Multi-Protocol SPDZ

Software to benchmark various secure multi-party computation (MPC) protocols such as SPDZ, SPDZ2k, MASCOT, Overdrive, BMR garbled circuits, Yao's garbled circuits, and computation based on three-party replicated secret sharing as well as Shamir's secret sharing (with an honest majority).

Contact

Filing an issue on GitHub is the preferred way of contacting us, but you can also write an email to mp-spdz@googlegroups.com (archive).

TL;DR (Binary Distribution on Linux or Source Distribution on macOS)

This requires either a Linux distribution originally released 2011 or later (glibc 2.12) or macOS High Sierra or later as well as Python 3 and basic command-line utilities.

Download and unpack the distribution, then execute the following from the top folder:

Scripts/tldr.sh
./compile.py tutorial
echo 1 2 3 4 > Player-Data/Input-P0-0
echo 1 2 3 4 > Player-Data/Input-P1-0
Scripts/mascot.sh tutorial

This runs the tutorial with two parties and malicious security.

TL;DR (Source Distribution)

On Linux, this requires a working toolchain and all requirements. On Ubuntu, the following might suffice:

apt-get install automake build-essential git libboost-dev libboost-thread-dev libsodium-dev libssl-dev libtool m4 python texinfo yasm

On MacOS, this requires brew to be installed, which will be used for all dependencies. It will execute the tutorial with two parties and malicious security.

make -j 8 tldr
./compile.py tutorial
echo 1 2 3 4 > Player-Data/Input-P0-0
echo 1 2 3 4 > Player-Data/Input-P1-0
Scripts/mascot.sh tutorial

Preface

The primary aim of this software is to run the same computation in various protocols in order to compare the performance. All protocols in the matrix below are fully implemented. In addition, there are further protocols implemented only partially, most notably the Overdrive protocols. They are deactivated by default in order to avoid confusion over security. See the section on compilation on how to activate them.

Protocols

The following table lists all protocols that are fully supported.

Security model	Mod prime / GF(2^n)	Mod 2^k	Bin. SS	Garbling
Malicious, dishonest majority	MASCOT	SPDZ2k	Tiny / Tinier	BMR
Covert, dishonest majority	CowGear / ChaiGear	N/A	N/A	N/A
Semi-honest, dishonest majority	Semi / Hemi / Soho	Semi2k	SemiBin	Yao's GC / BMR
Malicious, honest majority	Shamir / Rep3 / PS	Brain / Rep3 / PS	Rep3 / CCD	BMR
Semi-honest, honest majority	Shamir / Rep3	Rep3	Rep3 / CCD	BMR

Paper and Citation

The design of MP-SPDZ is described in this paper. If you use it for an academic project, please cite:

@misc{mp-spdz,
    author = {Marcel Keller},
    title = {{MP-SPDZ}: A Versatile Framework for Multi-Party Computation},
    howpublished = {Cryptology ePrint Archive, Report 2020/521},
    year = {2020},
    note = {\url{https://eprint.iacr.org/2020/521}},
}

History

The software started out as an implementation of the improved SPDZ protocol. The name SPDZ is derived from the authors of the original protocol.

This repository combines the functionality previously published in the following repositories:

• https://github.com/bristolcrypto/SPDZ-2
• https://github.com/mkskeller/SPDZ-BMR-ORAM
• https://github.com/mkskeller/SPDZ-Yao

Alternatives

There is another fork of SPDZ-2 called SCALE-MAMBA. The main differences at the time of writing are as follows:

• It provides honest-majority computation for any Q2 structure.
• For dishonest majority computation, it provides integration of SPDZ/Overdrive offline and online phases but without secure key generation.
• It only provides computation modulo a prime.
• It only provides malicious security.

More information can be found here: https://homes.esat.kuleuven.be/~nsmart/SCALE

Overview

For the actual computation, the software implements a virtual machine that executes programs in a specific bytecode. Such code can be generated from high-level Python code using a compiler that optimizes the computation with a particular focus on minimizing the number of communication rounds (for protocol based on secret sharing) or on AES-NI pipelining (for garbled circuits).

The software uses two different bytecode sets, one for arithmetic circuits and one for boolean circuits. The high-level code slightly differs between the two variants, but we aim to keep these differences a at minimum.

In the section on computation we will explain how to compile a high-level program for the various computation domains and then how to run it with different protocols.

The section on offline phases will explain how to benchmark the offline phases required for the SPDZ protocol. Running the online phase outputs the amount of offline material required, which allows to compute the preprocessing time for a particular computation.

Requirements

• GCC 5 or later (tested with up to 9) or LLVM/clang 5 or later (tested with up to 9). We recommend clang because it performs better.
• MPIR library, compiled with C++ support (use flag --enable-cxx when running configure)
• libsodium library, tested against 1.0.16
• OpenSSL, tested against and 1.0.2 and 1.1.0
• Boost.Asio with SSL support (libboost-dev on Ubuntu), tested against 1.65
• Boost.Thread for BMR (libboost-thread-dev on Ubuntu), tested against 1.65
• 64-bit CPU
• Python 3.5 or later
• NTL library for homomorphic encryption (optional; tested with NTL 10.5)
• If using macOS, Sierra or later

Compilation

1. Edit CONFIG or CONFIG.mine to your needs:

• By default, a CPU supporting AES-NI, PCLMUL, AVX2, BMI2, ADX is required. This includes mainstream processors released 2014 or later. For older models you need to deactivate the respective extensions in the ARCH variable.
• To benchmark online-only protocols or Overdrive offline phases, add the following line at the top: MY_CFLAGS = -DINSECURE
• PREP_DIR should point to a local, unversioned directory to store preprocessing data (the default is Player-Data in the current directory).
• For homomorphic encryption, set USE_NTL = 1.

2. Run make to compile all the software (use the flag -j for faster compilation using multiple threads). See below on how to compile specific parts only. Remember to run make clean first after changing CONFIG or CONFIG.mine.

Running computation

See Programs/Source/ for some example MPC programs, in particular tutorial.mpc. Furthermore, Read the Docs hosts a more detailed reference of the high-level functionality extracted from the Python code in the Compiler directory as well as a summary of relevant compiler options.

Compiling high-level programs

There are three computation domains, and the high-level programs have to be compiled accordingly.

Arithmetic modulo a prime

./compile.py [-F <integer bit length>] <program>

The integer bit length defaults to 64.

Note that in this context integers do not wrap around as expected, so it is the responsibility of the user to make sure that they don't grow too large. If necessary sint.Mod2m() can be used to wrap around manually.

The integer bit length together with the computation mandate a minimum for the size of the prime, which will be output by the compiler. It is also communicated to the virtual machine in the bytecode, which will fail if the minimum is not met.

Arithmetic modulo 2^k

./compile.py -R <integer bit length> <program>

Currently, most machines support bit lengths 64 and 72.

Binary circuits

./compile.py -B <integer bit length> <program>

The integer length can be any number up to a maximum depending on the protocol. All protocols support at least 64-bit integers.

Fixed-point numbers (sfix) always use 16/16-bit precision by default in binary circuits. This can be changed with sfix.set_precision. See the tutorial.

If you would like to use integers of various precisions, you can use sbitint.get_type(n) to get a type for n-bit arithmetic.

Mixed circuits

MP-SPDZ allows to mix computation between arithmetic and binary secret sharing in the same security model. In the compiler, this is used to switch from arithmetic to binary computation for certain non-linear functions such as comparison, bit decomposition, truncation, and modulo power of two, which are use for fixed- and floating-point operations. There are several ways of achieving this as described below.

Classic daBits

You can activate this by adding -X when compiling arithmetic circuits, that is ./compile.py -X [-F <integer bit length>] <program> for computation modulo a prime and ./compile.py -X -R <integer bit length> <program> for computation modulo 2^k.

Internally, this uses daBits described by Rotaru and Wood, that is secret random bits shared in different domains. Some security models allow direct conversion of random bits from arithmetic to binary while others require inputs from several parties followed by computing XOR and checking for malicious security as described by Rotaru and Wood in Section 4.1.

Extended daBits

Extended daBits were introduced by Escudero et al.. You can activate them by using -Y instead of -X. Note that this also activates classic daBits when useful.

Bristol Fashion circuits

Bristol Fashion is the name of a description format of binary circuits used by SCALE-MAMBA. You can access such circuits from the high-level language if they are present in Programs/Circuits. To run the AES-128 circuit provided with SCALE-MAMBA, you can run the following:

make Programs/Circuits
./compile.py aes_circuit
Scripts/semi.sh aes_circuit

This downloads the circuit, compiles it to MP-SPDZ bytecode, and runs it as semi-honest two-party computation 1000 times in parallel. It should then output the AES test vector 0x3ad77bb40d7a3660a89ecaf32466ef97. You can run it with any other protocol as well.

See the documentation for further examples.

Compiling and running programs from external directories

Programs can also be edited, compiled and run from any directory with the above basic structure. So for a source file in ./Programs/Source/, all SPDZ scripts must be run from ./. The setup-online.sh script must also be run from ./ to create the relevant data. For example:

spdz$ cd ../
$ mkdir myprogs
$ cd myprogs
$ mkdir -p Programs/Source
$ vi Programs/Source/test.mpc
$ ../spdz/compile.py test.mpc
$ ls Programs/
Bytecode  Public-Input  Schedules  Source
$ ../spdz/Scripts/setup-online.sh
$ ls
Player-Data Programs
$ ../spdz/Scripts/run-online.sh test

Dishonest majority

Some full implementations require oblivious transfer, which is implemented as OT extension based on https://github.com/mkskeller/SimpleOT.

Secret sharing

The following table shows all programs for dishonest-majority computation using secret sharing:

Program	Protocol	Domain	Security	Script
mascot-party.x	MASCOT	Mod prime	Malicious	mascot.sh
spdz2k-party.x	SPDZ2k	Mod 2^k	Malicious	spdz2k.sh
semi-party.x	OT-based	Mod prime	Semi-honest	semi.sh
semi2k-party.x	OT-based	Mod 2^k	Semi-honest	semi2k.sh
cowgear-party.x	Adapted LowGear	Mod prime	Covert	cowgear.sh
chaigear-party.x	Adapted HighGear	Mod prime	Covert	chaigear.sh
hemi-party.x	Semi-homomorphic encryption	Mod prime	Semi-honest	hemi.sh
soho-party.x	Somewhat homomorphic encryption	Mod prime	Semi-honest	soho.sh
semi-bin-party.x	OT-based	Binary	Semi-honest	semi-bin.sh
tiny-party.x	Adapted SPDZ2k	Binary	Malicious	tiny.sh
tinier-party.x	FKOS15	Binary	Malicious	tinier.sh

Semi and Semi2k denote the result of stripping MASCOT/SPDZ2k of all steps required for malicious security, namely amplifying, sacrificing, MAC generation, and OT correlation checks. What remains is the generation of additively shared Beaver triples using OT.

Similarly, SemiBin denotes a protocol that generates bit-wise multiplication triples using OT without any element of malicious security.

Tiny denotes the adaption of SPDZ2k to the binary setting. In particular, the SPDZ2k sacrifice does not work for bits, so we replace it by cut-and-choose according to Furukawa et al.

CowGear denotes a covertly secure version of LowGear. The reason for this is the key generation that only achieves covert security. It is possible however to run full LowGear for triple generation by using -s with the desired security parameter. The same holds for ChaiGear, an adapted version of HighGear. Option -T activates TopGear zero-knowledge proofs in both.

Hemi and Soho denote the stripped version version of LowGear and HighGear, respectively, for semi-honest security similar to Semi, that is, generating additively shared Beaver triples using semi-homomorphic encryption.

We will use MASCOT to demonstrate the use, but the other protocols work similarly.

First compile the virtual machine:

make -j8 mascot-party.x

and a high-level program, for example the tutorial (use -R 64 for SPDZ2k and Semi2k and -B <precision> for SemiBin):

./compile.py -F 64 tutorial

To run the tutorial with two parties on one machine, run:

./mascot-party.x -N 2 -I -p 0 tutorial

./mascot-party.x -N 2 -I -p 1 tutorial (in a separate terminal)

Using -I activates interactive mode, which means that inputs are solicitated from standard input, and outputs are given to any party. Omitting -I leads to inputs being read from Player-Data/Input-P<party number>-0 in text format.

Or, you can use a script to do run two parties in non-interactive mode automatically:

Scripts/mascot.sh tutorial

To run a program on two different machines, mascot-party.x needs to be passed the machine where the first party is running, e.g. if this machine is name diffie on the local network:

./mascot-party.x -N 2 -h diffie 0 tutorial

./mascot-party.x -N 2 -h diffie 1 tutorial

The software uses TCP ports around 5000 by default, use the -pn argument to change that.

Yao's garbled circuits

We use the implementation optimized for AES-NI by Bellare et al.

Compile the virtual machine:

make -j 8 yao

and the high-level program:

./compile.py -B <integer bit length> <program>

Then run as follows:

• Garbler: ./yao-party.x [-I] -p 0 <program>
• Evaluator: ./yao-party.x [-I] -p 1 -h <garbler host> <program>

When running locally, you can omit the host argument. As above, -I activates interactive input, otherwise inputs are read from Player-Data/Input-P<playerno>-0.

By default, the circuit is garbled in chunks that are evaluated whenever received.You can activate garbling all at once by adding -O to the command line on both sides.

Honest majority

The following table shows all programs for honest-majority computation:

Program	Sharing	Domain	Malicious	# parties	Script
replicated-ring-party.x	Replicated	Mod 2^k	N	3	ring.sh
brain-party.x	Replicated	Mod 2^k	Y	3	brain.sh
ps-rep-ring-party.x	Replicated	Mod 2^k	Y	3	ps-rep-ring.sh
malicious-rep-ring-party.x	Replicated	Mod 2^k	Y	3	mal-rep-ring.sh
replicated-bin-party.x	Replicated	Binary	N	3	replicated.sh
malicious-rep-bin-party.x	Replicated	Binary	Y	3	mal-rep-bin.sh
replicated-field-party.x	Replicated	Mod prime	N	3	rep-field.sh
ps-rep-field-party.x	Replicated	Mod prime	Y	3	ps-rep-field.sh
malicious-rep-field-party.x	Replicated	Mod prime	Y	3	mal-rep-field.sh
shamir-party.x	Shamir	Mod prime	N	3 or more	shamir.sh
malicious-shamir-party.x	Shamir	Mod prime	Y	3 or more	mal-shamir.sh
ccd-party.x	CCD/Shamir	Binary	N	3 or more	ccd.sh
malicious-cdd-party.x	CCD/Shamir	Binary	Y	3 or more	mal-ccd.sh

We use the "generate random triple optimistically/sacrifice/Beaver" methodology described by Lindell and Nof to achieve malicious security, except for the "PS" (post-sacrifice) protocols where the actual multiplication is executed optimistally and checked later as also described by Lindell and Nof. The implementations used by brain-party.x, malicious-rep-ring-party.x -S, malicious-rep-ring-party.x, and ps-rep-ring-party.x correspond to the protocols called DOS18 preprocessing (single), ABF+17 preprocessing, CDE+18 preprocessing, and postprocessing, respectively, by Eerikson et al. Otherwise, we use resharing by Cramer et al. for Shamir's secret sharing and the optimized approach by Araki et al. for replicated secret sharing. The CCD protocols are named after the historic paper by Chaum, Crépeau, and Damgård, which introduced binary computation using Shamir secret sharing over extension fields of characteristic two.

All protocols in this section require encrypted channels because the information received by the honest majority suffices the reconstruct all secrets. Therefore, an eavesdropper on the network could learn all information.

MP-SPDZ uses OpenSSL for secure channels. You can generate the necessary certificates and keys as follows:

Scripts/setup-ssl.sh [<number of parties>]

The programs expect the keys and certificates to be in Player-Data/P<i>.key and Player-Data/P<i>.pem, respectively, and the certificates to have the common name P<i> for player <i>. Furthermore, the relevant root certificates have to be in Player-Data such that OpenSSL can find them (run c_rehash Player-Data). The script above takes care of all this by generating self-signed certificates. Therefore, if you are running the programs on different hosts you will need to copy the certificate files.

In the following, we will walk through running the tutorial modulo 2^k with three parties. The other programs work similarly.

First, compile the virtual machine:

make -j 8 replicated-ring-party.x

In order to compile a high-level program, use ./compile.py -R 64:

./compile.py -R 64 tutorial

If using another computation domain, use -F or -B as described in the relevant section above.

Finally, run the three parties as follows:

./replicated-ring-party.x -I 0 tutorial

./replicated-ring-party.x -I 1 tutorial (in a separate terminal)

./replicated-ring-party.x -I 2 tutorial (in a separate terminal)

or

Scripts/ring.sh tutorial

The -I argument enables interactive inputs, and in the tutorial party 0 and 1 will be asked to provide three numbers. Otherwise, and when using the script, the inputs are read from Player-Data/Input-P<playerno>-0.

When using programs based on Shamir's secret sharing, you can specify the number of parties with -N and the maximum number of corrupted parties with -T. The latter can be at most half the number of parties.

BMR

BMR (Bellare-Micali-Rogaway) is a method of generating a garbled circuit using another secure computation protocol. We have implemented BMR based on all available implementations using GF(2^128) because the nature of this field particularly suits the Free-XOR optimization for garbled circuits. Our implementation is based on the SPDZ-BMR-ORAM construction. The following table lists the available schemes.

Program	Protocol	Dishonest Maj.	Malicious	# parties	Script
real-bmr-party.x	MASCOT	Y	Y	2 or more	real-bmr.sh
shamir-bmr-party.x	Shamir	N	N	3 or more	shamir-bmr.sh
mal-shamir-bmr-party.x	Shamir	N	Y	3 or more	mal-shamir-bmr.sh
rep-bmr-party.x	Replicated	N	N	3	rep-bmr.sh
mal-rep-bmr-party.x	Replicated	N	Y	3	mal-rep-bmr.sh

In the following, we will walk through running the tutorial with BMR based on MASCOT and two parties. The other programs work similarly.

First, compile the virtual machine. In order to run with more than three parties, change the definition of MAX_N_PARTIES in BMR/config.h accordingly.

make -j 8 real-bmr-party.x

In order to compile a high-level program, use ./compile.py -B:

./compile.py -B 32 tutorial

Finally, run the two parties as follows:

./real-bmr-party.x -I 0 tutorial

./real-bmr-party.x -I 1 tutorial (in a separate terminal)

or

Scripts/real-bmr.sh tutorial

The -I enable interactive inputs, and in the tutorial party 0 and 1 will be asked to provide three numbers. Otherwise, and when using the script, the inputs are read from Player-Data/Input-P<playerno>-0.

Online-only benchmarking

In this section we show how to benchmark purely the data-dependent (often called online) phase of some protocols. This requires to generate the output of a previous phase insecurely. You will have to (re)compile the software after adding MY_CFLAGS = -DINSECURE to CONFIG.mine in order to run this insecure generation.

SPDZ

The SPDZ protocol uses preprocessing, that is, in a first (sometimes called offline) phase correlated randomness is generated independent of the actual inputs of the computation. Only the second ("online") phase combines this randomness with the actual inputs in order to produce the desired results. The preprocessed data can only be used once, thus more computation requires more preprocessing. MASCOT and Overdrive are the names for two alternative preprocessing phases to go with the SPDZ online phase.

All programs required in this section can be compiled with the target online:

make -j 8 online

To setup for benchmarking the online phase

This requires the INSECURE flag to be set before compilation as explained above. For a secure offline phase, see the section on SPDZ-2 below.

Run the command below. If you haven't added MY_CFLAGS = -DINSECURE to CONFIG.mine before compiling, it will fail.

Scripts/setup-online.sh

This sets up parameters for the online phase for 2 parties with a 128-bit prime field and 128-bit binary field, and creates fake offline data (multiplication triples etc.) for these parameters.

Parameters can be customised by running

Scripts/setup-online.sh <nparties> <nbitsp> <nbits2>

To compile a program

To compile for example the program in ./Programs/Source/tutorial.mpc, run:

./compile.py tutorial

This creates the bytecode and schedule files in Programs/Bytecode/ and Programs/Schedules/

To run a program

To run the above program with two parties on one machine, run:

./Player-Online.x -N 2 0 tutorial

./Player-Online.x -N 2 1 tutorial (in a separate terminal)

Or, you can use a script to do the above automatically:

Scripts/run-online.sh tutorial

To run a program on two different machines, firstly the preprocessing data must be copied across to the second machine (or shared using sshfs), and secondly, Player-Online.x needs to be passed the machine where the first party is running. e.g. if this machine is name diffie on the local network:

./Player-Online.x -N 2 -h diffie 0 test_all

./Player-Online.x -N 2 -h diffie 1 test_all

The software uses TCP ports around 5000 by default, use the -pn argument to change that.

Honest-majority three-party computation of binary circuits with malicious security

Compile the virtual machines:

make -j 8 rep-bin

Generate preprocessing data:

Scripts/setup-online.sh 3

After compilating the mpc file, run as follows:

malicious-rep-bin-party.x [-I] -h <host of party 0> -p <0/1/2> tutorial

When running locally, you can omit the host argument. As above, -I activates interactive input, otherwise inputs are read from Player-Data/Input-P<playerno>-0.

BMR

This part has been developed to benchmark ORAM for the Eurocrypt 2018 paper by Marcel Keller and Avishay Yanay. It only allows to benchmark the data-dependent phase. The data-independent and function-independent phases are emulated insecurely.

By default, the implementations is optimized for two parties. You can change this by defining N_PARTIES accordingly in BMR/config.h. If you entirely delete the definition, it will be able to run for any number of parties albeit slower.

Compile the virtual machine:

make -j 8 bmr

After compiling the mpc file:

• Run everything locally: Scripts/bmr-program-run.sh <program> <number of parties>.
• Run on different hosts: Scripts/bmr-program-run-remote.sh <program> <host1> <host2> [...]

Oblivious RAM

You can benchmark the ORAM implementation as follows:

1. Edit Program/Source/gc_oram.mpc to change size and to choose Circuit ORAM or linear scan without ORAM.
2. Run ./compile.py -D gc_oram. The -D argument instructs the compiler to remove dead code. This is useful for more complex programs such as this one.
3. Run gc_oram in the virtual machines as explained above.

Benchmarking offline phases

SPDZ-2 offline phase

This implementation is suitable to generate the preprocessed data used in the online phase.

For quick run on one machine, you can call the following:

./spdz2-offline.x -p 0 & ./spdz2-offline.x -p 1

More generally, run the following on every machine:

./spdz2-offline.x -p <number of party> -N <total number of parties> -h <hostname of party 0> -c <covert security parameter>

The number of parties are counted from 0. As seen in the quick example, you can omit the total number of parties if it is 2 and the hostname if all parties run on the same machine. Invoke ./spdz2-offline.x for more explanation on the options.

./spdz2-offline.x provides covert security according to some parameter c (at least 2). A malicious adversary will get caught with probability 1-1/c. There is a linear correlation between c and the running time, that is, running with 2c takes twice as long as running with c. The default for c is 10.

The program will generate every kind of randomness required by the online phase except input tuples until you stop it. You can shut it down gracefully pressing Ctrl-c (or sending the interrupt signal SIGINT), but only after an initial phase, the end of which is marked by the output Starting to produce gf2n. Note that the initial phase has been reported to take up to an hour. Furthermore, 3 GB of RAM are required per party.

Benchmarking the MASCOT or SPDZ2k offline phase

These implementations are not suitable to generate the preprocessed data for the online phase because they can only generate either multiplication triples or bits.

HOSTS must contain the hostnames or IPs of the players, see HOSTS.example for an example.

Then, MASCOT can be run as follows:

host1:$ ./ot-offline.x -p 0 -c

host2:$ ./ot-offline.x -p 1 -c

For SPDZ2k, use -Z <k> to set the computation domain to Z_{2^k}, and -S to set the security parameter. The latter defaults to k. At the time of writing, the following combinations are available: 32/32, 64/64, 64/48, and 66/48.

Running ./ot-offline.x without parameters give the full menu of options such as how many items to generate in how many threads and loops.

Benchmarking Overdrive offline phases

We have implemented several protocols to measure the maximal throughput for the Overdrive paper. As for MASCOT, these implementations are not suited to generate data for the online phase because they only generate one type at a time.

Binary	Protocol
simple-offline.x	SPDZ-1 and High Gear (with command-line argument -g)
pairwise-offline.x	Low Gear
cnc-offline.x	SPDZ-2 with malicious security (covert security with command-line argument -c)

These programs can be run similarly to spdz2-offline.x, for example:

host1:$ ./simple-offline.x -p 0 -h host1

host2:$ ./simple-offline.x -p 1 -h host1

Running any program without arguments describes all command-line arguments.

Memory usage

Lattice-based ciphertexts are relatively large (in the order of megabytes), and the zero-knowledge proofs we use require storing some hundred of them. You must therefore expect to use at least some hundred megabytes of memory per thread. The memory usage is linear in MAX_MOD_SZ (determining the maximum integer size for computations in steps of 64 bits), so you can try to reduce it (see the compilation section for how set it). For some choices of parameters, 4 is enough while others require up to 8. The programs above indicate the minimum MAX_MOD_SZ required, and they fail during the parameter generation if it is too low.