ml-dsa
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20 hours ago •
itzmeanjan
itzmeanjan
Module-Lattice-based Digital Signature Standard (draft) standard by NIST
[!CAUTION] This Dilithium implementation is conformant with Dilithium specification @ https://pq-crystals.org/dilithium/data/dilithium-specification-round3-20210208.pdf. I also try to make it timing leakage free, using
dudect(see https://github.com/oreparaz/dudect) -based tests, but be informed that this implementation is not yet audited. If you consider using it in production, be careful !
dilithium
CRYSTALS-Dilithium: Post-Quantum Digital Signature Algorithm
Motivation
Dilithium is one of those post-quantum digital signature algorithms ( DSA ), which is selected by NIST for standardization. Dilithium's security is based on hardness of finding short vectors in lattice i.e. it's a lattice based Post Quantum Cryptographic (PQC) construction.
Dilithium DSA offers following three algorithms.
| Algorithm | What does it do ? |
|---|---|
| KeyGen | It takes a 32 -bytes seed, which is used for deterministically computing both public key and secret key i.e. keypair. |
| Sign | It takes a secret key and a N (>0) -bytes message as input, which is used for deterministically ( default )/ randomly ( in this case, you must supply 64 uniform random sampled bytes as seed ) signing message, producing signature bytes. |
| Verify | It takes a public key, N (>0) -bytes message and signature, returning boolean value, denoting status of successful signature verification operation. |
Here I'm maintaining Dilithium as a header-only, easy-to-use C++ library, offering key generation, signing & verification API for three NIST security level ( i.e. 2, 3, 5 ) parameters, as defined in table 2 of Dilithium specification. For more details see below.
[!NOTE] Find Dilithium specification @ https://pq-crystals.org/dilithium/data/dilithium-specification-round3-20210208.pdf, which you should refer to when understanding intricate details of this implementation.
[!NOTE] Follow progress of NIST PQC standardization effort here.
Prerequisites
- A C++ compiler with C++20 standard library such as
g++/clang++.
$ clang++ --version
Ubuntu clang version 17.0.2 (1~exp1ubuntu2.1)
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin
$ g++ --version
gcc (Ubuntu 13.2.0-4ubuntu3) 13.2.0
- System development utilities such as
make,cmake.
$ make --version
GNU Make 4.3
$ cmake --version
cmake version 3.25.1
- For testing correctness and compatibility of this Dilithium implementation, you need to globally install
google-testlibrary and headers. Follow guide @ https://github.com/google/googletest/tree/main/googletest#standalone-cmake-project, if you don't have it installed. - For benchmarking Dilithium algorithms, you must have
google-benchmarkheader and library globally installed. I found guide @ https://github.com/google/benchmark#installation helpful.
[!NOTE] If you are on a machine running GNU/Linux kernel and you want to obtain CPU cycle count for KEM routines, you should consider building
google-benchmarklibrary withlibPFMsupport, following https://gist.github.com/itzmeanjan/05dc3e946f635d00c5e0b21aae6203a7, a step-by-step guide. Find more aboutlibPFM@ https://perfmon2.sourceforge.net.
[!TIP] Git submodule based dependencies will generally be imported automatically, but in case that doesn't work, you can manually initialize and update them by issuing
$ git submodule update --initfrom inside the root of this repository.
Testing
For ensuring functional correctness ( & conformance with the Dilithium specification, assuming the reference implemenation @ https://github.com/pq-crystals/dilithium.git is correct ) of this Dilithium library implemenation, issue
[!NOTE] Dilithium Known Answer Tests are generated following the procedure, described in https://gist.github.com/itzmeanjan/d14afc3866b82119221682f0f3c9822d.
make -j # Run tests without any sort of sanitizers
make asan_test -j # Run tests with AddressSanitizer enabled
make ubsan_test -j # Run tests with UndefinedBehaviourSanitizer enabled
[13/13] Dilithium.ArithmeticOverZq (365 ms)
PASSED TESTS (13/13):
1 ms: build/test.out Dilithium.HintBitPolynomialEncodingDecoding
1 ms: build/test.out Dilithium.PolynomialEncodingDecoding
1 ms: build/test.out Dilithium.HashingToABall
1 ms: build/test.out Dilithium.NumberTheoreticTransform
6 ms: build/test.out Dilithium.Power2Round
16 ms: build/test.out Dilithium.MakingAndUsingOfHintBits
54 ms: build/test.out Dilithium.Dilithium3KeygenSignVerifyFlow
54 ms: build/test.out Dilithium.Dilithium2KeygenSignVerifyFlow
92 ms: build/test.out Dilithium.Dilithium3KnownAnswerTests
93 ms: build/test.out Dilithium.Dilithium5KeygenSignVerifyFlow
112 ms: build/test.out Dilithium.Dilithium2KnownAnswerTests
149 ms: build/test.out Dilithium.Dilithium5KnownAnswerTests
365 ms: build/test.out Dilithium.ArithmeticOverZq
You can run timing leakage tests, using dudect, execute following
[!NOTE]
dudectis integrated into this library implementation of Dilithium DSA to find any sort of timing leakages. It checks for constant-timeness of most of the vital internal functions. Though it doesn't check constant-timeness of functions which use uniform rejection sampling, such as expansion of public matrixAor sampling of the vectorss1,s2or hashing to a ball etc.
# Can only be built and run on x86_64 machine.
make dudect_test_build -j
# Before running the constant-time tests, it's a good idea to put all CPU cores on "performance" mode.
# You may find the guide @ https://github.com/google/benchmark/blob/main/docs/reducing_variance.md helpful.
timeout 10m taskset -c 0 ./build/dudect/test_dilithium2.out
timeout 10m taskset -c 0 ./build/dudect/test_dilithium3.out
timeout 10m taskset -c 0 ./build/dudect/test_dilithium5.out
[!TIP]
dudectdocumentation says iftstatistic is< 10, we're probably good, yes probably. You may want to readdudectdocumentation @ https://github.com/oreparaz/dudect. Also you might find the original paper @ https://ia.cr/2016/1123 interesting.
...
meas: 48.38 M, max t: +2.77, max tau: 3.99e-04, (5/tau)^2: 1.57e+08. For the moment, maybe constant time.
meas: 48.48 M, max t: +2.73, max tau: 3.93e-04, (5/tau)^2: 1.62e+08. For the moment, maybe constant time.
meas: 48.57 M, max t: +2.76, max tau: 3.96e-04, (5/tau)^2: 1.59e+08. For the moment, maybe constant time.
meas: 48.67 M, max t: +2.78, max tau: 3.99e-04, (5/tau)^2: 1.57e+08. For the moment, maybe constant time.
meas: 48.76 M, max t: +2.79, max tau: 3.99e-04, (5/tau)^2: 1.57e+08. For the moment, maybe constant time.
meas: 48.85 M, max t: +2.78, max tau: 3.97e-04, (5/tau)^2: 1.58e+08. For the moment, maybe constant time.
meas: 48.95 M, max t: +2.79, max tau: 3.98e-04, (5/tau)^2: 1.58e+08. For the moment, maybe constant time.
meas: 49.05 M, max t: +2.77, max tau: 3.95e-04, (5/tau)^2: 1.60e+08. For the moment, maybe constant time.
meas: 49.14 M, max t: +2.69, max tau: 3.84e-04, (5/tau)^2: 1.70e+08. For the moment, maybe constant time.
meas: 49.24 M, max t: +2.75, max tau: 3.92e-04, (5/tau)^2: 1.62e+08. For the moment, maybe constant time.
meas: 49.33 M, max t: +2.73, max tau: 3.89e-04, (5/tau)^2: 1.65e+08. For the moment, maybe constant time.
meas: 49.43 M, max t: +2.76, max tau: 3.93e-04, (5/tau)^2: 1.62e+08. For the moment, maybe constant time.
meas: 49.52 M, max t: +2.76, max tau: 3.92e-04, (5/tau)^2: 1.63e+08. For the moment, maybe constant time.
meas: 49.62 M, max t: +2.79, max tau: 3.97e-04, (5/tau)^2: 1.59e+08. For the moment, maybe constant time.
meas: 49.71 M, max t: +2.78, max tau: 3.94e-04, (5/tau)^2: 1.61e+08. For the moment, maybe constant time.
Benchmarking
Benchmarking key generation, signing and verification algorithms for various instantiations of Dilithium digital signature scheme can be done, by issuing
make benchmark -j # If you haven't built google-benchmark library with libPFM support.
make perf -j # If you have built google-benchmark library with libPFM support.
[!NOTE] Benchmarking expects presence of
google-benchmarkheaders and library in global namespace ( so that it can be found by the compiler ) i.e. header and library path must live on $PATH.
[!CAUTION] Ensure you've put all your CPU cores on performance mode before running benchmarks, follow guide @ https://github.com/google/benchmark/blob/main/docs/reducing_variance.md.
[!NOTE] Only deterministic signing procedure is benchmarked here, while signing random message of length 32 -bytes. One can benchmark non-deterministic signing procedure by explicitly passing truth value to template parameter of
sign(...)routine and they must supply a 64 -bytes uniform random sampled seed, when invokingsignroutine.
[!WARNING] Relying only on average timing measurement for understanding performance characteristics of Dilithium signing algorithm may not be a good idea, given that it's a post-quantum digital signature scheme of "Fiat-Shamir with Aborts" paradigm - broadly speaking, during signing procedure it may need to abort and restart again, multiple times, based on what message is being signed or what sort of random seed is being used for randomized signing. So it's a better idea to also compute other statistics such as minimum, maximum and median ( pretty useful ) when timing execution of signing procedure. In following benchmark results, you'll see such statistics demonstrating broader performance characteristics of Dilithium signing procedure for various parameter sets.
[!NOTE]
make perf- was issued when collecting following benchmarks. Notice, cycles column, denoting cost of executing Dilithium signature scheme routines in terms of CPU cycles. Follow this for more details.
On 12th Gen Intel(R) Core(TM) i7-1260P
Compiled with gcc version 13.2.0 (Ubuntu 13.2.0-4ubuntu3).
$ uname -srm
Linux 6.5.0-14-generic x86_64
2023-12-24T13:05:17+04:00
Running ./build/perf.out
Run on (16 X 4129.81 MHz CPU s)
CPU Caches:
L1 Data 48 KiB (x8)
L1 Instruction 32 KiB (x8)
L2 Unified 1280 KiB (x8)
L3 Unified 18432 KiB (x1)
Load Average: 0.33, 0.29, 0.27
--------------------------------------------------------------------------------------------------
Benchmark Time CPU Iterations CYCLES items_per_second
--------------------------------------------------------------------------------------------------
dilithium5_sign/32_mean 573 us 573 us 32 2.62244M 2.15221k/s
dilithium5_sign/32_median 570 us 570 us 32 2.56475M 1.77223k/s
dilithium5_sign/32_stddev 289 us 289 us 32 1.33162M 983.885/s
dilithium5_sign/32_cv 50.50 % 50.50 % 32 50.78% 45.72%
dilithium5_sign/32_min 266 us 266 us 32 1.22116M 584.246/s
dilithium5_sign/32_max 1712 us 1712 us 32 7.85984M 3.75566k/s
dilithium2_sign/32_mean 413 us 413 us 32 1.92469M 4.31431k/s
dilithium2_sign/32_median 239 us 239 us 32 1.10512M 4.18457k/s
dilithium2_sign/32_stddev 393 us 393 us 32 1.83463M 2.77288k/s
dilithium2_sign/32_cv 95.04 % 95.04 % 32 95.32% 64.27%
dilithium2_sign/32_min 114 us 114 us 32 531.5k 548.719/s
dilithium2_sign/32_max 1822 us 1822 us 32 8.53117M 8.78612k/s
dilithium3_keygen_mean 100 us 100 us 32 450.02k 10.0107k/s
dilithium3_keygen_median 98.4 us 98.4 us 32 450.223k 10.158k/s
dilithium3_keygen_stddev 3.72 us 3.72 us 32 1.44564k 370.119/s
dilithium3_keygen_cv 3.72 % 3.72 % 32 0.32% 3.70%
dilithium3_keygen_min 95.9 us 95.9 us 32 443.754k 9.43455k/s
dilithium3_keygen_max 106 us 106 us 32 452.294k 10.4222k/s
dilithium3_sign/32_mean 511 us 511 us 32 2.38786M 2.9922k/s
dilithium3_sign/32_median 346 us 346 us 32 1.60917M 2.89361k/s
dilithium3_sign/32_stddev 370 us 370 us 32 1.73198M 1.82606k/s
dilithium3_sign/32_cv 72.38 % 72.38 % 32 72.53% 61.03%
dilithium3_sign/32_min 171 us 171 us 32 798.767k 611.443/s
dilithium3_sign/32_max 1636 us 1635 us 32 7.65597M 5.86292k/s
dilithium3_verify/32_mean 104 us 104 us 32 472.594k 9.60074k/s
dilithium3_verify/32_median 102 us 102 us 32 472.428k 9.82254k/s
dilithium3_verify/32_stddev 3.54 us 3.55 us 32 870.197 322.901/s
dilithium3_verify/32_cv 3.40 % 3.40 % 32 0.18% 3.36%
dilithium3_verify/32_min 101 us 101 us 32 470.808k 9.0583k/s
dilithium3_verify/32_max 110 us 110 us 32 474.024k 9.92375k/s
dilithium5_keygen_mean 168 us 168 us 32 749.512k 5.96366k/s
dilithium5_keygen_median 167 us 167 us 32 748.591k 6.00453k/s
dilithium5_keygen_stddev 4.98 us 4.97 us 32 3.57104k 175.466/s
dilithium5_keygen_cv 2.96 % 2.96 % 32 0.48% 2.94%
dilithium5_keygen_min 162 us 162 us 32 743.167k 5.63682k/s
dilithium5_keygen_max 177 us 177 us 32 757.093k 6.18287k/s
dilithium2_verify/32_mean 64.0 us 64.0 us 32 292.642k 15.631k/s
dilithium2_verify/32_median 62.9 us 62.9 us 32 292.392k 15.9072k/s
dilithium2_verify/32_stddev 2.00 us 2.00 us 32 891.422 476.406/s
dilithium2_verify/32_cv 3.12 % 3.12 % 32 0.30% 3.05%
dilithium2_verify/32_min 62.4 us 62.4 us 32 291.515k 14.6913k/s
dilithium2_verify/32_max 68.1 us 68.1 us 32 294.848k 16.0265k/s
dilithium2_keygen_mean 59.0 us 59.0 us 32 269.516k 16.9619k/s
dilithium2_keygen_median 58.4 us 58.4 us 32 269.005k 17.1373k/s
dilithium2_keygen_stddev 1.96 us 1.96 us 32 2.73875k 546.274/s
dilithium2_keygen_cv 3.31 % 3.31 % 32 1.02% 3.22%
dilithium2_keygen_min 56.7 us 56.7 us 32 265.559k 15.6044k/s
dilithium2_keygen_max 64.1 us 64.1 us 32 275.942k 17.6457k/s
dilithium5_verify/32_mean 171 us 171 us 32 780.57k 5.8392k/s
dilithium5_verify/32_median 168 us 168 us 32 780.972k 5.95881k/s
dilithium5_verify/32_stddev 5.63 us 5.63 us 32 1.43262k 187.536/s
dilithium5_verify/32_cv 3.28 % 3.28 % 32 0.18% 3.21%
dilithium5_verify/32_min 167 us 167 us 32 778.381k 5.49484k/s
dilithium5_verify/32_max 182 us 182 us 32 782.668k 6.00273k/s
On ARM Cortex-A72 i.e. Raspberry Pi 4B
Compiled with gcc version 13.2.0 (Ubuntu 13.2.0-4ubuntu3).
$ uname -srm
Linux 6.5.0-1008-raspi aarch64
2023-12-24T15:13:38+05:30
Running ./build/perf.out
Run on (4 X 1800 MHz CPU s)
CPU Caches:
L1 Data 32 KiB (x4)
L1 Instruction 48 KiB (x4)
L2 Unified 1024 KiB (x1)
Load Average: 1.18, 1.24, 0.78
--------------------------------------------------------------------------------------------------
Benchmark Time CPU Iterations CYCLES items_per_second
--------------------------------------------------------------------------------------------------
dilithium5_verify/32_mean 982 us 982 us 32 1.76352M 1.01874k/s
dilithium5_verify/32_median 982 us 982 us 32 1.76347M 1.01877k/s
dilithium5_verify/32_stddev 0.618 us 0.417 us 32 447.3 0.432759/s
dilithium5_verify/32_cv 0.06 % 0.04 % 32 0.03% 0.04%
dilithium5_verify/32_min 981 us 981 us 32 1.76263M 1.01761k/s
dilithium5_verify/32_max 984 us 983 us 32 1.76433M 1.0195k/s
dilithium3_verify/32_mean 627 us 627 us 32 1.12594M 1.59563k/s
dilithium3_verify/32_median 627 us 627 us 32 1.12584M 1.59581k/s
dilithium3_verify/32_stddev 0.369 us 0.267 us 32 385.4 0.678979/s
dilithium3_verify/32_cv 0.06 % 0.04 % 32 0.03% 0.04%
dilithium3_verify/32_min 626 us 626 us 32 1.12543M 1.59386k/s
dilithium3_verify/32_max 628 us 627 us 32 1.12704M 1.59664k/s
dilithium2_verify/32_mean 397 us 397 us 32 712.872k 2.51939k/s
dilithium2_verify/32_median 397 us 397 us 32 712.921k 2.51999k/s
dilithium2_verify/32_stddev 1.01 us 0.901 us 32 514.135 5.66412/s
dilithium2_verify/32_cv 0.25 % 0.23 % 32 0.07% 0.22%
dilithium2_verify/32_min 396 us 396 us 32 711.708k 2.49017k/s
dilithium2_verify/32_max 402 us 402 us 32 713.909k 2.52477k/s
dilithium2_keygen_mean 332 us 332 us 32 596.912k 3.00994k/s
dilithium2_keygen_median 333 us 333 us 32 597.624k 3.00643k/s
dilithium2_keygen_stddev 1.30 us 1.31 us 32 2.40391k 11.8347/s
dilithium2_keygen_cv 0.39 % 0.39 % 32 0.40% 0.39%
dilithium2_keygen_min 329 us 329 us 32 591.684k 2.98561k/s
dilithium2_keygen_max 335 us 335 us 32 601.775k 3.03661k/s
dilithium3_sign/32_mean 4059 us 4058 us 32 7.29006M 365.449/s
dilithium3_sign/32_median 3304 us 3303 us 32 5.93442M 302.715/s
dilithium3_sign/32_stddev 2724 us 2723 us 32 4.89362M 233.459/s
dilithium3_sign/32_cv 67.10 % 67.11 % 32 67.13% 63.88%
dilithium3_sign/32_min 1171 us 1171 us 32 2.10391M 71.6582/s
dilithium3_sign/32_max 13957 us 13955 us 32 25.0762M 854.011/s
dilithium5_sign/32_mean 6044 us 6043 us 32 10.8539M 270.538/s
dilithium5_sign/32_median 4732 us 4732 us 32 8.49736M 211.344/s
dilithium5_sign/32_stddev 4933 us 4932 us 32 8.86216M 184.414/s
dilithium5_sign/32_cv 81.63 % 81.63 % 32 81.65% 68.17%
dilithium5_sign/32_min 1722 us 1722 us 32 3.09382M 38.8392/s
dilithium5_sign/32_max 25754 us 25747 us 32 46.2578M 580.602/s
dilithium5_keygen_mean 869 us 869 us 32 1.56125M 1.15065k/s
dilithium5_keygen_median 869 us 869 us 32 1.5609M 1.15112k/s
dilithium5_keygen_stddev 2.55 us 2.30 us 32 3.61955k 3.03513/s
dilithium5_keygen_cv 0.29 % 0.26 % 32 0.23% 0.26%
dilithium5_keygen_min 865 us 865 us 32 1.55393M 1.14095k/s
dilithium5_keygen_max 879 us 876 us 32 1.57077M 1.15633k/s
dilithium3_keygen_mean 548 us 547 us 32 983.306k 1.82668k/s
dilithium3_keygen_median 547 us 547 us 32 983.369k 1.82715k/s
dilithium3_keygen_stddev 0.969 us 0.740 us 32 698.653 2.45958/s
dilithium3_keygen_cv 0.18 % 0.14 % 32 0.07% 0.13%
dilithium3_keygen_min 547 us 547 us 32 981.931k 1.81604k/s
dilithium3_keygen_max 552 us 551 us 32 984.944k 1.82955k/s
dilithium2_sign/32_mean 2253 us 2253 us 32 4.04616M 666.05/s
dilithium2_sign/32_median 1763 us 1762 us 32 3.16635M 567.381/s
dilithium2_sign/32_stddev 1738 us 1738 us 32 3.12125M 385.348/s
dilithium2_sign/32_cv 77.14 % 77.13 % 32 77.14% 57.86%
dilithium2_sign/32_min 782 us 782 us 32 1.40548M 109.753/s
dilithium2_sign/32_max 9114 us 9111 us 32 16.3662M 1.27835k/s
On Apple M1 Max
Compiled with Apple clang version 15.0.0 (clang-1500.1.0.2.5).
$ uname -srm
Darwin 23.2.0 arm64
2024-01-22T19:57:30+04:00
Running ./build/bench.out
Run on (10 X 24 MHz CPU s)
CPU Caches:
L1 Data 64 KiB
L1 Instruction 128 KiB
L2 Unified 4096 KiB (x10)
Load Average: 1.61, 1.96, 2.34
---------------------------------------------------------------------------------------
Benchmark Time CPU Iterations items_per_second
---------------------------------------------------------------------------------------
dilithium5_verify/32_mean 182 us 182 us 32 5.48988k/s
dilithium5_verify/32_median 182 us 182 us 32 5.49471k/s
dilithium5_verify/32_stddev 0.790 us 0.792 us 32 23.356/s
dilithium5_verify/32_cv 0.43 % 0.43 % 32 0.43%
dilithium5_verify/32_min 182 us 182 us 32 5.36387k/s
dilithium5_verify/32_max 187 us 186 us 32 5.5007k/s
dilithium3_keygen_mean 103 us 102 us 32 9.75627k/s
dilithium3_keygen_median 103 us 103 us 32 9.75592k/s
dilithium3_keygen_stddev 0.113 us 0.113 us 32 10.7959/s
dilithium3_keygen_cv 0.11 % 0.11 % 32 0.11%
dilithium3_keygen_min 102 us 102 us 32 9.72649k/s
dilithium3_keygen_max 103 us 103 us 32 9.77192k/s
dilithium2_sign/32_mean 389 us 388 us 32 3.60304k/s
dilithium2_sign/32_median 308 us 308 us 32 3.28064k/s
dilithium2_sign/32_stddev 224 us 224 us 32 2.13469k/s
dilithium2_sign/32_cv 57.71 % 57.71 % 32 59.25%
dilithium2_sign/32_min 125 us 125 us 32 1.10009k/s
dilithium2_sign/32_max 910 us 909 us 32 7.99269k/s
dilithium3_verify/32_mean 113 us 112 us 32 8.89718k/s
dilithium3_verify/32_median 113 us 112 us 32 8.8974k/s
dilithium3_verify/32_stddev 0.062 us 0.044 us 32 3.48072/s
dilithium3_verify/32_cv 0.06 % 0.04 % 32 0.04%
dilithium3_verify/32_min 112 us 112 us 32 8.89127k/s
dilithium3_verify/32_max 113 us 112 us 32 8.90303k/s
dilithium2_verify/32_mean 69.5 us 69.4 us 32 14.4038k/s
dilithium2_verify/32_median 69.5 us 69.4 us 32 14.4049k/s
dilithium2_verify/32_stddev 0.055 us 0.026 us 32 5.49004/s
dilithium2_verify/32_cv 0.08 % 0.04 % 32 0.04%
dilithium2_verify/32_min 69.4 us 69.4 us 32 14.3902k/s
dilithium2_verify/32_max 69.7 us 69.5 us 32 14.4125k/s
dilithium2_keygen_mean 60.7 us 60.6 us 32 16.4974k/s
dilithium2_keygen_median 60.7 us 60.6 us 32 16.5101k/s
dilithium2_keygen_stddev 0.489 us 0.482 us 32 130.751/s
dilithium2_keygen_cv 0.81 % 0.80 % 32 0.79%
dilithium2_keygen_min 59.8 us 59.8 us 32 16.1638k/s
dilithium2_keygen_max 61.9 us 61.9 us 32 16.7339k/s
dilithium5_sign/32_mean 786 us 785 us 32 1.79488k/s
dilithium5_sign/32_median 574 us 573 us 32 1.74461k/s
dilithium5_sign/32_stddev 560 us 559 us 32 943.588/s
dilithium5_sign/32_cv 71.24 % 71.24 % 32 52.57%
dilithium5_sign/32_min 287 us 287 us 32 368.016/s
dilithium5_sign/32_max 2721 us 2717 us 32 3.48385k/s
dilithium3_sign/32_mean 498 us 497 us 32 3.06659k/s
dilithium3_sign/32_median 393 us 392 us 32 2.5494k/s
dilithium3_sign/32_stddev 366 us 365 us 32 1.76289k/s
dilithium3_sign/32_cv 73.44 % 73.44 % 32 57.49%
dilithium3_sign/32_min 190 us 189 us 32 621.66/s
dilithium3_sign/32_max 1611 us 1609 us 32 5.27815k/s
dilithium5_keygen_mean 168 us 167 us 32 5.97094k/s
dilithium5_keygen_median 168 us 168 us 32 5.96961k/s
dilithium5_keygen_stddev 0.495 us 0.469 us 32 16.7342/s
dilithium5_keygen_cv 0.30 % 0.28 % 32 0.28%
dilithium5_keygen_min 167 us 166 us 32 5.94183k/s
dilithium5_keygen_max 169 us 168 us 32 6.01249k/s
Usage
Dilithium is a header-only C++ library which is fairly easy-to-use. Let's see how to get started using it.
- Clone Dilithium repository.
cd
# Multi-step cloning and importing of submodules
git clone https://github.com/itzmeanjan/dilithium.git && pushd dilithium && git submodule update --init && popd
# Or do single step cloning and importing of submodules
git clone https://github.com/itzmeanjan/dilithium.git --recurse-submodules
- Write a program which makes use of Dilithium{2,3,5} {keygen, signing, verification} API ( all of these routines and constants, representing how many bytes of memory to allocate for holding public/ secret key and signature, live under
dilithium{2,3,5}::namespace ), while importing proper header file. - Finally compile your program, while letting your compiler know where it can find Dilithium (
./include) and Sha3 (./sha3/include) headers.
# Assuming `dilithium` was cloned just under $HOME
DILITHIUM_HEADERS=~/dilithium/include
SHA3_HEADERS=~/dilithium/sha3/include
g++ -std=c++20 -Wall -Wextra -pedantic -O3 -march=native -I $DILITHIUM_HEADERS -I $SHA3_HEADERS main.cpp
Dilithium API Usage Flow
Let's walk through an example, where I show you how to use Dilithium key generation, signing ( both deterministic and randomized ) and verification API for Dilithium2 instantiation. It should be pretty similar using other Dilithium instances.
- Let's begin by generating a Dilithium2 keypair, given 32 -bytes seed.
// main.cpp
// In case interested in using Dilithium3 or Dilithium5 API, import "dilithium3.hpp" or "dilithium5.hpp" and use keygen/ sign/ verify functions living either under `dilithium3::` or `dilithium5::` namespace.
#include "dilithium2.hpp"
#include <array>
#include "prng.hpp"
int main() {
std::array<uint8_t, 32> seed{};
std::array<uint8_t, dilithium2::PubKeyLen> pubkey{};
std::array<uint8_t, dilithium2::SecKeyLen> seckey{};
// Sample seed bytes from PRNG
//
// Be careful, read API documentation in include/prng.hpp
// before you consider using it in production.
prng::prng_t prng;
prng.read(seed);
dilithium2::keygen(seed, pubkey, seckey);
// ...
return 0;
}
- Given a Dilithium2 secret key and non-empty message M, sign it, computing signature.
int main() {
// Key Generation
// ...
std::array<uint8_t, 32> msg{};
std::array<uint8_t, dilithium2::SigLen> sig{};
// Sample a psuedo-random message, to be signed
prng.read(msg);
// Default behaviour is deterministic signing and you can safely pass
// an empty std::span (i.e. `{}`) for last parameter i.e. random seed.
dilithium2::sign(seckey, msg, sig, {});
// ...
return 0;
}
- In case you're interested in randomized signing, you may explicitly opt in ( at compile-time ) by passing TRUTH value for the only template parameter present in
signfunction definition, while also supplying a 64 -bytes uniform sampled random seed, when invokingsignprocedure.
int main() {
// Key Generation
// ...
// Deterministic Signing ( default )
// ...
// 64 -bytes random seed to be used for randomized
// message signing. We'll sample random seed bytes using PRNG.
std::array<uint8_t, 64> rnd_seed{};
prng.read(rnd_seed);
// You must pass a 64 -bytes random seed when you explicitly opt for randomized signing.
dilithium2::sign<true>(seckey, msg, sig, rnd_seed);
return 0;
}
- Verify signature, given public key, message M and the signature itself. It returns boolean truth value in case of successful signature verification otherwise it returns false.
int main() {
// Key Generation
// ...
// Deterministic Signing ( default )
// ...
// Randomized Signing ( explicit )
// ...
const bool flg = dilithium2::verify(pubkey, msg, sig);
assert(flg);
return 0;
}
I suggest you look at example program, which demonstrates how to use Dilithium2 API, similarly you can use Dilithium{3,5} API.
$ g++ -std=c++20 -Wall -Wextra -pedantic -O3 -march=native -I ./include -I ./sha3/include examples/dilithium2.cpp && ./a.out
Dilithium @ NIST security level 2
seed : e9ae2e74d3bc6e60b4966eb0da2f6d03c0d864ef7b001947d294d64cb90fd8c7
pubkey : 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
seckey : 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
message : 3d507521c3fb52591006a02be88a103e71c8dbc480e569012151413f5e00a1d8
signature : 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
verified : true
[!CAUTION] Before you consider using Psuedo Random Number Generator which comes with this library implementation, I strongly advice you to go through include/prng.hpp.
[!NOTE] Looking at the API documentation, in header files, can give you a good idea of how to use Dilithium API. Note, this library doesn't expose any raw pointer based interface, rather everything is wrapped under statically defined
std::span- which one can easily create fromstd::{array, vector}. I opt for using statically definedstd::spanbased function interfaces because we always know, at compile-time, how many bytes the seeds/ keys/ signatures are, for various different Dilithium instantiations. This gives much better type safety and compile-time error reporting.
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Module-Lattice-based Digital Signature Standard (draft) standard by NIST