## Entity authentication with a KEM

In cryptography, the process of authenticating a user (or app/service) is known as entity authentication or identification (to distinguish it from message authentication or data origin authentication). There are lots of ways to do this. In this post I’m going to talk about authentication schemes based on public key cryptography. It turns out that the notion of Key Encapsulation Mechanism (KEM), originally invented for (unauthenticated) public key encryption, also provides a unifying framework for thinking about public key entity authentication.

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## Book review: The Joy of Cryptography

Mike Rosulek, Oregon State University. Draft of January 3, 2021. Online: The Joy of Cryptography.

This is a freely-available book covering introductory material on cryptography. It’s suitable for anyone with undergraduate-level computer science knowledge. As is often the case in cryptography textbooks, there is a brief review of mathematical background in the first (or zeroth in this case) chapter: modular arithmetic, probability, and Big-O notation. This chapter is brief but clear and establishes the sense of humour that is present throughout the book.

Overall, the book is very well-presented and -written for the target audience—undergraduates and practitioners. I have read several other textbooks on cryptography, but most are pitched more at a level for post-graduate students (even if apparently meant for undergrads). The writing here is always clear and focused, without leaving out essential details. In particular, I appreciated that security definitions are immediately followed by a discussion on the implications of that definition, and what it leaves out.

On a more basic note, the book is beautifully presented, and clearly a lot of thought has gone into the typography, layout, and diagrams (which have a pleasing “tightness” in their use of space).

It is largely a book about symmetric cryptography—the first 12 chapters cover symmetric material in depth, and only the last 3 chapters cover public key crypto. I actually quite like this split, because I believe many topics that are discussed in the context of public key cryptography have simpler illustrations in a symmetric setting. Still, you may find the public key material a little sparse. There is no discussion of elliptic curves, for example, or the developing field of post-quantum cryptography. There is also no material on hybrid encryption (i.e., using a combination of public key and symmetric primitives) or the KEM-DEM paradigm, which I would expect in any modern textbook.

Update: Mike Rosulek pointed out that the book does actually cover hybrid encryption in the very last section: 15.4. Mea culpa.

It is a draft though, and the roadmap makes clear that some subjects that I would expect to be covered are coming at some point. On the other hand, by the time you finish the book you will know enough to tackle some of the more comprehensive textbooks, such as Smart’s Cryptography Made Simple (lol) or Boneh & Shoup’s Graduate Course in Applied Cryptography.

The presentation is also quite slow, taking its time to develop topics and foster a real understanding. This is both a blessing and a curse. The clarity is welcome, but you have to get to chapter 12 before authenticated encryption is covered. If you are looking to find out “what should I be using?”, waiting 12 chapters to find out might be testing your patience a little. But if you want to really understand the topics and what modern cryptography is all about, your patience is paid off.

The approach used in the book to proving security of schemes is that of code-based games. The idea here is that a cryptographic scheme is developed as a kind of abstract data type or API, defining the operations that can be performed: generating a key, encrypting a message, decrypting a message, and so on. An adversary is then given access to two implementations of that API: one that is our real encryption scheme, and another that is some idealised (typically random) implementation of it. The API the attacker sees is restricted in some ways compared to what a real user would see (e.g., the attacker doesn’t get to see the key used for encryption). A scheme is secure if we can prove that no “reasonable” adversary can tell which implementation they have been given any better than a blind guess.

A proof in this scheme proceeds by replacing parts of the real implementation with idealised versions, until we can show that an attack on the real implementation is equivalent to an attack on the idealised version. Such proofs are always relative to some basic set of primitives that we believe to be close enough to the ideal versions already (generally because a lot of smart people have tried to break them for a long time and failed).

This is a different approach than a lot of textbooks, which use this indistinguishability proof style only for encryption, and then switch to other types of games for other security notions (such as message authentication). I found this uniform presentation very helpful in understanding the material and in seeing connections between security notions that I hadn’t appreciated before. For example, this book is the first time I’ve appreciated the importance of the Decisional Diffie-Hellman assumption to proving pseudorandomness of key agreement, which then motivates the use of quadratic residues and safe primes. (Although it’s missing a follow-up discussion on the use of hashing/random oracles to relax this assumption, which is what you would do in practice—a chapter on hybrid encryption would help clarify this).

An another example, the book introduces secret sharing early on and then points out that a one-time pad can be viewed as a simple 2-of-2 secret sharing technique, a connection I had never made before. Secret sharing can be viewed as a type of multi-party computation (MPC) and the author is an MPC researcher and co-author of an MPC textbook. This specialist knowledge provides a welcome fresh perspective on otherwise familiar material.

Overall, this is an excellent introduction to cryptography, with a novel and engaging approach to security proofs. The draft status and some lacking topic areas make me hesitant to recommend it as your sole introduction to cryptography just yet, but as a companion to another textbook it has a lot to recommend it. Even if you believe you already know all this material, you may still gain a lot from reading it: I know I did.

## From KEMs to protocols

This is the third part of my series on Key Encapsulation Mechanisms (KEMs) and why you should care about them. Part 1 looked at what a KEM is and the KEM/DEM paradigm for constructing public key encryption schemes. Part 2 looked at cases where the basic KEM abstraction is not sufficient and showed how it can be extended to add support for multiple recipients and sender authentication. At the end of part 2, I promised to write a follow-up about tackling forward-secrecy and replay attacks in the KEM/DEM paradigm, so here it is. In this article we’ll go from simple one-way message encryption to a toy version of the Signal Protocol that provides forward secrecy and strong authentication of two (or more) parties.

WARNING: please pay attention to the word “toy” in the previous sentence. This is a blog post, not a thorough treatment of how to write a real end-to-end encrypted messaging protocol.

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## When a KEM is not enough

In my previous post, I described the KEM/DEM paradigm for hybrid encryption. The key encapsulation mechanism is given the recipient’s public key and outputs a fresh AES key and an encapsulation of that key that the recipient can decapsulate to recover the AES key. In this post I want to talk about several ways that the KEM interface falls short and what to do about it:

• As I’ve discussed before, the standard definition of public key encryption lacks any authentication of the sender, and the KEM-DEM paradigm is no exception. You often want to have some idea of where a message came from before you process it, so how can we add sender authentication?
• If you want to send the same message to multiple recipients, a natural approach would be to encrypt the message once with a fresh AES key and then encrypt the AES key for each recipient. With the KEM approach though we’ll end up with a separate AES key for each recipient. How can we send the same message to multiple recipients without encrypting the whole thing separately for each one?
• Finally, the definition of public key encryption used in the KEM/DEM paradigm doesn’t provide forward secrecy. If an attacker ever compromises the recipient’s long-term private key, they can decrypt every message ever sent to that recipient. Can we prevent this?

In this article I’ll tackle the first two issues and show how the KEM/DEM abstractions can be adjusted to cope with each. In a follow-up post I’ll then show how to tackle forward secrecy, along with replay attacks and other issues. Warning: this post is longer and has more technical details than the previous post. It’s really meant for people who already have some experience with cryptographic algorithms.

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## Hybrid encryption and the KEM/DEM paradigm

If you know a bit about public key cryptography, you probably know that you don’t directly encrypt a message with a public key encryption algorithm like RSA. This is for many reasons, one of which being that it is incredibly slow. Instead you do what’s called hybrid encryption: first you generate a random AES key (*) and encrypt the message with that (using a suitable authenticated encryption mode), then you encrypt the AES key with the RSA public key. The recipient uses their RSA private key to decrypt the AES key and then uses that to decrypt the rest of the message. This is much faster than trying to encrypt a large message directly with RSA, so pretty much all sane implementations of RSA encryption do this.