Hey! I'm David, a security engineer at the Blockchain team of Facebook, previously 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.

# Pollard's p-1 factorization algorithmposted March 2016

Got my graphic tablet back, needed to do a small video to get back into it so I made something on Pollard's p-1 factorization algorithm:

You can find the records on factoring with p-1 on loria.fr, the biggest prime factor found was of 66 digits (~220bits) using B1=10^8 and B2=10^10. But people have been using bigger parameters like B1=10^10 and B2=10^15. It doesn't really make sense to continue using p-1 after that, and more efficient algorithms that still have a complexity tied to the size of the smallest factor exist. The Elliptic Curve Method (Or Lenzstra factorization method) is one of them, and is carrying the same ideas as p-1 in the elliptic curves.

In the video I also don't talk about B2. This is if you have a factorization of p-1 that is B1-powersmooth, except for a large single prime. You can just set a B2 which would be larger than this last factor and try every factor between B1 and B2. There are some optimizations that exist to do that faster instead of doing it naively but this is it.

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# To_printposted March 2016

I have this list of papers that is accumulating in a folder. After an idea of @gtank I decided to print the whole (~150 two-sided pages) and go to my local UPS store to bind it for ~8\$.

I'm not posting the pdf I printed here, but I made it with pdfjoin --paper letterpaper --rotateoversize false *, could have used pdfbook to make it smaller but after printing a few pages I felt that it would be too small for my weary eyes (now I sound like an old bag).

here's the list of the papers I printed. They all seem like you can print them without going to jail.

# How are TLS records encrypted?posted March 2016

Once we are done with the handshake, both parties are now holding the same set of keys. Right after the Server sends its ChangeCipherSpec message it starts encrypting. Right after the Client sends his own ChangeCipherSpec he starts encrypting his messages as well.

The encrypted records still start with the type of record, the TLS version and the length of the following bytes in clear. The rest is encrypted.

We won't talk about compression because there are a bunch of vulnerabilities that should make you think twice about using compression. So here it is, null! (more about that)

Let's follow the TLS 1.2 RFC here

struct {
ContentType type;
ProtocolVersion version;
uint16 length;
select (SecurityParameters.cipher_type) {
case stream: GenericStreamCipher;
case block:  GenericBlockCipher;
} fragment;
} TLSCiphertext;

So as I said, we start with the type, the version and the length.

Right after the CipherSpecChange both parties will send an encrypted handshake message (a MAC of the whole transcript to authenticate the handshake), but most of the messages after will be encrypted Application data messages containing the real communications we want to protect.

0x17 is the byte for application data, then we have the TLS version, for TLS 1.2 it is 0x0303 (don't bother), then we have the length of the fragment which is described below for the AES-CBC case.

struct {
opaque IV[SecurityParameters.record_iv_length];
block-ciphered struct {
opaque content[TLSCompressed.length];
opaque MAC[SecurityParameters.mac_length];
};
} GenericBlockCipher;

So in the case of AES-128, you would have an IV of 16 bytes followed by the encrypted data.

Yup, the MAC is not here because it is encrypted. TLS is a MAC-then-encrypt construction (...), you can do encrypt-then-MAC in practice but through an extension (cf. RFC 7366).

The rest is pretty straight forward, after decryption of the block-ciphered structure you remove the padding, check the MAC, use the content.

Here's how the mac is used:

   MAC(MAC_write_key, seq_num +
TLSCompressed.type +
TLSCompressed.version +
TLSCompressed.length +
TLSCompressed.fragment);

where "+" denotes concatenation.

That's it for today!

# Distinction between the TLS PRF for the Master Key and for the Keysposted March 2016

I just realized one more thing in TLS that doesn't make sense (besides the fact that different versions of TLS have different PRFs). Here's RFC 5246 (the RFC on TLS 1.2) on how to use the PRF transform your pre-master key into a master key:

Here's the same RFC on how to use the PRF to transform your master key into your 4 or 6 keys:

Noticed anything?

Took me some time, the first takes the server random appended to the client random, while the second takes the client random appended to the server random. I'm willing to bet this is not to circumvent any attack but rather to confuse the implementer...

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# TLS, Pre-Master Secrets and Master Secretsposted March 2016

Everything you want to know about TLS 1.2 is in RFC 5246. But as you may know, if you've read RFCs before, it is not easy to parse (plus they have some sort of double spaces non-sense).

Before we can encrypt/MAC everything with keys to secure our connection, we need to go over a key exchange called the Handshake to safely agree on a set of keys for both parties to use. The handshake can currently use 5 different algorithms to do the key exchange: RSA, Diffie-Hellman, Elliptic Curve Diffie-Hellman and the ephemeral versions of the last two algorithms.

This blogpost is about what happens between this key exchange and the encryption/authentication of data.

## The Pre-Master Secret

The pre-master key is the value you directly obtain from the key exchange (e.g. $g^{ab} \pmod{p}$ if using Diffie-Hellman). Its length varies depending on the algorithm and the parameters used during the key exchange. To make things simpler, we would want a fixed-length value to derive the keys for any cipher suite we would want to use. This is the reason behind a pre master secret. The fixed-length value we'll call master secret. Here the RFC tells us how to compute it from the pre-master secret after having removed the leading zeros bytes.

master_secret = PRF(pre_master_secret, "master secret",
ClientHello.random + ServerHello.random)
[0..47];

The two random values ClientHello.random and ServerHello.random, sometimes called "nonces", are randomly generated and sent during the ClientHello of each parties. This is to bound the soon-to-be master key to this session. PRF stands for Pseudo-random function, basically some concrete construction that emulates a random oracle: given an input will produce an output computationally indistinguishable from a truly random sequence. But let's move on, and we will see later what exactly is that PRF.

## The Master Secret

A master secret is always 48 bytes. So now that we have a fixed length value, we can derive 4 keys from it:

• client_write_MAC_key
• server_write_MAC_key
• client_write_key
• server_write_key

As you can probably guess, MAC keys are for the authentication and integrity with whatever MAC algorithm you chose in the cipher suite, write keys are for the symmetric encryption.

Interestingly, two keys are generated for every purpose: one key per side. This is mostly by respect of good practices. Always segregate the use of your keys.

The symmetric ciphers chosen in the handshake will dictate how long these keys we generate need to be. Note that AEAD ciphers that combine both authentication and encryption will not need MAC keys but will need two other keys instead: client_write_IV and server_write_IV. This is because their MAC keys are directly derived from the encryption keys.

The same PRF we used on the pre-master key will be used on the master-key over and over until enough bytes have been created for the keys. From the section 6.3 of the RFC:

key_block = PRF(SecurityParameters.master_secret,
"key expansion",
SecurityParameters.server_random +
SecurityParameters.client_random);

The key_block value is then cut into enough keys.

That's it! Here's a recap:

Diffie-Hellman -> pre-master key -> 48bytes master key -> 4 variable-length keys.

## The PRF

OK. Now that we got a nice global view of the process, let's dig deeper. The PRF used in TLS 1.2 is discussed here. It is quite different from the PRF used in TLS 1.1, see here.

Remember, for example how it was used to transform the pre-master key into a master key:

master_secret = PRF(pre_master_secret, "master secret",
ClientHello.random + ServerHello.random)
[0..47];

This is how the PRF function is used:

PRF(secret, label, seed) = P_<hash>(secret, label + seed)

If you want to follow along with code, here's the relevant golang code

P_hash(secret, seed) = HMAC_hash(secret, A(1) + seed) +
HMAC_hash(secret, A(2) + seed) +
HMAC_hash(secret, A(3) + seed) + ...

where + indicates concatenation, A() is defined as:

A(0) = seed
A(i) = HMAC_hash(secret, A(i-1))

This was a copy/paste from the RFC. To make it clearer: We use the label string ("master secret" in our example) concatenated with the two peers' random values as a seed.

We then MAC the seed with our pre-master secret as the key. We use the first output. Iterating the MAC gives us the subsequent values that we can append to our output.

$u_0 = label + serverHello.random + clientHello.random$

$u_i = HMAC(secret, u_{i-1})$

$output = u_1 , u_2 , \cdots$ This goes on and on until the output is long enough to cover the 48 bytes of the master key (or the 4 keys if we're applying to PRF on the master key).

If P_256 is being used, then SHA-256 is being used. This means the output of HMAC will be 256 bits (32 bytes). To get the 48 bytes of the master key, two iterations are enough, and the remaining bytes can be discarded.

# Discrete logarithms in Multiplicative prime groupsposted March 2016

I'm doing some tests on how Pollard Rho performs. I implemented the thing in Sage here and it doesn't perform that well I found. Pollard Kangaroo is also bad, but that must come from my implementation (I didn't really go further here since I don't really need Kangaroo: I already know the order + the value I'm looking for is not in any particular interval)

old_rho is Pollard rho, rho_lambda is the mislabeled Pollard Kangaroo algorithm, trials is the simple enumeration.

I implemented the algorithm in Go, along some nice functions/variables that make Go's bignumber library a bit easier to tolerate. And guess what? What takes Sage 63 seconds to compute only take Go 5 seconds. The implementation is a copy/paste of what I did in Sage, no optimizations.

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# DROWN attack on OpenSSLposted March 2016

There is a new attack on OpenSSL. It's called DROWN.

Two problems:

1. in OpenSSL versions prior to January of this year, SSLv2 is by default not disabled. They thought that removing all the SSLv2 cipher suites from the default cipher string (back in 2010) would work but... nope. Even if not advertised in the serverHello, you can still do a handshake with whatever SSLv2 cipher you want. Another way of completely disabling SSLv2 exists, but it's recent and it is not the default option.

2. A padding oracle attack still exists in SSLv2. This is because of the export cipher suites. These weak ciphers and key lengths the USA government was forcing on OpenSSL so that people overseas could use it. So, these export cipher suites, nowadays they are bruteforce-able. It takes a few hours though, and a few hundred dollars, so no easy active MITM. It's a rather passive attack.

This is a cross-protocol attack. This means that you are a MITM, but you leave the client doing his thing on a TLS 1.2 or whatever SSLv3+ protocol. In the mean time though, you use the SSLv2 connection as an Oracle to recover the premaster-key (and thus the session key that is derived from it).

Three things:

1. The attack works on RSA handshakes. In the handshake (precisely in the clientKeyExchange) the client will encrypt his premaster-key with the server's RSA public key, this is what the attack decrypts. The server doesn't support RSA handshakes? You'll have to attack another server.

2. The server doesn't have to work with SSLv2. If another server (could even be a mail server) sharing the same RSA key and supporting SSLv2 exists, then you can use it as your oracle during the attack! Practical much?

3. To use the oracle, you need to first transform the RSA encrypted premaster-key into a valid SSLv2 RSA encrypted master-key. It is quite different, because of protocol differences, and you need to use quite a few tricks (trimmers!). It doesn't work all the time, around 1 out of 1,000 RSA encrypted premaster-key can be decrypted. This is often more than enough to steal the cookies and have consequences. If you're targeting a specific individual it can take time though, so to speed up these 1,000 handshakes just inject some javascript in a non-https webpage!

That's pretty much everything. I'm still going through the paper, trying to understand the math. There is a tool here to test your website. Another way of doing this (especially for internal servers) is to get an openssl version prior to january this year and do that on all of your subdomains/domains: openssl s_client -ssl2 -connect www.cryptologie.net:443

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# Briefs about cryptoposted March 2016

I'm posting a bunch of things all the time on twitter/facebook. There are mostly quotes of things I'm reading and that I find interesting.

here are the last ones:

# Checking your Diffie-Hellman parametersposted February 2016

I made a simple script to check for your DH modulus. Is it long enough? Is it a safe prime? I thought some non-cryptographers could benefit from such a tool, since usually all I have to do is fire up Sage and run some tests, but if you don't have Sage this can be tricky and annoying so...

Here's test_DHparams

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# NIST and Quantum Computersposted February 2016

A few weeks ago, NIST released a draft on their report on Post-Quantum Cryptography.

As we all know, some things are happening in the quantum computing world. Some are saying it will never work, some are saying it will but that it will take time until large enough quantum computers could break today's crypto.

So reading this paragraph taken from the NIST document, it can make sense on why we would want to move today to post-quantum crypto:

Historically, it has taken almost 20 years to deploy our modern public key cryptography infrastructure. It will take significant effort to ensure a smooth and secure migration from the current widely used cryptosystems to their quantum computing resistant counterparts. Therefore, regardless of whether we can estimate the exact time of the arrival of the quantum computing era, we must begin now to prepare our information security systems to be able to resist quantum computing.

Let's see where is this number coming from. SSL/TLS, its protocol or its implementation, its coverage or its efficiency, has been a huge mess so far:

• In 2009, 7 years ago, moxie introduced SSLStrip at Blackhat, a technique to render https completely useless without preloaded HSTS.

• It's only in 2013, 3 years ago, that facebook finally made the whole app https-only just blows my mind. And that's not thinking of the myriad of companies, commerce, banks and other websites that were all accessible through http back then.

• Nowdays most websites are still vulnerable to moxie's 2009 attack. Think about it, TLS is supposed to protect the communications against a passive and an active attacker on the network. In the passive case, I think it succeeded (in most cases). In the active case? Even HSTS or HPKP can still be somehow circumvented. Only browsers are fully capable of protecting us nowadays.

• And this is ignoring all the horrible implementations flaws like heartbleed, the broken cert validations of browsers, the broken basicConstraints of most CAs...

We could also talk about the deprecation of md5 and sha1, but sleevi does that better than me:

• 1996, 20 years ago, researches recommend to switch from md5 to sha1 because of recent advances.

• 2013, 17 years after the recommendation, Apple finally removes its support for MD5 in certificates.

• We're still in the middle of deprecating sha1, and it's a mess.

(there's also a graphical timeline made by ange: here)

Or what about the deprecation of DES? Or RC4? Or 1024 bit DH? ..

To come back to the NIST's report, here's a nice table of the impact of quantum computing on today's algorithms:

sums up pretty well what djb wrote:

Imagine that it's fifteen years from now. Somebody announces that he's built a large quantum computer. RSA is dead. DSA is dead. Elliptic curves, hyperelliptic curves, class groups, whatever, dead, dead, dead.

Contrarily to the european initiative PQCrypto, they seem to imply that they will recommend lattice-based crypto whenever their new suite B will be done. I find hard to trust any system's security proof that rely on lattice's theorical bounds because as it is known with LLL, BKZ and others: practical results are way better than these theorical limits. I don't know much about lattice crypto though, and I would you out to this paper in my to read list: Lattice-based crypto for beginners.

They agree on Hash-based signatures (which are explained in a 4 posts series on my blog), which is timy because a new version of the RFC draft for XMSS has came out, which might be the most polished hash-based signature system out there (although it is stateful unlike SPHINCS).

The paper ends on these wise words that explains how security estimation works (and has always worked):

We note that none of the above proposals have been shown to guarantee security against all quantum attacks. A new quantum algorithm may be discovered which breaks some of these schemes. However, this is similar to the state today. Although most public-key cryptosystems come with a security proof, these proofs are based on unproven assumptions. Thus the lack of known attacks is used to justify the security of public-key cryptography currently in use.

To talk about quantum computing advances, I don't know much about it but here are some notes:

• Shor’s algorithm (the one that breaks everything) was born on 1994.

• Late 1990s, error correcting codes and threshold theorems for quantum computing. Quantum computing might be possible?

• 2011, "the world's first commercially available quantum computer" is released by D-Wave. I believe this angered many people because this wasn't really quantum computing.

• 2015, Google and NASA have D-wave computers.

To finish this blogpost, a few things I remember from last month Real World Crypto conference:

• Tanja asked the first speaker presenting the blackphone about quantum crypto. His response: "post-quantum right now is marketing". People laughed.

• On day 3, str4d announced that they wanted to move to post-quantum algorithms for i2p (a thing like Tor). People did not receive that as a good news. I heard people quoting djb's "crypto should be boring" line.

There is definitely a skepticism in the crypto world about quantum computing, as there is a gold rush into designing new post-quantum crypto.

# Moxie and TLSposted February 2016

I have mix feelings about the UI of Signal, but watching these two videos from Moxie at different periods of TLS' life, I now have a brand new admiration for the person.

The two videos are here:

In both of them, he start talking about his sslsnif: a great tool that you can find here, written in C++ and that allows you to serve clients fallacious certs taking advantage of browsers vulnerabilities (such as not checking for basic constraints fields in the certificate chain, stop reading subject names after null bytes, etc...)

Another tools that he released, coded in python, is sslstrip. The thing takes advantage of the fact that almost no one types https:// in the address bar when navigating to a website directly. A man-in-the-middle attacker can stop the redirection to https and serve the entire website either through http or through another https website which url looks similar to the victim's website (thanks to some unicode to make it look like victimwebsite.com/something/?yourwebsite.com where yourwebsite.com is the real website being visited).

https' only purpose is to defend against man-in-the-middle attacker. Because of sslstrip, any active attacker on the network can render https completely useless. Security measures to prevent that, that I can think of, are HSTS headers, preloaded HSTS (see chrome's one) and HPKP (see Tim Taubert's blogpost.)

Both tools needs arpspoof to create the man-in-the-middle position. I've discovered another tool that seems to combine all of these tools in one (but I'm not really sure what's the difference here): bettercap. It looks good also.

Other tools I've discovered, that verify how the client's handle certificate verification: one is developed by some coworker and is called tlspretense, the other one is called x509test and seem to be pretty popular too. I have no idea how both these tools perform, I guess I will check that next time I to.

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# Openssl dhparam 2048posted February 2016

While some people would tell you to use pre-defined Diffie-Hellman parameters (rfc3526, rfc5114) so as not to mess something up during the generation (hum hum socat), others would tell you "hey, look at what happened with logjam and their hardcoded DH params!" and will point you to openssl dhparam to generate your own customized parameters.

But how does it really work? The high level code is less than 450 lines long. It's basically taking several stuff into consideration ("do you want to use 2 or 5 as a generator? No? Let's use 2 anyway!") and then using some DH_generate_parameters() function that will set up everything.

Inside of that function relies probable_prime_dh_safe() that will look for a particular kind of prime as the DH modulus: a safe prime.

A safe prime looks like this: $2p + 1$ with $p$ a prime as well.

Actually, we can also say that the function looks for a Sophie Germain prime, a prime $p$ such that $2p+1$ is also prime.

Like that you know =), safe primes are not the same as Sophie Germain prime, but they are closely related.

(Not to mix up with strong prime as well, which are just primes that have some properties.)

Another important thing: the function returns a safe probable primes. That is a function that has not been mathematically proven to be a prime (but is close enough).

To do that, first a random number is generated. Then a test called the Miller-Rabin test is applied to the random number a number of time. The more you run the test and the more certainty you get that it is a prime, if the test ever fail you know for sure that the number is not a prime (so a composite) and you need to regenerate a new random number and start again with the tests.

How many times should you apply this test? Enough time so that if everyone tried to generate primes every second for the lifespan of earth we would still have low chances to find a false positive (a composite number passing X tests of Miller-Rabin). The numbers used in OpenSSL come from the table 4.4 in the Handbook of Applied Cryptography.

(To make the test faster a bunch of small primes are first tested as divisors, then some trial divisions follow.)

# How does the Mersenne's Twister work?posted February 2016

Someone asked that question on reddit, and so I replied with a high level answer that should provide a clear enough view of the algorithm:

From a high level, here's what a PRNG is supposed to look like:

you start with a seed (if you re-use the same seed you will obtain the same random numbers), you initialize it into a state. Then, every time you want to obtain a random number, you transform that state with a one-way function $g$. This is because you don't want people to find out the state out of the random output.

You want another random number? You first transform the state with a one way function $f$: this is because you don't want people who found out the state to be able to retrieve past states (forward secrecy). And then you use your function $g$ again to output a random number.

Mersenne Twister (MT) is like that, except:

• your first state is not used to output any random numbers
• a state allows you to output not only one, but 624 random numbers (although this could be thought as one big random number)
• the $g$ function is reversible, it's not a one-way function, so MT it is not a cryptographically secure PRNG.

With more details, here's what MT looks like:

the $f$ function is called "twist", the $g$ function is called "temper". You can find out how each functions work by looking at the working code on the wikipedia page of MT.

# How to backdoor Diffie-Hellman, lessons learned from the Socat non-prime primeposted February 2016

The socat thingy created some interest in my brain and I'm now wondering how to build a NOBUS (Nobody But Us) backdoor inside Diffie-Hellman and how to reverse it if it's not a proper NOBUS.

Ways to do that is to imagine how the DH non-prime modulus could have been generated to allow for a backdoor. For it to be a NOBUS it should not easily be factorable, but for it to allow a Pohlig-Hellman attack it must have a B-smooth order with B small enough for the adversary to compute the discrete log in a subgroup of order B.

I'm currently summing up my research in the open on a github repo: How to backdoor Diffie-Hellman, lessons learned from the Socat non-prime prime. If anyone is interested in any parts of this research (factorizing the modulus, thinking of ways to build the backdoored modulus, ...) please shoot me a message :)

If you go on the github repository you will see an already working proof of concept that explains each of the steps (generation, attack)

This proof of concept underlines one of the ways the malicious committer could have generated the non-prime modulus $p = p_1 p_2$ with both $p_i$ primes such that $p_i - 1$ are smooth. The attack works, but I'm thinking about ways of reversing such a non-prime modulus that would disable the NOBUS property of the backdoor. Spoiler alert: Pollard's p-1 factorization algorithm.

Anyway, if you're interested in contributing to that research, or if you have any comments that could be useful, please shoot me a message =)

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