Ars Technica has a really good article with details:

Ultimately, the architects settled on a creative solution. Rather than bolt KEM onto the existing double ratchet, they allowed it to remain more or less the same as it had been. Then they used the new quantum-safe ratchet to implement a parallel secure messaging system.

Now, when the protocol encrypts a message, it sources encryption keys from both the classic Double Ratchet and the new ratchet. It then mixes the two keys together (using a cryptographic key derivation function) to get a new encryption key that has all of the security of the classical Double Ratchet but now has quantum security, too...

Similarly, quantum factorisation is performed using sleight-of-hand numbers that have been selected to make them very easy to factorise using a physics experiment and, by extension, a VIC-20, an abacus, and a dog. A standard technique is to ensure that the factors differ by only a few bits that can then be found using a simple search-based approach that has nothing to do with factorisation…. Note that such a value would never be encountered in the real world since the RSA key generation process typically requires that |p-q| > 100 or more bits [9]. As one analysis puts it, “Instead of waiting for the hardware to improve by yet further orders of magnitude, researchers began inventing better and better tricks for factoring numbers by exploiting their hidden structure” [10]...

The first new algorithm Microsoft added to SymCrypt is called ML-KEM. Previously known as CRYSTALS-Kyber, ML-KEM is one of three post-quantum standards formalized last month by the National Institute of Standards and Technology (NIST). The KEM in the new name is short for key encapsulation. KEMs can be used by two parties to negotiate a shared secret over a public channel. Shared secrets generated by a KEM can then be used with symmetric-key cryptographic operations, which aren’t vulnerable to Shor’s algorithm when the keys are of a sufficient size...

After three rounds of evaluation and analysis, NIST selected four algorithms it will standardize as a result of the PQC Standardization Process. The public-key encapsulation mechanism selected was CRYSTALS-KYBER, along with three digital signature schemes: CRYSTALS-Dilithium, FALCON, and SPHINCS+.

These algorithms are part of three NIST standards that have been finalized:

• FIPS 203: Module-Lattice-Based Key-Encapsulation Mechanism Standard
• FIPS 204: Module-Lattice-Based Digital Signature Standard
• FIPS 205: Stateless Hash-Based Digital Signature Standard...

The mathematics of lattice cryptography revolve around combining sets of vectors—that’s the lattice—in a multi-dimensional space. These lattices are filled with multi-dimensional periodicities. The ...

There’s a lot of detail in the Apple blog post, and more in Douglas Stabila’s security analysis.

I am of two minds about this. On the one hand, it’s probably premature to switch to any particular post-quantum algorithms. The mathematics of cryptanalysis for these lattice and other systems is still rapidly evolving, and we’re likely to break more of them—and learn a lot in the process—over the coming few years. But if you’re going to make the switch, this is an excellent choice. And Apple’s ability to do this so efficiently speaks well about its algorithmic agility, which is probably more important than its particular cryptographic design. And it is probably about the right time to worry about, and defend against, attackers who are storing encrypted messages in hopes of breaking them later on future quantum computers...

Now, computer scientist Oded Regev has a significant speed-up to Shor’s algorithm, at the cost of more storage.

Details are in this article. Here’s the result:

The improvement was profound. The number of elementary logical steps in the quantum part of Regev’s algorithm is proportional to ...

This process has been going on since 2016, and since that time there has been a huge increase in quantum technology and an equally large increase in quantum understanding and interest. Yet seven years later, we have only four algorithms, although last week NIST announced that a number of other candidates are under consideration, a process that is expected to take “several years.

The delay in developing quantum-resistant algorithms is especially troubling given the time it will take to get those products to market. It generally takes four to six years with a new standard for a vendor to develop an ASIC to implement the standard, and it then takes time for the vendor to get the product validated, which seems to be taking a troubling amount of time...

Researchers have just published a side-channel attack—using power consumption—against an implementation of the algorithm that was supposed to be resistant against that sort of attack.

The algorithm is not “broken” or “cracked”—despite headlines to the contrary—this is just a side-channel attack. What makes this work really interesting is that the researchers used a machine-learning model to train the system to exploit the side channel...

We have long known from Shor’s algorithm that factoring with a quantum computer is easy. But it takes a big quantum computer, on the orders of millions of qbits, to factor anything resembling the key sizes we use today. What the researchers have done is combine classical lattice reduction factoring techniques with a quantum approximate optimization algorithm. This means that they only need a quantum computer with 372 qbits, which is well within what’s possible today. (The ...

Current quantum computers are still toy prototypes, and the engineering advances required to build a functionally useful quantum computer are somewhere between a few years away and impossible. Even so, we already know that that such a computer could potentially factor large numbers and compute discrete logs, and break the RSA and Diffie-Hellman public-key algorithms in all of the useful key sizes...

It was just broken, really badly.

We present an efficient key recovery attack on the Supersingular Isogeny Diffie­-Hellman protocol (SIDH), based on a “glue-and-split” theorem due to Kani. Our attack exploits the existence of a small non-scalar endomorphism on the starting curve, and it also relies on the auxiliary torsion point information that Alice and Bob share during the protocol. Our Magma implementation breaks the instantiation SIKEp434, which aims at security level 1 of the Post-Quantum Cryptography standardization process currently ran by NIST, in about one hour on a single core...

For general encryption, used when we access secure websites, NIST has selected the CRYSTALS-Kyber algorithm. Among its advantages are comparatively small encryption keys that two parties can exchange easily, as well as its speed of operation.

For digital signatures, often used when we need to verify identities during a digital transaction or to sign a document remotely, NIST has selected the three algorithms CRYSTALS-Dilithium...

The NSA already has classified quantum-resistant algorithms of its own that it developed over many years, said Joyce. But it didn’t enter any of its own in the contest. The agency’s mathematicians, however, worked with NIST to support the process, trying to crack the algorithms in order to test their merit.

“Those candidate algorithms that NIST is running the competitions on all appear strong, secure, and what we need for quantum resistance,” Joyce said. “We’ve worked against all of them to make sure they are solid.”...

Finally, we calculate the number of physical qubits required to break the 256-bit elliptic curve encryption of keys in the Bitcoin network within the small available time frame in which it would actually pose a threat to do so. It would require 317 × 10⁶ physical qubits to break the encryption within one hour using the surface code, a code cycle time of 1 μs, a reaction time of 10 μs, and a physical gate error of 10^-3. To instead break the encryption within one day, it would require 13 × 10...