Thursday, November 10, 2016

Study Group - All Your Queries Are Belong to Us: The Power of File-Injection Attacks on Searchable Encryption

Published at USENIX 2016 and written by Yupeng Zhang, Jonathan Katz, and Charalampos Papamanthou from University of Maryland this paper proves again how difficult is to get proper security in searchable encryption.

Some of you may wonder why I chose this paper. Reason is that I wanted to get a grasp of how research looks like outside of the MPC area at a top security conference. And what other conference to pick than this year's USENIX?

After I finished reading this paper I realised a cool thing: that you need little to none a priori knowledge about Searchable Encryption (SE) [1]. Fortunately I didn't find myself diving into the literature of SE in order to fully understand the ideas presented there - how many crypto papers have you read and still be able to say this once you're done with them?

Let's introduce first the notion of SE. The setup consists in 2 parties, a client and an encrypted mail server. The client wants to obtain all e-mails which have 'alice' and 'work' as tags. He then computes a token in a deterministic manner, then the server replies with the e-mails  corresponding to the query:



To ease the presentation we can assume that instead of e-mails the server will respond with identifiers (urls).

An adversary wins if he breaks the token privacy, namely it recovers keywords from tokens.

A file injection attack is when an attacker has encrypted e-mails of his choice on the server and can see the tokens which are queried by the client. More simple, the server behaves in a honest-but-curious way and stores encrypted e-mails of his choice by spamming the client.

From a first sight this seems harmless. But if you combine it with some leaked e-mails (lots of them these days) the authors managed to have a 70% recovery rate for a keyword from a token with only 20% of leaked files. This was tested using an Enron email dataset of ~30000 emails. [2]

To understand how this works it's important to look at a simpler attack:

Binary search attack.

Suppose an adversary has a keyword universe $K$, with size $|K|$ a power for $2$. He can inject $\log_2{|K|}$ files and then recover keywords from every token with $100\%$ success.

Each file $F_i$, $i \in [0, \log_2{|K|}] $ contains keyword $k_j$ for which $j$ has $i$'th most significative bit equal to $1$. To see why this works let's look at an example inspired from the paper.

In this case we have $|K| = 8$. Black cells are the keywords inserted in file $i$. Since the search result is $F_1$, $F_3$ ($101$) we conclude that the token was derived from the keyword $k_5$.
A server which inserts one email per day can execute an attack in $2$ weeks time for $10,000$ keyword universe.

Countermeasure.

One crucial observation is that we insert $|K|/2$ keywords per file. So one can think that limiting the keywords per file to some threshold $T << |K|/2$ will fix the problem.

The threshold countermeasure turns out to be ineffective. The authors proved this with an attack which works almost like the first one after splitting the keywords in chunks of size $2T$. (denoted as hierarchical search attack).

For $|K| = 5,000$ and $T=200$ an adversary should can inject $131$ files in order to recover keywords for every token.

Attacks with partial leakage.

Sometimes e-mails leak. And when that happens...SE is almost useless against adaptive injection attacks as the authors prove. We say adaptive because it needs a forward insecure SE scheme.

What if we have want to recover keywords from multiple tokens?

The idea is to compute the distribution for each keyword $f^*(k)$ from the leaked files. Then the assumption that $f^*(k)$ is close to the queried tokens distribution $f(t)$ is made - which will turn true latter.
Next, a small candidate set of keywords is chosen (also called ground truth) and files are injected using the binary search attack. The rest of the tokens are recovered by taking joint distributions with previous tokens.

For more attacks and countermeasures which fail I strongly recommend reading the article [3].

At the end of the article one can wonder if building secure SE schemes would really help against these access pattern attacks...The authors suggest that an interesting direction is to look into lower bounds on these attacks but this seems a far from trivial task.

[1] http://crypto.stanford.edu/~dabo/pubs/papers/encsearch.pdf
[2] https://www.cs.cmu.edu/~./enron/
[3] https://eprint.iacr.org/2016/172

Friday, November 4, 2016

What is SPDZ? Part 3: SPDZ specifics

This blog post is the final in a series of three in which we describe what SPDZ is. The first part can be found here, and the second here.

In this part, we consider what it is that makes SPDZ SDPZ.


SPDZ MPC

Using the term SPDZ is perhaps a little misleading. There are actually a few protocols which we call SPDZ because they are really just improvements on the original. As the online phase of SPDZ is already ‘pretty good’, optimisations to the protocol normally involve improving the offline phase.

In what follows, we talk about the methods used in SPDZ for preprocessing (generating the triples) and then explain some improvements that have been made.

Original SPDZ

Here we will outline some of the methods in the original SPDZ protocol, as in [5].

Preprocessing
In the first SPDZ paper, the authors show how to use somewhat homomorphic encryption (SHE) to generate the triples. This is a (pretty expensive) public key cryptosystem in which the encryption operation is ring homomorphic - that is, \[\textsf{Enc}_{\textsf{pk}}(a)\boxplus\textsf{Enc}_{\textsf{pk}}(b)=\textsf{Enc}_{\textsf{pk}}(a+b)\] and \[\textsf{Enc}_{\textsf{pk}}(a) \boxdot \textsf{Enc}_{\textsf{pk}}(b) = \textsf{Enc}_{\textsf{pk}}(a \cdot b)\]
(For those who know about FHE: we can do MPC with FHE, as you’d hope; however, at the time of writing no FHE scheme which is competitive with current secret-sharing MPC currently exists. SPDZ is what the authors call the ‘ideal use of FHE in MPC’: raw material is computed locally so that the online computations are minimal and communication overhead is small.)

During the offline phase, we also generate lots of random values which are used by the parties to provide input to the circuit.

Sacrificing
In order to check that the adversary has not introduced errors into the triples, for each triple we want to use we have to ‘sacrifice’ a triple by taking another triple from the preprocessing and checking that the third element is the product of the first two using Beaver’s method outlined in the previous post. Fortunately the method by which triples are generated is very efficient.

ZKPoPKs
Zero-Knowledge Proofs of Plaintext Knowledge (ZKPoPKs) are a way of ensuring parties encrypt as they should: when encrypting in the SHE scheme, there are conditions on the plaintext and noise that need to be met to ensure the ciphertext is well-formed.

MACs
Before any preprocessing computation takes place, the parties agree on some MAC key $\alpha$ with which they MAC all their data and which they secret-share amongst themselves so that no individual party knows the MAC key.

This MAC key is used to MAC all the data in the circuit. For each secret-shared value, there is also a secret-shared MAC on that value.

After the circuit has been evaluated, the parties open the secret corresponding to the circuit output and also the MAC on it, and check that the MAC is correct. If an actively corrupt party cheats at any point in the circuit evaluation, the final MAC check reveals this has happened. Note that this means the parties could be computing on invalid data.

In the original paper, each party also MACked the global MAC key with a personal MAC key. This was not needed in subsequent works.

Updated SPDZ: SDPZ2

While the online phase of SPDZ was (almost) the same, in SPDZ2 [4] the authors made significant changes to the offline phase of the original protocol. We outline the major changes here.

They solved an open problem left by the first paper in giving a secure protocol which generated a shared SHE decryption key.

They also offered various performance enhancements, including the introduction of ‘squares’ and ‘bits’ in the preprocessing, which allowed faster online computations of specific operations in the circuit.

A major improvement was allowing MACs to be checked on data before the end of the computation. This involved checking the MAC of a public value that depended on the underlying data. The big advantage of this is that the preprocessed data MACked under the same key can still be used after this MAC check, but not in the original protocol since the check reveals the key.

The new protocol had covert and malicious variants, the former having lesser parameter requirements. A covertly secure protocol ensures the adversary wins with probability at most $1/n$ for some $n \in \mathbb{N}$. The reason for this was that it was believed more closely to match the real-world situation.

New SPDZ: MASCOT

The most recent improvements to the SPDZ protocol is a work known as MASCOT [3]. As with SPDZ2, the online phase of SPDZ was left as it was; this paper improves on the offline phase, showing how to use oblivious transfer (OT) to achieve much faster preprocessed data than by using SHE.  It is authored by Bristol’s own Marcel, Emmanuela and Peter, and appeared at CCS last week.

(Oblivious transfer is a process in which one party inputs two strings and a second party chooses exactly one; this is done in such a way that the first party doesn’t know which string the second chose, and the second does not learn anything about the string it didn’t choose.)

The authors of MASCOT note that the heavy machinery of public-key cryptography required in the original SPDZ paper to generate the preprocessing (namely, the use of SHE) prevents the communication from being a bottleneck since the local computation takes so long.

The protocol is actively secure and performs significantly faster than previous SPDZ2 implementations, with much lower communication overhead.

The Future of SPDZ

MASCOT has only just been released, and there are already whispers of a new paper. Watch this space...


References

[1] B. Applebaum, Y. Ishai, and E. Kushilevitz. How to garble arithmetic circuits. 52nd FOCS, pp120–129. IEEE Computer Society Press, 2011
[2] D. Beaver. Efficient Multiparty Protocols using Circuit Randomisation. In J. Feigenbaum, editor, CRYPTO, volume 576 of Lecture Notes in Computer Science, pp420-432, Springer, 2012.
[3] R. Bendlin, I. Damgard, C. Orlandi, and S. Zakarias. Semi-homomorphic encryption and multiparty computation. In EUROCRYPT, pp169-188, 2011.
[4] I. Damgard, M. Keller, E. Larraia, V. Pastro, P. Scholl, N. P. Smart. Practical covertly secure MPC for dishonest majority - or: Breaking the SPDZ limits. In ESORICS (2013), J. Crampton, S. Jajodia, and K. Mayes, Eds., vol. 8134 of Lecture Notes in Computer Science, Springer, pp. 1–18.
[5] I. Damgard, V. Pastro, N. P. Smart, and S. Zakarias. Multiparty computation from somewhat homomorphic encryption. In Advances in Cryptology – CRYPTO 2012, volume 7417 of LNCS, pp643–662. Springer, 2012.
[6] I. Damgard and S. Zakarias. Constant-overhead secure computation of
boolean circuits using preprocessing. In TCC, pp621-641, 2013.
[7] M. Keller and E. Orsini and P. Scholl. MASCOT: Faster Malicious Arithmetic Secure Computation with Oblivious Transfer. Cryptology ePrint Archive, Report 2016/505, 2016.
[8] J. Buus Nielsen, P. Nordholt, C. Orlandi, and S. Burra. A new approach to practical active-secure two-party computation. In Reihaneh Safavi-Naini and Ran Canetti, editors, Advances in Cryptology CRYPTO 2012, volume 7417 of Lecture Notes in Computer Science, pp681-700. Springer Berlin Heidelberg, 2012.
[9] A. Shamir. How to Share a Secret. In Communications of the ACM, Volume 22 Issue 11, Nov. 1979, pp612-613.
[10] A. Yao. How to generate and exchange secrets. In SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp162–167. IEEE, 1986.

Thursday, November 3, 2016

Study Group - Dedup est Machina

The title of the paper is 'Dedup Est Machina: Memory Deduplication as an Advanced Exploitation Vector'.

I saw the paper being presented at the IEEE Security and Privacy Symposium, and it was one of the more enjoyable talks due to the nature of the exploit, and the enthusiasm of Erik Bosman presenting it. It also won a 'PWNIE' award for the 'most innovative research on hacking' at Black Hat USA this year.

The idea of the paper is simple - to combine the Memory Deduplication exploit with a Rowhammer attack.

They use this to craft a JavaScript-based attack against the Microsoft Edge browser, to gain arbitrary memory read and write access in the browser.

Memory Deduplication


The idea is to reduce the total memory footprint of a running system by sharing memory pages with identical content across independent processes.

A good example of this is the page cache in modern operating systems. The page cache stores a single cached copy of file system contents in memory, and shares the copy across different processes.

In running processes, two or more pages with the same content in run-time memory are always deduplicated, even if the pages are completed unrelated, and their equivalence is fortuitous.

So, in order to keep a single copy of a number of identical pages, the memory deduplication system needs to perform three tasks:
  1. Detect. The system needs to detect memory pages with the same content, usually done at regular and predetermined intervals during normal system operations
  2. Only Keep One. We only want to keep one physical copy, and return the others to the memory allocator; to do this, the deduplication system updates the page-table entries of the owning processes, so that the virtual addresses now point to a single shared copy; these entries are also marked as read-only to support copy-on-write semantics
  3. Create Private Copy on Write. Specifically, when one of the owning processes writes to the read only page, a copy-on-write page fault occurs; at this point, the memory deduplication system can create a private copy of the page, and map it into the corresponding page table entries of the faulting process
On Windows 8.1 onwards, it's known as memory combining.

Memory Deduplication as a Side Channel


So you may have spotted how you could exploit this as a side channel. Writing to a shared page from any of the owning processes results in a page fault and a subsequent page copy.

Due to these additional expensive operations, a write to a shared page takes significantly longer compared to a write to a regular page (by significantly longer, we're talking up to one order of magnitude). This timing difference provides an attacker with a side channel to detect whether a given page exists in the system.

To do this, the attacker has four easy steps:
  1. Craft a page with the exact same content as the target page
  2. Wait for the memory deduplication system to do its thing
  3. Write to the crafted page
  4. Time how long it takes for the write operation to finish
If the write takes longer than a write to a non-deduplicated page, the attacker can conclude that a page with the same content exists.

So just using this, an attacker could detect whether a user was visiting a particular web page, or running a particular program.

Of course, false positives are possible due to noise etc, so the attacker can use redundancy (repeated attempts) to disclose the intended information in a reliable way.

At the moment, the memory deduplication side channel is looking a little bit pathetic, as it's a slow single-bit side channel that can only be used for leaking a limited number of bits from a victim process.

However, we can abuse memory deduplication to get some stronger primitives.

Primitives

Primitive #1: Alignment Probing



First up, we have a primitive that allows a byte-by-byte probing of secret information by controlling its alignment in memory. So, this primitive is applicable when the secret data has weak alignment properties.

Imagine in this case, the secret is a password stored in memory immediately after a blob of attacker-provided input bytes. What the attacker can do is provide some more input bytes, which effectively push the second part of the secret out of the target page. Now the attacker can brute force the first part of the secret (e.g. one byte) using deduplication.

All the attacker needs to do now is provide fewer input bytes to bring another portion of the secret back into the page. Just like before, the attacker uses deduplication to get the rest of the secret. For larger secrets, you just keep repeating these steps until all of the secret is known.

This alignment probing primitive is apparently very effective, and is used to disclose code pointers in Microsoft Edge, and a password hash in nginx.

Primitive #2: Partial Reuse



When the secret has strong memory alignment properties, the first primitive falls through.

Up next is a primitive that can be used when attacker-controlled input can partially overwrite stale secret data with predictable reuse properties. User-space memory allocators encourage memory reuse and do not normally zero out deallocated buffers for performance reasons.

This means that a target application often reuses the same memory page and selectively overwrites the content of a reused page with new data.

So, same set up as before. This time, when the attacker gives a larger input, it overwrites the first part of the secret. Now the attacker can brute force the second part using memory deduplication.

Once this is known, the attacker can brute force the first part of the secret, by deduplicating against a page without an overwritten secret.

This primitive is apparently fairly common in practice, and is used in the paper to break heap ASLR (Address Space Layout Randomisation) in nginx.

Primitive #3: Birthday Heap Spray

Last one. This primitive can leak a secret even when the attacker has no control over memory alignment or reuse. It relies on a generative approach that revolves around the birthday paradox, which we all know and love.

This primitive is applicable when the attacker can force the application to controllably spray target secrets over memory.
Up until now we have only assumed there is one secret we want to leak.

If a partially masked secret has $P$ possible values, we use memory deduplication to perform $1 \times P$ comparisons between the $P$ probe pages and the single target page - essentially brute forcing the secret.

For large $P$, a very large amount of memory is required, for the attack, and for the redundancy. However, if the attacker can cause the target application to generate many secrets, memory deduplication provides a much stronger primitive than brute forcing.

For instance, an attacker can generate a large number of secret heap pointers by creating a large number of objects from JavaScript, each referencing another object.

So, assume the attacker causes the application to generate $S$ such pages, each with a different secret pointer. The attacker now also creates $P$ probe pages, with $P$ being roughly the same size as $S$.

Each probe page uses the same encoding as the secret pages, except that, not knowing the secret pointers, the attacker needs to 'guess' their values. Each probe page contains a different guessed value. The idea is to find at least one of the probe pages matching any of the secret pages. So we have a classic Birthday Problem, where the secret and probe values are the birthdays.

Since memory deduplication compares any page with any other page in each deduplication pass, it automatically tests all $P$ probe pages against the $S$ target secret pages.

A hit on any of the $P$ possible values immediately exposes a target secret. This birthday primitive reduces the memory requirements of the attack by a factor of $S$. It's especially useful when leaking the location of randomised pointers. For exploitation purposes, it's not important which pointer the attacker hits, as long as at least one of them is hit.

This primitive is used to leak a randomised heap pointer in Microsoft Edge, and also used to craft a reliable Rowhammer exploit.

Rowhammer Exploit

Anyway, now we get to the good bit; the Rowhammer exploit. The Rowhammer bug was first publicly disclosed in June 2014 by Kim et al. However, it was not until March 2015 that someone published a working Linux kernel privilege escalation exploit using Rowhammer.

Essentially, Rowhammer is DRAM vulnerability that allows an attacker to flip bits in a (victim) memory page by repeatedly reading from other (aggressor) memory pages. These bit flips are deterministic, meaning that once a vulnerable memory location is identified, it is possible to reproduce the bit flip patterns by reading again the same set of aggressor pages.

There are two variations of the Rowhammer attack. Single-sided Rowhammer repeatedly activates a single row to corrupt its neighbouring rows' cells. Double-sided Rowhammer targets a single row by repeatedly activating both its neighbours rows. Although double-sided is more effective than single-sided in practice, it requires you to know not only the location of what you want to target, but also the two adjacent memory pages. Therefore, the authors of this paper choose to use the single-sided Rowhammer attack.

So, if you want to use the Rowhammer attack to do something useful, rather than to just corrupt random memory pages, then first you need to find a vulnerable memory location. To do this, you can allocate a large array filled with doubles. If you do it right, you can end up with an encoded double value such that all bits are set to 1.

Then, you find some eviction sets so you can bypass the cache, and you hammer 32 pages at a time (though it doesn't say where they get the number 32 from). By hammer, I mean you read from each page two million times before moving to the next page. Once you've gone through all 32 pages, you check the entire Array for bit flips. After scanning a sufficient number of pages, you will know which bits can be flipped at which offsets.

Next, you want to determine what to place in these vulnerable memory locations to craft the exploit. The idea behind the Rowhammer attack in this paper, is to place some data in the array which, after a bit flip, can yield a reference to a controlled counterfeit object.



So once we find out which bit is flipped, we can store a valid object reference at the vulnerable memory location and pivot to a counterfeit object with Rowhammer.

So that's the attack - using memory deduplication to leak the code and heap pointers, and Rowhammer to pivot them to a counterfeit object.  In practice, the paper says it takes about 30 minutes for the two deduplication passes and 508MB of memory. It also takes 1GB of memory to find bit flips, and 32MB to find the cache eviction sets. The timing of the Rowhammer depends on the vulnerable DRAM chips, so they say it varies from seconds to hours in practice.


Mitigation

So finally, the last part of the paper talks about how one would prevent attacks like this. A fairly obvious observation would be to just no deduplicate, as that prevents all exploits like this. However, deduplication contributes heavily to using physical memory efficiently, so we would like to keep it if possible.

This paper suggests that you only deduplicate zero pages. A zero page is a page with a starting address of zero, or rather, the series of memory addresses at the absolute beginning of a computer's address space. The paper doesn't go into much detail about why, but claims they've done experiments to show that it's nearly as efficient as full deduplication, but entirely eliminates the possibility for any attacks targeting memory deduplication.

Fin

So that's the end of paper, they finish by saying memory deduplication is dangerous in the wrong hands, and they've been working with their contacts at Microsoft to find an immediate solution.

I had a look round on the internet but no-ones has published anything further since the paper, so who knows?

Friday, October 28, 2016

Study Group: Towards Sound Fresh Re-keying with Hard (Physical) Learning Problems

MathJax TeX Test Page
The first study group on side-channel analysis in the academic year 2016/2017 is "Towards Sound Fresh Re-keying with Hard (Physical) Learning Problems" from Stefan Dziembowski, Sebastian Faust, Gottfried Herold, Anthony Journault, Daniel Masny and Francois-Xavier Standaert, and recently published at Crypto this year. There are mainly two reasons why I chose to present it: it is somehow close to my personal research and, generally speaking, I like to read of different areas influencing each other. In this particular case, learning problems which are usually associated to post-quantum cryptography (more specifically to lattice-based cryptography) have been used to prove a security result in leakage-resilient cryptography.

In one sentence and very loosely speaking, the content of the paper which has been covered in the study group can be summarised as: the authors construct a fresh re-keying scheme and show it is secure under the LPN assumption. The remaining of this blog post is structured as a number of questions and answers on the previous statement that will hopefully clarify and detail its meaning.

Q: what is a fresh re-keying scheme?
A: intuitively, a fresh re-keying scheme is a set of cryptographic algorithms which allows the generation of session keys from a master secret key and some publicly known randomness. A session key is used to encrypt a message and then discarded. The following diagram nicely represents what is going on.

Protecting block ciphers (or other cryptographic primitives) against DPA is an expensive procedure that introduces overhead in the design and whose security is often based on heuristic arguments. Such attacks exploit the dependence of multiple power traces with the same secret key to retrieve it. Hence, fresh re-keying schemes represent a solution that try to structurally avoid the threat by using each session key only once. Both the receiver and the sender can compute the same session key by means of the shared master secret key and some randomness associated with the ciphertext. At that point, the underlying block cipher should retain security only against SPA, while the algorithm the session key is computed by should resist DPA to prevent the adversary to gain information about the master secret key thanks to leakage. The overall scheme is designed to be more efficient than a one which applies DPA countermeasures directly to the block cipher because the GenSK algorithm is built to be easier to protect.

Q: what is the LPN assumption?
A: the Learning Parity with Noise (LPN) problem asks to find a vector $\mathbf{k}\in\mathbb{Z}_2^n$ given query access to an oracle that outputs $$ (\mathbf{r},<\mathbf{r},\mathbf{k}>\oplus e) \in \mathbb{Z}_2^n\times\mathbb{Z}_2 $$ where $\mathbf{r}\in\mathbb{Z}_2^n$ is a uniformly random vector and $e\leftarrow \mathcal{B}_\tau$ is an error bit drawn from the Bernoulli distribution of a fixed parameter $0<\tau<1/2$, i.e. $\mathbb{P}(e=1)=\tau$. The decisional version, instead, asks to distinguish the above distribution from the uniform one on $\mathbb{Z}_2^n\times\mathbb{Z}_2$. The first step in the overall proof is to define a similar version of such a problem, what the author call the Learning Parity with Leakage (LPL) problem, in which everything is the same bar the distribution of the noise, which is now taken to be the Gaussian distribution over the reals. Note that the second component of the LPL distribution is then a real number. It is shown that: $$ \text{LPN is hard} \Rightarrow \text{LPL is hard}. $$

Q: what does it mean for a fresh re-keying scheme to be secure?
A: since we deal with an environment where leakage exists, we should specify both what kind of security model the proof follows and what kind of leakage the adversary is allowed to query.

Security can be stated as the indistinguishability game in the above picture. In the real part on the left, the adversary plays with and queries a real oracle, that is to say an oracle that generates all keys and randomness according to the scheme to be secured. Hence also the leakage is computed on the sensitive variables. Instead, the right-hand side depicts the random game: an ideal oracle generates keys at random, which are then passed to a simulator which computes leakage on them. For the security claim to be valid, the (computational) adversary shouldn't distinguish the oracles she is playing with.
The leakage model is the $t$-noisy probing model. If you imagine a circuit, such a model can be depicted as the adversary being allowed to put probes on up to $t$ wires and to read a noisy version of the values being carried on them. In the specific case of their construction, since there isn't a circuit from which the adversary can get leakage from, the authors specify a set of variables on which noisy bit-selector functions are applied.

Q: which is their fresh re-keying scheme?
A: even if the scheme, called $\Pi_{noisy}$ in the paper, is composed of a set of algorithms, I only specify the core of GenSK, which is responsible for generating the session key. I assume the inputs are some randomness $R$ and a set of shares of the master secret key $\{msk_i\}_{i\leq d}$, which are needed in order to secure the algorithm against DPA. $$ \begin{align*} 1.\ & u_i \leftarrow R\cdot msk_i \\ 2.\ & u \leftarrow \sum_{i\leq d} u_i \\ 3.\ & sk \leftarrow H(u) \\ \end{align*} $$ The sum in the second line is computed iteratively as $((\dots (u_1 +u_2)+u_3)+\dots )+u_d$ for security reasons, while the $H$ in the third line is an hash function modelled as a random oracle. In the game, the adversary is given $sk$ and $R$ and can query leakage on a certain amount of bits of: $msk_i$, $R\cdot msk_i$ and $\sum u_i$. A further step not specified in the previous list is the application of a refreshing algorithm to the shares of the master secret key, in such a way that they look like new shares when the next session key is generated. Finally, the author prove the following: $$ \text{LPL is hard} \Rightarrow \Pi_{noisy} \text{ is secure}. $$

For many other details, the paper is available on ePrint.

What is SPDZ? Part 2: Circuit Evaluation

This blog post is the second in a series of three in which we describe what SPDZ is. The first part can be found here and the third here.

In this part we discuss how we perform the additions and multiplications of shared secret values as in the SPDZ protocol.

The Preprocessing Model

The preprocessing model is a framework in which we assume the existence of a 'trusted dealer' who distributes 'raw material' to the parties before any circuit evaluation takes place. The reason for doing this is that if we have the 'raw material' already in place, the circuit can be evaluated very efficiently.

We realise this by splitting the protocol into two phases: a 'preprocessing' (or 'offline') phase where we generate the preprocessed data, and an 'online' phase where we evaluate the circuit. (The term 'offline' is a little misleading, since the parties still communicate.)

Evaluating the Circuit

We now suppose that the parties have agreed on some arithmetic circuit they want to evaluate jointly.

Providing input

The first step of the circuit evaluation is to read in values from (some of) the parties. In SPDZ, this means getting each party with input, $P_i$ with $x^i$ (the superscript $i$ is an index to avoid confusion with $x_i$ which is a share of $x$), to secret-share input $x^i$ to the other parties.

To do this, we need to use a share of a random value: each party $P_i$ holds some $r_i$ and the value $r = \sum_i r_i$ is uniformly random and not known by any party. First, every party sends their share $r_j$ of $r$ to party $P_i$. This is called a 'partial opening' because $P_i$ now discovers the value of $r$ (by summing the shares).

Party $P_i$ then broadcasts the value $x^i - r$. Party $P_1$ sets its share of $x^i$ as $x_1^i=r_1 + x^i - r$, and $P_j$ for $j > 1$ sets its share of $x^i$ as $x_i^j=r_j$. (In practice, it is not always $P_1$ who does a different computation.)

Thus we have turned $x^i$ into $\langle x^i \rangle$.

Addition

Suppose we want to compute some arithmetic circuit on our data; that is, some series of additions and multiplications. We have the following very useful properties of an additive sharing scheme:
  • If we have some secret-shared field elements $\langle a \rangle$ and $\langle b \rangle$, so party $P_i$ has $a_i$ and $b_i$, each party can locally compute $a_i + b_i$, and hence, since $\sum_i a_i + \sum_i b_i = \sum_i (a_i+b_i)$, they obtain a secret sharing $\langle a + b \rangle$ of the value $a+b$.
  • If each party multiplies its share by some public value $\alpha$, then since $\sum_i \alpha a_i = \alpha \sum_i a_i = \alpha a$ the parties obtain a secret sharing $\langle \alpha a \rangle$ of the value $\alpha a$.

In other words, the secret-sharing is linear; this is good because it means there is no communication cost for doing these operations. Unfortunately, we have to do a bit more work to multiply secret-shared elements.

Multiplication

In 1991, Donald Beaver [2] observed the following: suppose we want to compute $\langle x y \rangle$ given some $\langle x \rangle$ and $\langle y \rangle$, and we already have three secret-shared values, called a ‘triple’, $\langle a \rangle$, $\langle b \rangle$ and $\langle c \rangle$ such that $c = a b$. Then note that if each party broadcasts $x_i-a_i$ and $y_i - b_i$, then each party $P_i$ can compute $x-a$ and $y-b$ (so these values are publicly known), and hence compute \[ z_i=c_i + (x-a) b_i + (y-b) a_i + (x-a)(y-b) \] Now observe that if we add all of these up, we get \[\sum_i z_i = c + (x-a)b + (y-b)a + (x-a)(y-b) = xy\] and so we have a secret sharing $\langle z \rangle$ of $xy$.

The upshot is that if we have lots of triples, then we can perform many multiplications on secret-shared data and hence we can compute any arithmetic circuit. (Note that a triple cannot be reused because this would reveal information about the secrets we are trying to multiply - i.e. since $x-a$ and $y-b$ are made public in the process above)

Importantly, observe that not only do these triples not depend on the input secret-shared values they are used to multiply, they are also completely independent of the circuit to be evaluated. This means that we can generate these triples at any point prior to evaluating the circuit. The values of $a$, $b$ and $c$ are not known by any parties when generated - each party only knows a share of each of 'some values' for which they are told this relation holds.

Moreover, if these triples are generated in the offline phase at some point before the circuit evaluation, since addition, scalar multiplication and field multiplication are inexpensive in terms of communication and computation, the online phase is both highly efficient and information-theoretically secure.

Combining the above, we have now established roughly how SPDZ evaluates an arithmetic circuit.

Next time: In the final part, we will look at what makes SDPZ different from other MPC protocols which achieve the same thing.

References 

[1] B. Applebaum, Y. Ishai, and E. Kushilevitz. How to garble arithmetic circuits. 52nd FOCS, pp120–129. IEEE Computer Society Press, 2011
[2] D. Beaver. Efficient Multiparty Protocols using Circuit Randomisation. In J. Feigenbaum, editor, CRYPTO, volume 576 of Lecture Notes in Computer Science, pp420-432, Springer, 2012.
[3] R. Bendlin, I. Damgard, C. Orlandi, and S. Zakarias. Semi-homomorphic encryption and multiparty computation. In EUROCRYPT, pp169-188, 2011.
[4] I. Damgard, M. Keller, E. Larraia, V. Pastro, P. Scholl, N. P. Smart. Practical covertly secure MPC for dishonest majority - or: Breaking the SPDZ limits. In ESORICS (2013), J. Crampton, S. Jajodia, and K. Mayes, Eds., vol. 8134 of Lecture Notes in Computer Science, Springer, pp. 1–18.
[5] I. Damgard, V. Pastro, N. P. Smart, and S. Zakarias. Multiparty computation from somewhat homomorphic encryption. In Advances in Cryptology – CRYPTO 2012, volume 7417 of LNCS, pp643–662. Springer, 2012.
[6] I. Damgard and S. Zakarias. Constant-overhead secure computation of
boolean circuits using preprocessing. In TCC, pp621-641, 2013.
[7] M. Keller and E. Orsini and P. Scholl. MASCOT: Faster Malicious Arithmetic Secure Computation with Oblivious Transfer. Cryptology ePrint Archive, Report 2016/505, 2016.
[8] J. Buus Nielsen, P. Nordholt, C. Orlandi, and S. Burra. A new approach to practical active-secure two-party computation. In Reihaneh Safavi-Naini and Ran Canetti, editors, Advances in Cryptology CRYPTO 2012, volume 7417 of Lecture Notes in Computer Science, pp681-700. Springer Berlin Heidelberg, 2012.
[9] A. Shamir. How to Share a Secret. In Communications of the ACM, Volume 22 Issue 11, Nov. 1979, pp612-613.
[10] A. Yao. How to generate and exchange secrets. In SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp162–167. IEEE, 1986.

Friday, October 21, 2016

What is SPDZ? Part 1: MPC Circuit Evaluation Overview

This blog post is the first in a series of three in which we look at what MPC circuit evaluation is, an outline of how MPC protocols in the so-called 'preprocessing model' work, and finally the specifics of SPDZ. They will come in weekly instalments.

In this part, we will introduce the idea of MPC circuit evaluation.

Introduction

If you do research in the field of cryptography, at some point you’ve quite possibly come across the curiously named SPDZ ('speedz'). The aim of this blog post is to explain what it is and why it’s used. In order to keep this post as short and accessible as possible, lots of the details are omitted, and where new concepts are introduced, they are kept superficial.

We start by defining secure multi-party computation (MPC): MPC is a way by which multiple parties can compute some function of their combined secret input without any party revealing anything more to the other parties about their input other than what can be learnt from the output.

Let’s make this more concrete: suppose there are two millionaires who want to know which of them has more money without revealing exactly how much money they have. How can they do this? Clearly we can do it with MPC, providing it exists.

Thankfully, MPC does exist. It is used in many different contexts and has various applications, ranging from the 'simple' and specific such as oblivious transfer (more on this later), to the relatively general-purpose functionality of joint circuit computation.  SDPZ is an MPC protocol allowing joint computation of arithmetic circuits.

Circuit Garbling vs Secret Sharing

There are two main constructions of MPC protocols for circuit evaluation: circuit garbling and secret sharing.

The answer to the so-called millionaire’s problem was first found in the 1980s with Yao’s garbled circuits [10]. As circuit garbling is somewhat parallel to the MPC model we work with in SPDZ, we will not discuss it here.

Contrasting this, the SPDZ protocol is a secret-sharing-based MPC protocol.

Secret-Sharing-Based MPC

Whereas circuit garbling involves encrypting and decrypting keys in a specific order to emulate a circuit evaluation (originally a Boolean circuit, but now arithmetic circuits too [1]), SPDZ instead ‘secret shares’ inputs amongst all parties and uses these shares to evaluate a circuit.

SPDZ is neither the first nor the only secret-sharing-based MPC protocol. Other well known constructions include BDOZ [3], TinyOT [8] and MiniMAC [6]. MASCOT [7] can be seen as an oblivious-transfer-based version of SPDZ. This will be discussed in a little more detail later on.

What is secret sharing?

Suppose I have some field element $a \in \mathbb{F}$, split it up ‘at random’ (uniformly) into two pieces, $a = a_1 + a_2$, and give party $P_1$ the value $a_1$ and $P_2$ the value $a_2$. Neither party knows the value $a$, but together they can recover it. We will write $\langle a \rangle$ to mean that the value $a$ is secret-shared between all parties (i.e. for each i, party $P_i$ has $a_i$, where $\sum_i a_i = a$).

Of course, there are different ways of secret sharing data (e.g. the analogous multiplicative sharing $a = a_1 \cdot a_2$, and also more complicated schemes like Shamir’s [9]), but it turns out that the additive scheme is particularly useful for MPC applications, as we shall see.

The basic overview of secret-sharing MPC of arithmetic circuits (SSMPCoAC?) is the following: 
  1. The parties first secret-share their inputs; i.e. input $x^i$ is shared so that $\sum_j x_j^i = x^i$ and party $P_j$ holds $x_j^i$ (and $P_i$ which provides input is included in this sharing, even though it knows the sum).
  2. The parties perform additions and multiplications on these secret values by local computations and communication of certain values (in methods specified below). By construction, the result of performing an operation is automatically shared amongst the parties (i.e. with no further communication or computation).
  3. Finally, the parties 'open' the result of the circuit evaluation. This last step involves each party sending their 'final' share to every other party (and also performing a check that no errors were introduced by the adversary along the way).
These are the steps we follow in a few different MPC circuit evaluation protocols, as we have discussed. The way we compute the circuit differs (slightly) with the protocol.

Next time: In the next part in this series, we will see how to use these secret-shared values to evaluate an arithmetic circuit as in the SDPZ protocol.

 

References

[1] B. Applebaum, Y. Ishai, and E. Kushilevitz. How to garble arithmetic circuits. 52nd FOCS, pp120–129. IEEE Computer Society Press, 2011
[2] D. Beaver. Efficient Multiparty Protocols using Circuit Randomisation. In J. Feigenbaum, editor, CRYPTO, volume 576 of Lecture Notes in Computer Science, pp420-432, Springer, 2012.
[3] R. Bendlin, I. Damgard, C. Orlandi, and S. Zakarias. Semi-homomorphic encryption and multiparty computation. In EUROCRYPT, pp169-188, 2011.
[4] I. Damgard, M. Keller, E. Larraia, V. Pastro, P. Scholl, N. P. Smart. Practical covertly secure MPC for dishonest majority - or: Breaking the SPDZ limits. In ESORICS (2013), J. Crampton, S. Jajodia, and K. Mayes, Eds., vol. 8134 of Lecture Notes in Computer Science, Springer, pp. 1–18.
[5] I. Damgard, V. Pastro, N. P. Smart, and S. Zakarias. Multiparty computation from somewhat homomorphic encryption. In Advances in Cryptology – CRYPTO 2012, volume 7417 of LNCS, pp643–662. Springer, 2012.
[6] I. Damgard and S. Zakarias. Constant-overhead secure computation of
boolean circuits using preprocessing. In TCC, pp621-641, 2013.
[7] M. Keller and E. Orsini and P. Scholl. MASCOT: Faster Malicious Arithmetic Secure Computation with Oblivious Transfer. Cryptology ePrint Archive, Report 2016/505, 2016.
[8] J. Buus Nielsen, P. Nordholt, C. Orlandi, and S. Burra. A new approach to practical active-secure two-party computation. In Reihaneh Safavi-Naini and Ran Canetti, editors, Advances in Cryptology CRYPTO 2012, volume 7417 of Lecture Notes in Computer Science, pp681-700. Springer Berlin Heidelberg, 2012.
[9] A. Shamir. How to Share a Secret. In Communications of the ACM, Volume 22 Issue 11, Nov. 1979, pp612-613.
[10] A. Yao. How to generate and exchange secrets. In SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp162–167. IEEE, 1986.


Thursday, October 6, 2016

Study Group: Crying Wolf: An Empirical Study of SSL Warning Effectiveness

Today's study group was on the now a little dated paper of 2009 'Crying Wolf: An Empirical Study of SSL Warning Effectiveness' [1], which was published at USENIX. In cryptography research, it is easy to overlook implementation and usability and instead focus on theory. As is succinctly explained in Randall Munroe's well-known comic, the weaknesses in our cryptographic solutions are seldom in the constructions themselves, but in their real-world application.

This paper explores the use and design of warnings which modern (!) browsers present to a user when SSL certificates cannot be verified, and in particular the user's reaction to them. There is little point in a cryptographically secure system of authentication if the end user ignores and proceeds past warnings when presented with them. The authors suggests that when browsers 'cry wolf' upon encountering SSL errors, users become desensitised over time, learn to ignore these warnings, and thus become susceptible to having their data stolen.

What is SSL?

(The initiated can skip this.)
SSL stands for Secure Sockets Layer, and is a method by which a client can access a web server securely. The SSL Handshake protocol uses a so-called SSL certificate to verify a server's authenticity to a client. An SSL certificate specifies whom the certificate was issued to, whom it was issued by, the period of validity and the server's public key. (Old SSL protocols have been superseded by TLS, but the principles involved are essentially the same.) At a very high level, the protocol proceeds as follows:
  1.  The client sends a 'hello' message to the server, requesting content.
  2.  The server sends the client its certificate, which contains its public key.
  3.  The client checks that the certificate is valid.
  4.  If the check passes, the client generates a session key, encrypts using the server's public key, and sends this to the server. If the check fails, the client aborts.
  5.  The server decrypts the session key using its secret key.
The client and the server can now encrypt all data sent between them using the (symmetric) session key.

What can go wrong?

If the certificate is invalid, the client aborts. The problems this study considers are:
  •  Expired certificate: the certificate is no longer valid.
  •  Unknown certificate authority: the issuing authority is not known.
  •  Domain mismatch: the domain of the web server and the certificate's listed domain do not match.
If one of the above occurs, the web browser will alert the user. The purpose of the study was to assess the effectiveness of the browser in conveying the severity of the problem to the user: strong warnings where the risks are small cause people to assume high-risk situations given the same warning are just as innocuous.

The Studies

Survey

Using a survey, the authors gathered data from 409 web users on their reactions to SSL warnings and their overall comprehension of the risks involved in ignoring them.

They found that context (i.e. the type of website visited) made little difference to whether or not a user would heed the warnings.

According to the data, respondents who understood 'Domain mismatch' and 'Unknown certificate authority' warnings were less likely to proceed than those who did not, whereas those who understood certificate expiry errors were more likely to proceed. In fact, the experimenters found that users consistently rated risk of an expired certificate lower than the other two errors.

The authors additionally report some wonderful responses from users, including:
  •  'I use a Mac, so nothing bad would happen'
  •  'Since I use FreeBSD, rather than Windows, not much [risk]'
  •  'On my Linux box, nothing significantly bad would happen'

Laboratory Experiment

A set of 100 participants were asked to use four websites to complete different tasks. One website was a banking website with an invalid certificate, one a library website with an invalid certificate, and two were other sites used as dummies.

The participants were shown either Internet Explorer 7 (IE7), Firefox 2 (FF2), Firefox 3 (FF3), or one of two newly-designed SSL warnings. The IE7 warning is whole page but requires just one click to ignore. The FF2 warning is a pop-up window but also only requires one click to ignore. The first version of the FF3 warning needed 11 steps. 'They made the original version of the warning so difficult for users to override, that only an expert could be likely to figure out how to do it.' The first new design was multi-page and asked users to specify the nature of the website they were visiting, presenting severe warnings for websites requiring a high level of security and milder warnings otherwise. The second new design was similar to the FF3 warning but 'looked more severe'. Images can be found in the paper.

For the library website, the IE7, FF2 and multi-page warnings did not prevent people from proceeding compared to the FF3 warning, and the single-page warning was similar to the previous warnings.

For the banking website, the two new warnings did prevent people from accessing the website, but no more than the FF3 warning. The new warnings and the FF3 warning outperformed the IE7 and FF2 warnings in preventing people from accessing the website.

Conclusions

In conclusion, the authors say that the average user does not understand the dangers of SSL warnings, and as such the decision of whether or not to proceed should essentially be made for them by the browser in most cases.

More recently, Chrome recently redesigned its SSL warnings due to the large proportion of users who simply ignored all SSL warnings [2].

To see different SSL warnings in your current browser, visit badssl.com.

References 

[1] Crying Wolf: An Empirical Study of SSL Warning Effectiveness by Joshua Sunshine, Serge Egelman, Hazim Almuhimedi, Naha Atri and Lorrie Faith Cranor. In Proceedings of the 18th Conference on USENIX Security Symposium, 2009; link.
[2] Improving SSL Warnings: Comprehension and Adherence by Adrienne Porter Felt, Alex Ainslie, Robert W. Reeder, Sunny Consolvo, Somas Thyagaraja, Alan Bettes, Helen Harris and Jeff Grimes. In CHI 2015; link.