is a pure-Go implementation of ML-KEM-768 optimized for correctness and readability. ML-KEM (formerly known as Kyber, renamed because we can’t have nice things) is a post-quantum key exchange mechanism in the process of being standardized by NIST and adopted by most of the industry.

The package amounts to ~500 lines of code, plus 200 lines of comments, and 650 lines of tests. It has no dependencies except for It’s meant for upstreaming into the Go standard library (initially as an internal-only package used in an opt-in crypto/tls experiment) and was designed to provide high security assurance through ease of review, simplicity, and thorough testing.

I livecoded part of its development on Twitch, and you can watch the replay on YouTube.

Unlike most other implementations, this code was not ported from the reference pq-crystals library, but written from scratch not having ever closely read other codebases. This was an intentional exercise in spec validation, to show it is possible to produce an interoperable implementation from the specification alone.

The FIPS 203 document turned out to be an excellent implementation guide, with detailed pseudo-code, exhaustive definitions, and consistent type information. (This is something I would like to ask of any large specification document: define your types and use them and denote them!) To make the code both easier to review and better as a learning resource, function and variable names, and even operation ordering, are carefully picked to mirror the FIPS specification.

The specification actually requires fairly limited math background, but to facilitate the work of implementers, I wrote up Enough Polynomials and Linear Algebra to Implement Kyber.

Beyond that, the only parts left as an exercise to the reader were

  1. implementing arithmetic modulo the prime 3329;
  2. concretely implementing the compress and decompress functions mapping values [0, 3329) to and from [0, 2ᵈ); and
  3. ensuring constant time operations.

Modulo arithmetic was reasonably easy, as we all collectively learned a lot about finite field arithmetic through years of RSA and elliptic curve implementations. The small prime actually makes the task feel unnaturally simple.

Compression and decompression turned out to be the most difficult part of the project. The specification defines them in abstract terms as fractions and rounding rules—“just” compute (2ᵈ/q)·x or (q/2ᵈ)·y and round to the closest integer—but in practice we need to implement them with constant time arithmetic and bitwise operations! In my public comments I pointed out that having each implementation figure out a strategy is risky and redundant. I was more correct than I thought: it turned out that the reference implementation and ~every implementation ported from it used a division which depending on compiler optimizations and platform might result in a DIV instruction, which is variable-time even when the divisor is fixed. This package was unaffected, because it used Barrett reduction from the start, like BoringSSL.

You can read the rest of my formal public comments on the pqc-forum mailing list.

Readability was a major goal of the implementation, and it was pursued even especially for complex functions like compression and decompression. A readable implementation has two purposes: first, it allows effective review, both during the code review process and later by interested researchers, improving security; second, it serves as an educational resource for the next generation of maintainers and cryptography engineers (or curious nerds). Reading the Go cryptography standard library is how I got started on the path that led me here, so it is especially important to me to preserve and improve it as a learning resource. It’s obviously subjective, but I believe this to be the most understandable public ML-KEM/Kyber implementation. Compare for example our compression/decompression functions with the reference implementation.

Sometimes improving readability and reviewability means making code longer and less reusable: for example for ML-KEM-768 we need to serialize 1-, 4-, 10-, and 12-bit integers in a packed format. A universal 1-to-12 bit encoder and decoder is a pretty gnarly piece of code to write correctly, but each of those four sizes are actually pretty easy to write a dedicated encoder/decoder for.[1] This is why we have ringCompressAndEncode1/4/10 etc. instead of a single universal function. This also made it easy to work some special required checks into the 12-bit decoder.

This, by the way, was only possible because we targeted ML-KEM-768 specifically, or we’d have had to implement 5- and 11-bit encodings, as well. ML-KEM is specified at three security levels (-512, -768, and -1024). However, the Kyber team recommends using -768 over -512 for a more conservative security margin against novel cryptanalysis, while -1024 exists only for the same reasons 256-bit security levels exist: compliance and blind strength matching. Most protocols being tested or standardized coalesced around ML-KEM-768, so targeting only that improves not only readability, but also security (because there are fewer moving parts), and performance (because we can optimize allocation sizes, iteration counts, and encoding algorithms) at little to no cost.

After readability, testing is the main component in this package’s high security assurance strategy. Besides checking that key generation, encapsulation, and decapsulation round-trip correctly, and maintaining a test coverage of 95%+, we

  • ensure interoperability with test vectors obtained from NIST and other implementations;
  • exhaustively test every input combination for base field arithmetic operations (addition, subtraction, and multiplication modulo 3329) against expected values computed trivially with variable-time operations;
  • exhaustively test compression and decompression against math/big.Rat (contributed by David Buchanan);
  • test that pre-computed constants match their definition;
  • check that incorrect lengths (both long and short) cause the appropriate error for every input of every function;
  • run an extensive set of reusable test vectors we developed (see below);
  • run test vectors provided by Sophie Schmieg which will be eventually included in Wycheproof.

Our test vectors are designed to be reusable by other implementations, and are published as part of the CCTV project along with detailed intermediate values for testing and debugging each intermediate step and partial algorithm, which we used during development. There are different sets of tests vectors, each designed to reach different edge cases.

  • Negative test vectors provide invalid encapsulation keys, where the coefficients are higher than 3329. These were often requested, since all the test vectors from the Kyber and NIST teams are for regular, correct inputs. These vectors individually test every value from 3329 to 2¹²-1 and every coefficient location, sharing the remaining coefficients so they compress from 1–3 MiB down to 12–28 KiB.

  • “Unlucky” vectors require an unusually large number of XOF reads. Kyber samples a matrix from a portion of public keys[2] with rejection sampling: it gets a random value between 0 and 2 ¹²-1 and checks if it’s less than 3329, if not, it tries again. The amount of bytes needed to sample a matrix depends on how lucky you get with the sampling, and that’s a random function of the public key component. These vectors are regular public keys and require reading more than 575 bytes from the SHAKE-128 XOF in SampleNTT, which would ordinarily happen with probability 2⁻³⁸. Sophie’s vectors were bruteforced further, and require up to 591 bytes.

    At this point I would like to thank our detection and response team for not killing my job(s) hashing vast amounts of random seeds and looking for zeroes in the output. — Sophie Schmieg

  • Special vectors fail if strcmp is used in ML-KEM.Decaps. In ML-KEM.Decaps the ciphertext is compared with the output of K-PKE.Encrypt for implicit rejection. If an implementation were to use strcmp() for that comparison it would fail to reject some ciphertexts if a zero byte terminates the comparison early. This one I hope is going to sit as a silent trap for years—who would use strcmp() in cryptographic code—and then ruthlessly kill a vulnerability, because of course someone will.

  • Accumulated vectors (derived from the reference pq-crystals implementation) allow testing randomly reachable edge cases without checking in large amounts of data. The reference implementation of Kyber includes a test_vectors.c program that generates 300MB of random vectors. I had no intention of checking in the output or compiling C, but since they are just randomly generated vectors, we can regenerate them in our tests from the deterministic RNG (SHAKE-128 with an empty input) and check they hash to an expected value. We can even take it further, and produce hashes for a million random tests, beyond the 10k they generate.

I am happy to report that none of the tests, many introduced after completion of the implementation, identified any issues in There is at least one reported instance of the negative vectors identifying a defect in a major implementation, though.

Performance is not a primary goal (neither of this package nor of the Go cryptography packages) but the package needs to be fast enough to be useful. Thankfully, ML-KEM is pretty fast, to the point that this simple implementation is competitive with our assembly-optimized P-256 and X25519 implementations.

To compare apples to apples, note that we need to compare the whole operation that each side needs to perform for key establishment: for ECDH, two scalar multiplications (one of them by the fixed base point); for KEMs, key generation and decapsulation on one side, and encapsulation on the other. ECDH is symmetrical, ML-KEM key establishment is not.

The ECDH benchmarks below already include the two scalar multiplications, while the mlkem768 benchmarks are split as key generation and decapsulation under “Alice” and encapsulation under “Bob”. Since decapsulation includes a full encryption (to check the resulting ciphertext matches the input), Alice takes a lot longer than Bob: the latter does an encryption, while the former does an encryption, a decryption, and a key generation.

All in all, “Bob” is as fast as our X25519 or P-256, while “Alice” takes less than twice. Compared to some of the fastest ML-KEM implementations out there (BoringSSL and libcrux), this package takes approximately double the time. For such a simple and unoptimized implementation, this is more than satisfactory.

goos: darwin
goarch: arm64
cpu: Intel(R) Core(TM) i5-7400 CPU @ 3.00GHz
pkg: crypto/ecdh
                         │   sec/op    │
ECDH/P256-8                49.43µ ± 0%
ECDH/X25519-8              77.46µ ± 0%

                         │   sec/op    │
RoundTrip/Alice-8          109.4µ ± 0%
RoundTrip/Bob-8            56.19µ ± 0%

goos: linux
goarch: amd64
pkg: crypto/ecdh
                         │   sec/op    │
ECDH/P256-4                78.88µ ± 1%
ECDH/X25519-4              115.6µ ± 2%

                         │   sec/op    │
RoundTrip/Alice-4          223.8µ ± 2%
RoundTrip/Bob-4            114.7µ ± 1%

The performance wasn’t entirely free. In general, I followed high-performance Go programming patterns, trying for example to minimize heap allocations. Next, I reworked the x/crypto/sha3 package so it could be used without any heap allocation thanks to the mid-stack inlining trick. However, I haven’t merged those changes yet and they are not included in the benchmarks above, because they have a negative effect on Apple M2 processors. No idea why yet.

goos: darwin
goarch: arm64
                  │   sec/op    │   sec/op     vs base                │
RoundTrip/Alice-8   109.4µ ± 0%   121.3µ ± 1%  +10.91% (p=0.000 n=10)
RoundTrip/Bob-8     56.19µ ± 0%   59.94µ ± 2%   +6.66% (p=0.000 n=10)

goos: linux
goarch: amd64
                  │   sec/op    │   sec/op     vs base               │
RoundTrip/Alice-4   223.8µ ± 2%   218.6µ ± 1%  -2.32% (p=0.000 n=10)
RoundTrip/Bob-4     114.7µ ± 1%   109.5µ ± 0%  -4.57% (p=0.000 n=10)

The one successful optimization was complaining about the confusing result above on the Gophers Slack #performance channel, which sniped Josh Bleecher Snyder into contributing a couple changes :)

There is some low hanging fruit still: key generation and decapsulation both sample a matrix from the same value, and since the two are usually done sequentially on the Alice side, the matrix could be stored saving around 10% time. There might be an opportunity to save a copy in the sha3 read path, too. After that, it’s a matter of optimizing the field implementation.

If you got this far, you might want to follow me on Bluesky at or on Mastodon at

Bonus track: using a ML-KEM implementation as Kyber v3

NIST made a few small changes to the Round 3 submission of Kyber. They are summarized in Section 1.3 of the FIPS draft.

However, there are a few experimental protocols defined in terms of Kyber v3 (or “draft00”), including the main deployed PQ TLS key exchange. Do we have to make a separate package to support them?

Luckily, no we don’t.

One change adds some validation for an edge case (non-canonical coefficient encodings in public keys) that was undefined in Kyber. Honest implementations will not produce such keys, so we can reject them as specified in the FIPS draft. It will make it possible to fingerprint our implementation as Kyber-on-ML-KEM but will be otherwise harmless.

One change removed a hashing step applied to CSPRNG input. Since those bytes are random, it’s impossible for any party to tell the difference.

The final change is the major one, and the trickiest. The ciphertext used to be hashed into the shared secret. This difference would prevent interoperability. However, the mixing happens as an additional key derivation, which was entirely removed in ML-KEM, which instead returns the value K as-is. This means we can run ML-KEM to generate the shared secret K and then apply

SHAKE-256(K || c)[:32]

to generate the Kyber shared secret. No need to break the ML-KEM abstraction.

There’s one wrinkle: both Kyber and ML-KEM perform implicit rejection in Decapsulate by hashing a secret with the ciphertext and returning that as the shared secret. If we do the key derivation above on top of ML-KEM, we’ll hash the ciphertext twice for implicit rejections. That’s ok, because the output of implicit rejection is unpredictable by design, not an interoperation target.

The picture

In Berlin there's an old closed airport, Tempelhof, which is now a public park. Walking down the taxiways (pictured) or along the centrelines of the 09L/27R and 09R/27L crossed-out runways is kinda unsettling, at least for me. ("Should I be speaking with Ground or Tower? Can I enter this runway?") Fun fact, in 2010 a single-engine plane forgot to switch fuel tank and did an emergency landing on 27L. Closed runways are the best bad places to land, after all.

A cement taxiway pictured at sunset, from the middle of the yellow centreline. The airport terminal is visible on the horizon, and a patch of grass on the left.

This work was funded by a Google Open Source Security Subsidy and by my awesome clients—Sigsum, Latacora, Interchain, Smallstep, Ava Labs, Teleport, and Tailscale—who, through our retainer contracts, get face time and unlimited access to advice on Go and cryptography.

Here are a few words from some of them!

Latacora — We wrote about password hashing with delegation, a somewhat less known password hashing primitive. It's a PBKDF with a special property, that allows offloading hashing computation to a potentially untrusted server. In this blog post, we describe this primitive and discuss its applicability in the context of End-to-End Encrypted (E2EE) backup systems.

Teleport — For the past five years, attacks and compromises have been shifting from traditional malware and security breaches to identifying and compromising valid user accounts and credentials with social engineering, credential theft, or phishing. Teleport Identity Governance & Security is designed to eliminate weak access patterns through access monitoring, minimize attack surface with access requests, and purge unused permissions via mandatory access reviews.

Ava Labs — We at Ava Labs, maintainer of AvalancheGo (the most widely used client for interacting with the Avalanche Network), believe the sustainable maintenance and development of open source cryptographic protocols is critical to the broad adoption of blockchain technology. We are proud to support this necessary and impactful work through our ongoing sponsorship of Filippo and his team.

  1. The minimum common multiple of 1/4/10/12 with 8 is less than 64, so we can pack a few values in a uint64, and then serialize that. The result is IMHO pretty readable. ↩︎

  2. IIUC the matrix could have been hardcoded but is instead derived from a seed in the key instead to bypass debate on how the hardcoded matrix was generated, and any backdoor concerns. My somewhat spicy opinion is that we’ll come to see this as a mistake, and a case of failing to define parameters. If the matrix was hardcoded ML-KEM would be faster and simpler. For example, there would be no need for these tests at all, and the matrix derivation typo in the spec draft couldn’t have happened. Maybe some deployments can just specify and use a profile of ML-KEM that fixes the matrix seed. ↩︎