CVE-2021-40528: On the (in)security of ElGamal in OpenPGP - Part I

Jul 20, 2021 Authors: Luca De Feo, Bertram Poettering, Alessandro Sorniotti

Topic: cryptanalysis

On the (in)security of ElGamal in OpenPGP – Part I #

In this two-parts post we dissect the paper “On the (in)security of ElGamal in OpenPGP”, to be presented at ACM CCS’21.

TL;DR: we found two types of vulnerabilities in the way OpenPGP implementations handle ElGamal encryption. We call the first type cross-configuration attacks, where two otherwise secure PGP implementations interact in an insecure way, leading to plaintext recovery given a single ciphertext. Luckily, the majority of PGP users seems to be unaffected, nevertheless we found more than 2000 vulnerable public keys. The second type of vulnerability, to be described in Part II of this post, is a classical side-channel vulnerability leading to secret key recovery, however we argue there that the cross-configuration scenario makes it worse.

This write-up is intended for a technical audience who wants to understand the attacks without the clout of academic papers. If you’re simply interested in understanding whether you might be affected, jump straight to the FAQ.

50 shades of ElGamal #

In 1985, Taher ElGamal described the first public key encryption scheme based on discrete logarithms. The scheme represented an alternative to the patented RSA cryptosystem and has since become a staple of any course on cryptography. For such a venerable, well known and, after all, simple scheme, you’d expect little room for interpretation or doubt. And yet, do you think you could fully specify all the details without making mistakes?

Ask two cryptographers to implement ElGamal, and chances are they will make radically different choices in the way parameters are set up. In the best case, the two implementations will simply be incompatible. In the worst case, they may be compatible, but… Let’s pretend we are one of those cryptographers tasked with implementing ElGamal, and let’s go over the various choices in front of us.

Prime modulus generation #

We are talking about classic ElGamal, not variants based on elliptic curves. The first thing we need is a finite field with a hard discrete logarithm problem. Given the current state of discrete log cryptanalysis, this means a 2048 bit prime at the very least. We have two choices, already:

• Some standardized prime, such as those defined in RFC 2526;
• Generate a prime such that p – 1 contains a large enough prime factor.

The first option is probably okay, but some users may be worried about backdoored primes. Another argument against standard primes is that it facilitates bulk interception. Although it would be possible in principle to define trustworthy standard primes, as a matter of fact they do not appear to be very popular for ElGamal.

The second option leads to more choices. The goal is to generate a prime p such that p – 1 contains one large prime factor, call it q. Here is a list of the most sensible choices:

• Generate a safe prime, i.e., such that q = (p – 1)/2 is prime;
• Generate a DSA-like prime, i.e., first choose q, then try p = qf + 1 for random f until a prime is found.
• Generate a Lim–Lee prime1, i.e., one such that all prime factors of (p – 1)/2 are large.

How large is “large” depends on the target security. NIST’s FIPS-186 specification of DSA requires q to be 224 or 256 bits long, for a prime p of 2048 bits. Of course, in safe primes q is as large as it can be. Why, exactly, q needs to be large is something we will come back to soon.

Defining a group generator #

The multiplicative group of integers modulo p contains several subgroups: one for each divisor of p – 1. Of these, the most important one is the subgroup of order q. To instantiate ElGamal we need to specify a group generator, call it g, but again we have at least two choices:

• A generator of the full group of invertible integers modulo p, which has order p – 1;
• A generator of the prime order subgroup of order q.

For efficiency, it may be interesting to find a small g. In the first case this is always possible: start from g = 2, and increase until a generator of the full group is found, which should happen pretty fast. In the second case this is only possible for safe primes: g = 4 always works for safe primes, while for the two other kinds of prime the subgroup of order q is unlikely to contain any small elements other than 1.

Creating a key pair #

The last step in key generation is to choose a secret exponent x and form the public key X = gx mod p. For mathematical soundness, x should be a uniformly random integer between 1 and the order of g (so, either p – 1 or q), thus ensuring that the public key is a uniformly random element of the group generated by g.

In some cases it is possible to forego some mathematical rigor and draw x from a smaller interval. We shall call “short exponents” secrets drawn this way. This speeds up key generation and decryption, however we shall see that it is a risky choice. In no case shall x be drawn from too small an interval, lest key recovery become feasible via exhaustive search or Pollard’s Lambda algorithm.

Encryption #

To encrypt to a public key X a message M, assumed to be an integer modulo p, draw a random “ephemeral” exponent y, compute Y = gy mod p, and form the ciphertext (Y, M·Xy mod p).

The exponent y can be sampled from the same intervals as the secret key x. Use of short exponents in this case is even riskier, as we shall see.

Decryption #

Decryption is the only routine where every choice is pre-determined. Upon receiving a ciphertext (Y, Z), the original message is recovered as M = Z/Yx mod p.

Note how each decryption involves an exponentiation by the secret exponent x. This will be important later when we discuss side-channel attacks.

ElGamal in PGP #

OpenPGP is a very popular standard aimed to promote consumable, interoperable email security. The standard is defined in RFC 4880, which dictates message formats, operations and certain cryptographic aspects of the standard. According to section 9.1 of the RFC, ElGamal is the only public key encryption algorithm required for an implementation of the OpenPGP standard, RSA being only recommended:2

Implementations MUST implement DSA for signatures, and Elgamal for encryption. Implementations SHOULD implement RSA keys.

So, what flavour of ElGamal do you think the OpenPGP standard mandates? The RFC is quite vague in that respect, only specifying the public key format, and pointing to external references for algorithmic details. Unsurprisingly, implementations have freely interpreted the standard, leading to several interoperable but diverging realizations.

To get a picture of what variants of ElGamal are implemented in the OpenPGP ecosystem, we studied three of the most popular RFC 4880 compliant implementations: GPG, Crypto++, and the Go standard library. Given that many lesser-known and closed-source libraries also contribute to the ecosystem, we further analyzed a dump of OpenPGP public keys obtained from an official key server.

OpenPGP key format #

The only common denominator among all libraries is the key format, which is thoroughly specified in RFC 4880 (Key ID 0x10). In particular, an OpenPGP ElGamal subkey is a triple made of:

• the prime modulus p,
• the (sub)group generator g,
• the public key gx.

An ElGamal secret key consists of the subkey data above together with the secret exponent x.

Without surprise, we found all libraries follow the standard in this respect.

GPG’s Libgcrypt #

Libgcrypt is the C library doing the crypto heavy-lifting for GPG. The roots of its ElGamal implementation go as far back as 1997. The relevant ways in which Libgcrypt interprets the standard are as follows:

• It generates Lim–Lee primes, with the size q of the smallest odd factor of p – 1 chosen as a function of the size of p. In particular, when p is 2048 bits long, q is 225 bits long.

• It takes g to be the smallest integer ≥ 2 that generates the full group of invertible integers modulo p.

• It uses short exponents both for the secret key x and the ephemeral exponent y. The size of x and y is roughly 3/2 that of q, with y usually ending up being a few bits longer than x. For p of 2048 bits, x has 340 bits and y has 344.

The algorithm to generate Lim–Lee primes was published in 1997, at a time when people started being wary of attacks based on small subgroups (more on them later), and when safe primes were still considered to be too expensive to generate. It is thus not so surprising that Libgcrypt uses this relatively little-known prime modulus generation. The choice of using both small generators and short exponents appears to have been dictated mainly by efficiency considerations.

Crypto++ #

Crypto++ has implemented ElGamal since version 1.0. Source code history only goes as far back as version 5.0, released in 2002, and by that time the ElGamal implementation had already crystallized. These are the choices made by Crypto++ by default:

• The modulus p is a random safe prime p = 2q + 1.
• The generator g is the smallest quadratic residue, i.e., the smallest generator of the subgroup of order q, i.e., one of 2, 3 or 4.
• It uses short exponents both for x and y. Both are sampled uniformly at random between 1 and 2n, with n depending on the size of p. When p is 2048 bits long, n = 226.

Go standard library #

Go only implements ElGamal encryption and decryption, but no key generation, it is thus not fully RFC 4880 compliant, for better or for worse. The only choice it has to make is the size of the ephemeral exponent y, and it makes a pretty boring (i.e., safe) one: y is sampled uniformly between 0 and p – 1.

OpenPGP public keys in the wild #

Open source libraries only make a fraction of the OpenPGP ecosystem. To get a richer picture, we obtained a public key dump from https://pgp.key-server.io/dump/, dated Jan 15, 2021, containing 2,721,869 keys, out of which 835,144 had ElGamal subkeys.

It is impossible to know everything about how a library implements ElGamal by only looking at the public keys it produces. For example, it is (hopefully) impossible to know what intervals x and y are sampled from. However, just by looking at p and g, some information can still be gained.

Safe primes are easy to recognize, as it is sufficient to test whether (p – 1)/2 is prime. For other primes, even a partial factorization of p – 1 can give a deal of information. Note that it is in general difficult to completely factor integers of 2048 bits. We classified primes into four categories:

• Safe primes.
• Non-safe primes for which we could not find any factor other than 2. These are likely to be Lim–Lee primes.
• Non-safe primes for which we could complete the factorization of p – 1. These have unusually large q (within 99% the size of p), and for this reason we call them quasi-safe primes.
• Non-safe primes for which we could find some non-trivial factors, but we couldn’t complete the factorization. These are likely to have been generated à la DSA.

An interesting finding is that the vast majority of public keys only used one of 16 “standard” primes. None of these primes appears to have been defined in an RFC, and, while we could track some of them down, we’re still unable to explain all of them.

A few more bribes of information can be gleaned from g. First, we ran a quadratic residuosity test. For safe primes this tells everything: if g is a square it generates the subgroup of order q, if not it generates the full group. For other primes, where we could find non-trivial factors, we ran higher residuosity tests, giving at least a hint to what group g generates. For all categories of primes, we found the full spectrum of possibilities:

• g generates the full group (or at least nothing contradicts this hypothesis),
• g generates the subgroup of order q (or at least nothing contradicts this hypothesis),
• none of the above.3

Summary #

That was pretty wild! Let’s try to systematize our findings via a sort of ElGamal bingo card. In the table below we classify both libraries and keys found in the wild according to how they select parameters p, g, x and y. For some entries, we do not have enough information to pin down the exact features, but we give our best guess based on the data we have.

Prime category

Generated group size

Short exponents?

Quantity

SP

LL

DSA

QS

p – 1

q

other

short x

short y

total

s. 2016

Libgcrypt

×

×

×

×

–

–

Crypto++

×

×

×

×

–

–

Go

–

–

–

–

–

–

–

–

–

Safe prime I

×

×

–

–

472,518

783

Safe prime II

×

×

–

–

107,339

219

Lim–Lee I

?

?

–

–

211,271

6003

Lim–Lee II

?

?

–

–

47

24

Quasi-safe I

×

×

–

–

15,592

89

Quasi-safe II

×

×

–

–

20

3

Quasi-safe III

×

×

–

–

26,199

125

DSA-like I

×

?

–

–

828

810

DSA-like II

×

?

–

–

27

26

DSA-like III

×

×

–

–

1,304

1300

Table 1: Features of ElGamal implementation found in libraries and public keys in the wild. The last two columns give the number of non-expired keys in total and since 2016. Legend: × = has feature, ? = likely has feature, – = not applicable; SP = Safe prime, LL = Lim–Lee prime, DSA = DSA-like prime, QS = Quasi-safe prime.

As messy as this may look, we haven’t uncovered any specific issues so far. All the configuration in the table are sound, interoperable, and safe… if taken in isolation! Notice however the two lines in bold red in the table: DSA-like I and III. We shall now see why these two “perfectly fine” types of keys are to be considered vulnerable in the OpenPGP ecosystem.

Computing discrete logarithms #

The attacks we are going to describe require minimal understanding of discrete logarithm computations. In their simplest form, they are a combination of the Pohlig–Hellman algorithm for discrete logarithms in groups of composite order, and of Shanks’ Baby-step giant-step algorithm.4

As usual, we are given g and h = gz, and our goal is to find z. If z is drawn from a set Z of size N, Baby-step giant-step finds it in roughly N½ operations. Its strength is in requiring roughly the same amount of work no matter how complicated the set Z is. For example, we will use it in the next part of this blog post to find z when some of its bits are known via a side-channel.

Pohlig–Hellman will be useful in a slightly more involved scenario, based on two key assumptions:

1. The order of the group generated by g contains some small factors,
2. z is a short exponent.

To fix ideas, let’s say that p is a 2048 bits prime, that (p – 1) = 2qf₀f₁f₂··· with fi “small”, that g generates the full group of invertible integers modulo p, and that z is 224 bits long. The bitlength of z is supposedly set large enough that none of the known discrete log algorithms could find z in a feasible amount of time. In particular, Baby-step giant-step, with its square-root complexity, would roughly take 2¹¹² operations.

Without going into details, the Pohlig–Hellman algorithm reduces the problem of computing discrete logs in the group of order p – 1 to that of computing discrete logs in each of the groups of order 2, q, f₀, f₁, etc. Another way to look at it is that, by solving a discrete logarithm in the subgroup of order fi, Pohlig–Hellman finds log₂(fi) bits of z.5 If z were 2048 bits long, this wouldn’t help us at all: after learning

1 + log₂(f₀) + log₂(f₁) + ···

bits we would still be left with log₂(q) unknown bits, which, assuming p was constructed correctly, would still be too much.

But we assumed that z is 224 bits long. Then, by solving a discrete log in each of the “small” subgroups, we are left with only

u = 224 – 1 – log₂(f₀) – log₂(f₁) – ···

unknown bits. If this number is small enough, we can finish off the computation with Baby-step giant-step, which takes roughly 2u/2 operations.

Cross-configuration attacks on OpenPGP #

The algorithms above highlight some dangerous combinations of ElGamal options. Specifically, if after key generation:

1. p – 1 contains small factors,
2. g generates the full group of invertible elements, or at least a subgroup with enough small factors in the order,
3. x is a short exponent,

then secret key recovery may be possible at a significantly lower cost than intended. This attack was already described in the 90s,6 so it is no surprise that it does not apply to any of the libraries we analyzed: indeed, both safe primes and Lim–Lee primes block it, because their only small factor is 2.7

But the same idea can be applied to the ephemeral exponent y, and it appears that this risk was overlooked in the OpenPGP ecosystem. In practice, this leads to plaintext recovery in a context where two OpenPGP libraries, one sender and one receiver, interact, and:

1. The receiver’s public key defines a prime such that p – 1 contains small factors;
2. The receiver’s public key defines a generator g that generates the full group of invertible elements, or at least a subgroup with enough small factors in the order;
3. The sender’s library uses short ephemeral exponents y.

After computing the discrete log y as previously sketched, recovering the plaintext message is an easy exercise.

Looking at Table 1 we see that two libraries, Libgcrypt and Crypto++, can play the role of the sender, and that two types of public keys, those named “DSA-like I” and “DSA-like III” can play the role of the receiver in this attack.8 Let that sink in: any message sent by GPG or Crypto++ in the past 20-something years to one of 2,132 registered public PGP keys had weak or nonexistent security! Fortunately, that’s a quite small number of keys, but it accounts for a good fraction of keys registered since 2016, and it is impossible to say how many more unregistered keys may be at risk.

The feasibility of the attack depends on the number and size of the small factors in the group order, as well as on the size of the short exponents. So, for example, Crypto++ encrypted messages are always more vulnerable than GPG messages. The CCS ‘21 paper contains a careful analysis of the computational effort expected for each of the affected keys. As a proof of concept, we picked the weakest key we could find in the key dump and we encrypted a message to it using Crypto++. We were able to recover the plaintext in 2.5 hours on a single Intel E5-2640 core.

Disclosure timeline and mitigations #

• Feb 24: Notified Libgcrypt developers.
• May 1: Notified owners of affected public keys, with suggestion to migrate to RSA or ECC encryption keys if possible.
• May 24: CVE-2021-33560 assigned.
• Jun 2: Libgcrypt 1.8.8 released, fixes the vulnerability by sampling large ephemeral exponents y with as many bits as p.
• Jul 10: Notified Crypto++ developers.
• Jul 10: Paper online.

FAQ #****What kind of vulnerability did you find? #

In this first part we described a plaintext recovery attack on OpenPGP ciphertexts encrypted with ElGamal. Only certain combinations of sender and receiver software are exposed. We found that GPG (via Libgcrypt) and Crypto++ are affected when acting as sender, while Go is not.

We could not identify a specific library that would be affected when acting as receiver, but an analysis of registered PGP public keys shows that such libraries exist. Any message encrypted to their keys by one of the weak sender libraries is at risk of being exposed.

What is the attack scenario? #

This is a mathematical attack, thus it only requires interception of ciphertexts. For example, ciphertexts may be gathered through a data breach, or by snooping on an insecure network.

Is the attack practical? #

Running times for the attack vary depending on the sender’s software and the receiver’s public key. They can go from a few hours on commodity hardware to several CPU-years.

How many people are affected? #

The attack is a combination of a specific behaviour of the software on the sender side and certain mathematical properties of the public key of the recipient of the encrypted transmission. While the weakness appears to be very common on the sender side, we only found 2,132 registered public keys to be affected, among more than 2 millions. It is likely that the vulnerability only affects a small proportion of all OpenPGP communications, however we cannot know how frequent the weakness is among unregistered public keys.

Am I affected? #

If you are a GPG (Libgcrypt) or Crypto++ user, the messages you send or have sent may be at risk. Update to the latest version of GPG. A fix for Crypto++ is upcoming. If you are unsure which library you are using, or if you want to check whether your software has been patched, you can use the tool we provide here.

However, public keys generated by GPG and Crypto++ are not at risk, so you do not need to revoke your ElGamal keys if you know they were generated by one of these software.

If you are unsure which software generated your ElGamal key, read on.

How do I tell if my ElGamal key is affected? #

It takes some computational resources to tell, with a reasonable degree of confidence, whether a public key is affected or not. Thus we are not able at the moment to provide a simple tool to test your keys.

If you cannot confirm that your key was safely generated, then we recommend that you revoke it. To generate a new key we recommend either GPG or Crypto++, and/or to use a different algorithm altogether (e.g., RSA or ECC).

Endnotes #

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On the (in)security of ElGamal in OpenPGP - Part II