# Four papers from Columbia at Crypto 2018

CS researchers worked with collaborators from numerous universities to explore and develop cryptographic tools. The papers were presented at the 38th Annual International Cryptology Conference.

**Correcting Subverted Random Oracles**

Authors: Alexander Russell *University of Connecticut*, Qiang Tang *New Jersey Institute of Technology*, Moti Yung *Columbia University*, Hong-Sheng Zhou *Virginia Commonwealth University*

Many cryptographic operations rely on cryptographic hash functions. The structure-less hash functions behave like what is called a random oracle (random looking public function). The random oracle methodology has proven to be a powerful tool for designing and reasoning about cryptographic schemes, and can often act as an effective bridge between theory and practice.

These functions are used to build blockchains for cryptocurrencies like Bitcoin, or to perform “password hashing” – where a system stores a hashed version of the user’s password to compare against the value that is presented by the user: it is hashed, and compared to the previously stored computed value in order to identify the user. These functions garble the input to look random, and it is hard from the output to find the input or any other input that maps to the same output result.

The paper is in the area of implementations of cryptographic functions that are subverted (aka implemented incorrectly or maliciously). Shared one of the researchers, Moti Yung from Columbia University,” We deal with a hash function that some entries in the function table are corrupted by the opponent – a bad guy who implemented it differently than the specification. The number of corruption is relatively small to the size of the function. What we explored is what can be done with it by the bad guys and can we protect against such bad guys.”

First, the paper shows devastating attacks are possible with this modification of a function: the bad guys can control the blockchain, or allow an impersonator to log in on behalf of a user, respectively, in the above applications.

Then, to counter the attack the researchers demonstrated how to manipulate the function (apply it a few times and combine the results in different ways) and how the new function prevents attacks.

In this paper, the focus is on the basic problem of correcting faulty—or adversarially corrupted—random oracles, so that they can be confidently applied for such cryptographic purposes.

“The construction is not too complex,” said Yung, a computer science professor. “The proof that it works, on the other hand, required the development of new techniques that show that the manipulation produces a hash function that behaves like a random oracle in spite of the manipulation by the bad guy.”

Authors: Lucas Kowalczyk *Columbia University*, Tal Malkin *Columbia University*, Jonathan Ullman *Northeastern University*, Daniel Wichs *Northeastern University*

How can one know if the way they are using data is protecting the privacy of the individuals to whom the data belongs? What does it mean to protect an individual’s privacy exactly? Differential privacy is a mathematical definition of privacy that one can objectively prove one’s usage satisfies. Informally, if data is used in a differentially private manner, then any one individual’s data cannot have significant influence on the output behavior.

For example, Apple uses differentially private data collection to collect information about its users to improve their experience, while now being able to additionally give them a mathematical guarantee about how their information is being used.

Differential privacy is a nice notion, but there is a question: is it always possible to use data in a differentially private manner? “Our paper is an impossibility result,” said Lucas Kowalczyk, a PhD student at Columbia University. “We exhibit a setting, an example of data and desired output behavior, for which we prove that differential privacy is not possible to achieve. The specific setting is constructed using tools from cryptography.”

**Proofs of Work From Worst-Case Assumptions**

Authors: Marshall Ball *Columbia University*, Alon Rosen *IDC, Herzliya*, Manuel Sabin *UC Berkeley*, Prashant Nalini Vasudevan *MIT*

A long standing goal in the area is constructing NP-Hard cryptography. This would involve proving a “win-win” theorem of the following form: either some cryptographic scheme is secure or the satisfiability problem can be solved efficiently. The advantage of a theorem like this, is that if the scheme is broken it will yield efficient algorithms for a huge number of interesting problems.

Unfortunately, there hasn’t been much progress beyond uncovering certain technical barriers. In the paper the researchers proved win-wins of the following form: either the cryptographic scheme is secure or there are faster (worst-case) algorithms than are currently known for satisfiability and other problems (violating certain conjectures in complexity theory).

“In particular, the cryptography we construct is something called a Proof of Work,” said Maynard Marshall Ball, a fourth year PhD student. “Roughly, a Proof of Work can be seen as a collection of puzzles such that a random one requires a certain amount of computational “work” to solve, and solutions can be checked with much less work.”

These objects appear to be inherent to cryptocurrencies, but perhaps a simpler and older application comes from spam prevention. To prevent spam, an email server can simply require a proof of work in order to receive an email. In this way, it is easy to solve a single puzzle and send a single email, but solving many puzzles to send many emails is infeasible.

**A Simple Obfuscation Scheme for Pattern-Matching with Wildcards**

Authors: Allison Bishop *Columbia University and IEX*, Lucas Kowalczyk *Columbia University*, Tal Malkin *Columbia University*, Valerio Pastro *Columbia University and Yale University*, Mariana Raykova *Yale University*, Kevin Shi *Columbia University*

Suppose a software patch is released which addresses a vulnerability triggered by some bad inputs. The question is how to publicly release the patch so that all users can apply it, without revealing which bad inputs trigger the vulnerability – to ensure that an attacker will not take advantage of the vulnerability on unpatched systems.

This can be addressed by what cryptographers call “program obfuscation” — producing a program that anyone can use, but no one can figure out the internal working of, or any secrets used by the program. Obfuscation has many important applications, but it is difficult to achieve, and in many cases was shown to be impossible. For cases where obfuscation is possible, existing solutions typically utilize complex and inefficient techniques and rely on cryptographic assumptions that are not well understood.

The researchers discovered a surprisingly simple and efficient construction that can obfuscate programs for pattern-matching with wildcards, motivated for example by the software patching application above. What this means is that a program can be released to check for any given input whether it matches some specific pattern (e.g., the second bit is 1, the third bit is 0, the sixth bit is 0, etc).

“Crucially, our construction allows you to publicly release such a program, while keeping the actual pattern secret,” said Tal Malkin, a computer science professor, “So people can use the program and information is kept safe.”