### Post-Quantum Cryptography

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Chrome was augmented with the ability to include a dummy, arbitrarily-sized extension in the TLS ClientHello fixed number of bytes of random noise. After taking into account the performance and key size offered by different types key-exchange schemes, he concluded that constructs based on structured lattices may be most suitable for future use in TLS. However, Langley also observed a peculiar phenomenon; client connections measured at 95th percentile had much higher latency than the median. It means that in those cases, isogeny-based systems may be a better choice.

In our experiment, we want to more thoroughly evaluate and ascribe root causes to these unexpected latency increases. We would particularly like to learn more about the characteristics of those networks: what causes increased latency? We want to answer key questions, like:.

Our experiment will involve both server- and client-side performance statistics collection from real users around the world all the data is anonymized. Cloudflare is operating the server-side TLS connections. Additionally, Chrome will always include X for servers that do not support post-quantum key exchange.

The post-quantum key exchange will only be negotiated in TLS version 1. To this end, we will perform follow-up experiments based on per-client information we collect server-side.

### Post-Quantum Cryptography

As a first-pass analysis, we will investigate whether the slowed-down clients share common network features, like common ASes, common transit networks, common link types, and so on. To determine this, we will run a traceroute from vantage points close to our servers back toward the clients not overloading any particular links or hosts and study whether some client locations are subject to slowdowns for all destinations or just for some. Be warned: the details of PQ cryptography may be quite complicated.

In some cases it builds on classical cryptography, and in other cases it is completely different math. It would be rather hard to describe details in a single blog post. Instead, we are giving you an intuition of post-quantum cryptography, rather than provide deep academic-level descriptions. Nevertheless, settle in for a bit of an epic journey because we have a lot to cover. As PKE, it also allows agreement on a secret, but in a slightly different way.

The idea is that the session key is an output of the encryption algorithm, conversely to public key encryption schemes where session key is an input to the algorithm. In a KEM, Alice generates a random key and uses the pre-generated public key from Bob to encrypt encapsulate it. This results in a ciphertext sent to Bob. Bob uses his private key to decrypt decapsulate the ciphertext and retrieve the random key. The idea was initially introduced by Cramer and Shoup.

Experience shows that such constructs are easier to design, analyze, and implement as the scheme is limited to communicating a fixed-size session key. The key exchange KEX protocol, like Diffie-Hellman , is yet a different construct: it allows two parties to agree on a shared secret that can be used as a symmetric encryption key.

For example, Alice generates a key pair and sends a public key to Bob. He then sends his public key to Alice who can now generate the same shared secret. Alice performs key generation and sends the public key to Bob. Bob uses it to encapsulate a symmetric session key and sends it back to Alice. Alice decapsulates the ciphertext received from Bob and gets the symmetric key. This is actually what we do in our experiment to make integration with the TLS protocol less complicated.

He described this implementation in detail here. NTRU is a cryptosystem based on a polynomial ring. One can add polynomials in the ring in the usual way, by simply adding theirs coefficients modulo some integer. In other words, polynomial ring arithmetic is very similar to modular arithmetic , but instead of working with a set of numbers less than N , you are working with a set of polynomials with a degree less than N. First, we generate a pair of public and private keys. HRSS brings an improvement to this issue since it ensures that those inverses always exist, making key generation faster than as proposed initially in NTRU.

The goal of the blinder is to generate different ciphertexts per encyption. NTRU cryptosystem is a grandfather of lattice-based encryption schemes. His work evolved into a whole area of research with the goal of creating more practical, lattice-based cryptosystems. The picture below visualizes lattice as points in a two-dimensional space. However, the lattices used in cryptography have higher dimensions, say above , as well as highly non-orthogonal basis vectors. On these instances, the problems get extremely hard to solve.

The main improvements are:. Read more about SIDH in our previous post. The math behind SIDH is related to elliptic curves. An elliptic curve is a set of points that satisfy a specific mathematical equation.

An interesting fact about elliptic curves is have a group structure. That is, the set of points on the curve have associated a binary operation called point addition. The set of points on the elliptic curve is closed under addition.

## Quantum-Safe Cryptography (QSC)

Thus, adding two points results in another point that is also on the elliptic curve. If we can add two different points on a curve, then we can also add one point to itself. Multiplication of scalars is commutative. It turns out that carefully if choosing an elliptic curve "correctly", scalar multiplication is easy to compute but extremely hard to reverse.

## ETSI - Quantum - Safe Cryptography, Computing Cryptography

This problem is suitable for cryptographic purposes. Alice and Bob agree on a secret key as follows. To agree on a shared secret, each party multiplies their private key with the public key of the other party.

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The result of this is the shared secret. There is a vast theory behind elliptic curves. An introduction to elliptic curve cryptography was posted before and more details can be found in this book. Each curve has a number that can be associated to it. This number is not unique per curve, meaning many curves have the same value of j-invariant , but it can be viewed as a way to group multiple elliptic curves into disjoint sets. We say that two curves are isomorphic if they are in the same set, called the isomorphism class.

The j-invariant is a simple criterion to determine whether two curves are isomorphic. When it comes to isogeny , think about it as a map between two curves. There may exist many of those mappings, each curve used in SIDH has small number of isogenies to other curves. Natural question is how do we compute such isogeny. Here is where the kernel of an isogeny comes. The kernel uniquely determines isogeny up to isomorphism class. Formulas for calculating isogeny from its kernel were initially given by J.

## Post-Quantum Cryptography

There are two isomorphism classes on the picture above. But curious readers can find a number of academic research papers available on the Internet. For SIDH to work, we need a big set of elements and something secret that will act on the elements from that set. When we talk about Isogeny Based Cryptography , as a topic distinct from Elliptic Curve Cryptography , we usually mean algorithms and protocols that rely fundamentally on the structure of isogeny graphs.

An example of such a small graph is pictured below. Each vertex of the graph represents a different j-invariant of a set of supersingular curves. The edges between vertices represent isogenies converting one elliptic curve to another. As you can notice, the graph is strongly connected, meaning every vertex can be reached from every other vertex.

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In the context of isogeny-based crypto, we call such a graph a supersingular isogeny graph. As the graph is strongly connected, it is possible to walk a whole graph by starting from any vertex, randomly choosing an edge, following it to the next vertex and then start the process again on a new vertex. Such a way of visiting edges of this graph is called a random walk. The random walk is a key concept that makes isogeny based crypto feasible.

When you look closely at the graph, you can notice that each vertex has a small number of edges incident to it, this is why we can compute the isogenies efficiently. The key question is - where does the security of the scheme come from exactly? In order to get it, it is necessary to visit a couple hundred vertices. What it means in practice is that secret isogeny of large degree is constructed as a composition of multiple isogenies of small, prime degree. Which means, the secret isogeny is:. This property and properties of the isogeny graph are what makes some of us believe that scheme has a good chance to be secure.

The security level of a system depends on value n - the number of steps taken during the walk. The random walk is a core process used when both generating public keys and computing shared secrets. Isogeny is also applied to points P and Q. The core idea in SIDH is to compose two random walks on an isogeny graph of elliptic curves in such a way that the end node of both ways of composing is the same.

The triple forms a public key which is exchanged between parties. The picture below visualizes the operation.