Attacking RSA exponentiation with fault injection

A new paper, “Fault-Based Attack of RSA Authentication” (pdf) by Pellegrini et al, is making the rounds. The general idea is that an attacker can disrupt an RSA private key operation to cause an invalid signature to be returned, then use that result to extract the private key. If you’re new to fault injection attacks on RSA, I previously wrote an intro that should help.

The main concept to grasp is that public key crypto is brittle. In the case of RSA’s CRT operation, a single bit error in one multiplication result is enough to fully compromise your private key. We’ve known this since 1997. The solution is simple: validate every signature with the public key before returning it to the caller.

The authors noticed something curious. OpenSSL does verify signatures it generates before returning them, but if it detects a problem, it does not just return an error. It then tries again using a different exponentiation process, and then returns that signature without validating it.

Think about this for a moment. What conditions could cause an RSA private key operation to compute an invalid answer? An innocent possibility is cosmic radiation, bad RAM, etc. In this case, all computations should be considered unreliable and any retried operation should be checked very carefully. The other and more likely possibility is that the system is under attack by someone with physical proximity. In this case, OpenSSL should generate a very obvious log message and the operation should not be retried. If it is, the result should be checked very carefully.

For whatever reason, the OpenSSL programmers decided to retry with fixed-window exponentiation and trust that since there were no published fault attacks for it, they didn’t have to validate its result. This is a foolhardy attitude — not something you want to see in your crypto library. There had been many other fault injection attacks against various components or implementation approaches for RSA, including right-to-left exponentiation. There was no reason to consider left-to-right exponentiation invulnerable to this kind of attack.

Fixed-window exponentiation is a form of sliding window exponentiation. This is just a table-based optimization, where a window (say, 3 bits wide) is moved across the exponent, computing the final result incrementally. While this may be resistant to some timing attacks (but not cache timing or branch prediction attacks), there is nothing that would prevent fault injection attacks.

Indeed, it turns out to be vulnerable. The authors generate a few thousand signatures with single bit-flips in some window of the signature. Then they compare the faulty signatures to a correct signature over the same message. They compute the value for that portion of the private exponent since there are only 2w possibilities for that location if w is the window size in bits. This is repeated until enough of the private key is known.

The method they used to create the faulty signatures was a bit artificial. They built a SPARC system on an FPGA running Linux and OpenSSL. They then decreased the power supply voltage until multiplies started to fail. Since multiplication logic is a relatively long chain, it is often one of the first things to fail. However, a more interesting hardware result would be to attempt this kind of attack on an actual server because FPGAs work differently than ASICs. It might require careful targeting of the right power pins on the CPU. Since power pins are numerous in modern systems, this may be more effective than only modulating the system power supply.

This was a nice attack but nothing earth-shattering. The only thing I was floored by (yet again), was the willingness of crypto implementers to perform unsafe operations in the face of an almost certain attack. Shame on OpenSSL.

2 thoughts on “Attacking RSA exponentiation with fault injection

  1. I agree that the method used in this attack was a bit artificial, but it’s also been used in a more realistic attack on a commercial ARM9 CPU (specifically the widely-used ARM926EJ-S) running Linux, see (although as with the FPGA-CPU attack, they also disabled blinding and signature-verification in their attack).

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