
   },
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
   "s": "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",
   "t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
  }
 ],
 "gX": {
  "x": "A2A4C6D8A47A6889D5A188923C7701D4776AB5748920D077C9D747C53AFED33B",
  "y": "747AE5DF5936B3DCBF2F6F6ADD7B2B2F17159BC8CCA3A3B0DBF4ED9CDF2D1902",
  "curve": "secp256k1"
 }
}

Sign

Refer to the test case for more details.

Important Tips

  • A reliable broadcasting channel is needed. Messaging is left to the application layer, which is not provided by this library. There should be a mechanism in your transport that allows "reliable broadcasting", which means that parties can broadcast messages to other parties so that everyone is guaranteed to receive the same message. This can be done, for example, by sharing and comparing hashes of received messages.
  • Refer to the revised version of GG18 but not the old one. This revised version of GG18 fixes some crucial details in the protocol. The proof of the protocol described in the previous version is not correct, though no attack has been shown that exploits the bug in the proof.
  • The ZK proof of GG20 refers to the revised GG18 and CMP.
  • There some typos in the CMP[3]. More details in "Modify MPC-CMP as a Threshold Signature Scheme"

Security Audit

The GG18 protocol originates from a previous repository developed by Safeheron and audited by Kudelski Security. There are some updates after the audit due to the attacks on the protocol[4] [5] [6].

The CMP protocols has been audited by LeastAuthority in 2023 and we have upgrade the protocol according to the opinions of the LeastAuthority, and this is the audit report which can be found in LeastAuthority website.

Reference

[1] GG18: Fast Multiparty Threshold ECDSA with Fast Trustless Setup
[2] GG20: One Round Threshold ECDSA with Identifiable Abort
[3] CMP: UC Non-Interactive, Proactive, Threshold ECDSA with Identifiable Aborts
[4] A Note on the Security of GG18
[5] Alpha-Rays: Key Extraction Attacks on Threshold ECDSA Implementations
[6] Attacking Threshold Wallets

Development Process & Contact

This library is maintained by Safeheron. Contributions are highly welcomed! Besides GitHub issues and PRs, feel free to reach out by mail.

About

This project is a C++ implementation of {t,n}-Threshold Signature Scheme

www.safeheron.com

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mpc ecdsa ecdsa-signature safeheron mpc-wallet mpc-tss

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