Hey! I'm David, the author of the Real-World Cryptography book. I'm a crypto engineer at O(1) Labs on the Mina cryptocurrency, previously I was the security lead for Diem (formerly Libra) at Novi (Facebook), and a security consultant for the Cryptography Services of NCC Group. This is my blog about cryptography and security and other related topics that I find interesting.

# Factoring RSA Keys With TLS Perfect Forward Secrecy posted September 2015

Note: This is a blogpost I initialy wrote for the NCC Group blog here.

Here's the story:

Florian Weimer from the Red Hat Product Security team has just released a technical report entitled "Factoring RSA Keys With TLS Perfect Forward Secrecy"

## Wait what happened?

A team of researchers ran an attack for nine months, and from 4.8 billion of ephemeral handshakes with different TLS servers they recovered hundreds of private keys.

The theory of the attack is actually pretty old, Lenstra's famous memo on the CRT optimization was written in 1996. Basicaly, when using the CRT optimization to compute a RSA signature, if a fault happens, a simple computation will allow the private key to be recovered. This kind of attacks are usually thought and fought in the realm of smartcards and other embedded devices, where faults can be induced with lasers and other magical weapons.

The research is novel in a way because they made use of Accidental Faults Attack, which is one of the rare kind of remote side-channel attacks.

This is interesting, the oldest passive form of Accidental Fault Attack I can think of is Bit Squatting that might go back to 2011 at that defcon talk.

## But first, what is vulnerable?

Any library that uses the CRT optimization for RSA might be vulnerable. A cheap countermeasure would be to verify the signature after computing it, which is what most libraries do. The paper has a nice list of who is doing that.

 Implementation Verification cryptlib 3.4.2 disabled by default GnuPG 1.4.1.8 yes GNUTLS see libgcrypt and Nettle Go 1.4.1 no libgcrypt 1.6.2 no Nettle 3.0.0 no NSS yes ocaml-nocrypto 0.5.1 no OpenJDK 8 yes OpenSSL 1.0.1l yes OpenSwan 2.6.44 no PolarSSL 1.3.9 no

But is it about what library you are using? Your server still has to be defective to produce a fault. The paper also have a nice table displaying what vendors, in their experiments, where most prone to have this vulnerability.

 Vendor Keys PKI Rate Citrix 2 yes medium Hillstone Networks 237 no low Alteon/Nortel 2 no high Viprinet 1 no always QNO 3 no medium ZyXEL 26 no low BEJY 1 yes low Fortinet 2 no very low

If you're using one of these you might want to check with your vendor if a firmware update or other solutions were talked about after the discovery of this attack. You might also want to revoke your keys.

Since the tests were done on a broad scale and not on particular machines, it is obvious that more are vulnerable to this attack. Also only instances connected to internet that offered TLS on port 443 were tested. The vulnerability could potentially exist in any stack using this CRT optimization with RSA.

The first thing you should do is asses where in your stack the RSA algorithm is used to sign. Does it use CRT? If so, does it verifies the signature? Note that the blinding techniques we talked about in one of our cryptography bulletin (may first of this year) will not help.

## What can cause your server to produce such erroneous signatures

They list 5 reasons in the paper:

• old or vulnerable libraries that have broken operations on integer. For example CVE-2014-3570 the square operations of OpenSSL was not working properly for some inputs

• race conditions, when applications are multithreaded

• arithmetic unit of the CPU is broken by design or by fatigue

• corruption of the private key

• errors in the CPU cache, other caches or the main memory.

Note that at the end of the paper, they investigate if a special hardware might be the cause and end up with the conclusion that several devices leaking the private keys were using Cavium hardware, and in some cases their "custom" version of OpenSSL.

## I'm curious. How does that work?

### RSA-CRT

Remember, RSA signature is basically $y = x^d \pmod{n}$ with $x$ the message, $d$ the private key and $n$ the public modulus. Also you might want to use a padding system but we won't cover that here. And then you can verify a signature by doing $y^e \pmod{n}$ and verify if it is equal to $x$ (with $e$ the public exponent).

CRT is short for Chinese Remainder Theorem (I should have said that earlier). It's an optimization that allows to compute the signatures in $\mathbb{Z}_p$ and $\mathbb{Z}_q$ and then combine it into $\mathbb{Z}_n$ (remember $n = pq$). It's way faster like that.

So basically what you do is:

$$\begin{cases} y_p = x^d \pmod{p} \\ y_q = x^d \pmod{q} \end{cases}$$

and then combine these two values to get the signature:

$$y = y_p q (q^{-1} \pmod{p}) + y_q p (p^{-1} \pmod{q}) \pmod{n}$$

And you can verify yourself, this value will be equals to $y_p \pmod{p}$ and $y_q \pmod{q}$.

### The vulnerability

Let's say that an error occurs in only one of these two elements. For example, $y_p$ is not correctly computed. We'll call it $\widetilde{y_p}$ instead. It is then is combined with a correct $y_q$ to produce a wrong signature that we'll call $\widetilde{y}$ .

So you should have:

$$\begin{cases} \widetilde{y} = \widetilde{y}_p \pmod{p}\\ \widetilde{y} = y_q \pmod{q} \end{cases}$$

Let's notice that if we raise that to the power $e$ and remove $x$ from it we get:

$$\begin{cases} \widetilde{y}^e - x = \widetilde{y}^e_p - x = a \pmod{p}\\ \widetilde{y}^e - x = y_q^e - x = 0 \pmod{q} \end{cases}$$

This is it. We now know that $q \mid \widetilde{y}^e - x$ while it also divides $n$. Whereas $p$ doesn't divide $\widetilde{y}^e - x$ anymore. We just have to compute the Greatest Common Divisor of $n$ and $\widetilde{y}^e - x$ to recover $q$.

### The attack

The attack could potentially work on anything that display a RSA signature. But the paper focuses itself on TLS.

A normal TLS handshake is a two round trip protocol that looks like this:

The client (the first person who speaks) first sends a helloClient packet. A thing filled with bytes saying things like "this is a handshake", "this is TLS version 1.0", "I can use this algorithm for the handshake", "I can use this algorithm for encrypting our communications", etc...

Here's what it looks like in Wireshark:

The server (the second person who speaks) replies with 3 messages: a similar ServerHello, a message with his certificate (and that's how we authenticate the server) and a ServerHelloDone message only consisting of a few bytes saying "I'm done here!".

A second round trip is then done where the client encrypts a key with the server's public key and they later use it to compute the TLS shared key. We won't cover them.

Another kind of handshake can be performed if both the client and the server accepts ephemeral key exchange algorithms (Diffie-Hellman or Elliptic Curve Diffie-Hellman). This is to provide Perfect Forward Secrecy: if the conversations are recorded by a third party, and the private key of the server is later recovered, nothing will be compromised. Instead of using the server's public key to compute the shared key, the server will generate a ephemeral public key and use it to perform an ephemeral handshake. Usualy just for this session or for a limited number.

An extra packet called ServerKeyExchange is sent. It contains the server's ephemeral public key.

Interestingly the signature is not computed over the algorithm used for the ephemeral key exchange, that led to a long series of attack which recently ended with FREAK and Logjam.

By checking if the signature is correctly performed this is how they checked for the potential vulnerability.

## I'm a researcher, what's in it for me?

Well what are you waiting for? Go read the paper!

But here are a list of what I found interesting:

We implemented a crawler which performs TLS handshakes and looks for miscomputed RSA signatures. We ran this crawler for several months. The intention behind this configuration is to spread the load as widely as possible. We did not want to target particular servers because that might have been viewed as a denial-of-service attack by individual server operators. We assumed that if a vulnerable implementation is out in the wild and it is somewhat widespread, this experimental setup still ensures the collection of a fair number of handshake samples to show its existence. We believe this approach—probing many installations across the Internet, as opposed to stressing a few in a lab—is a novel way to discover side-channel vulnerabilities which has not been attempted before.

• they used public information to choose what to target, like scans.io, tlslandscape and certificate-transparency.

• Some TLS servers need a valid Server Name Indication to complete a handshake, so connecting on port 443 of random IPs should not be very efficient. But they found that it was actually not a problem and most key found like that were from weird certificates that wouldn't even be trusted by your browser.

• To avoid too many DNS resolutions they bypassed the TTL values and cached everything (they used PowerDNS for that)

• They guess what devices were used to perform the TLS handshakes from what was written in the x509 certificates in the subject distinguished name field or Common Name field

• They used SSL_set_msg_callback() (see doc) to avoid modifying OpenSSL.
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# Explanation of my paper: Timing and Lattice Attacks on a Remote ECDSA OpenSSL Server: How Practical Are They Really? posted September 2015

I gave a talk about my paper at the NCC Group office in Chicago and recorded myself.

If you have any questions, you think something was not clear, badly explained, etc... I'll take any feedback since this is going to be my master defense in two weeks.

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# Timing and Lattice Attacks on a Remote ECDSA OpenSSL Server: How Practical Are They Really? posted August 2015

It's a timing attack on an vulnerable version of OpenSSL. In particular its ECDSA signature with binary curves.

There was an optimization right before the constant-time scalar multiplication of the nonce with the public point. That leads to a timing attack that leaks the length of the ephemeral keys of an Openssl server's signatures.

In this paper I explain how to setup such an attack, how to use lattices to recover the private key out of just knowing the lengths of the nonces of a bunch of signatures taken during an ephemeral handshake.

If this doesn't make sense to you just read the paper :D

Also everything is on this github repo. You can reproduce my setup for a vulnerable server and an attacker. Patch and tools are there. If you end up getting better results than the ones in the paper, well tell me!

Also here's a demo:

EDIT: it's on the ePrint archive as well now.

# Key Ceremony posted August 2015

There was a DNSSEC KSK (Key Signing Key) Ceremony. I guess it most not be too far away from what is a Key Ceremony

In public-key cryptography and computer security, a root key ceremony is a procedure where a unique pair of public and private root keys is generated. Depending on the certificate policy, the generation of the root Keys may require notarization, legal representation, witnesses and ‘key holders’ to be present, as the information on the system is a responsibility of the parties. The 'best practice' is to follow the SAS 70 standard for root key ceremonies.

The actual Root Key-Pair generation is normally conducted in a secure vault that has no communication or contact with the outside world other than a single telephone line or intercom. Once the vault is secured, all personnel present must prove their identity using at least two legally recognized forms of identification. Every person present, every transaction and every event is logged by the lawyer in a Root Key Ceremony Log Book and each page is notarized by the notary. From the moment the vault door is closed until it is re-opened, everything is also video recorded. The lawyer and the organization’s two signatories must sign the recording and it too is then notarized.

Finally, as part of the above process, the Root Key is broken into as many as twenty-one parts and each individual part is secured in its own safe for which there is a key and a numerical lock. The keys are distributed to as many as twenty-one people and the numerical code is distributed to another twenty-one people.

It's not that interesting, but I was just curious so I watched the footage of what it is here: http://data.iana.org/ksk-ceremony/21/KSK21-CAM1.mp4

It's boring.

But if you have nothing better to do, or you are curious like me, well here you go

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# I got a job! posted August 2015

Hey you! So mmm, I don't know if you've been reading my blog for long, but it all started when I got accepted at the University of Bordeaux' Cryptography Master. At first it was just a place where I would talk about my (then) new life in Bordeaux and what I was doing in class.

2 years and 287 blog posts later, here I am, still blogging and still in school. But not for long! Well not in school for long, I'm still gonna blog don't worry.

So yeah, the big news is, I'll be starting full time as a security consultant for the Cryptography Services team of NCC Group in November!

Woop woop!

Pardon? You are here for the crypto? ah umm, wait, I have this:

It's from this paper: Practical realisation and elimination of an ECC-related software bug attack by B.B.Brumley, Barbosa, Page and Vercauteren.