Papers by Ranjit Jhala

On Subnormal Floating Point and Abnormal Timing

Most of the time, people think the floating-point unit has no security significance, simply because you don’t use the floating-point unit for anything with security significance. Cryptography, authentication, access control, process isolation, virtualization, trusted paths, it’s all done with integers. Usually not even negative integers.

Today’s paper presents some situations where that is not the case: where the low-level timing behavior of floating-point arithmetic is, in fact, security-critical. The most elemental of these situations involves displaying stuff on the screen—nowadays, everything on the screen gets run through a 3D rendering pipeline even if it looks completely flat, because you have the hardware just sitting there, and that process intrinsically involves floating point. And there’s an API to tell you how long it took to render the previous frame of animation because if you were actually animating something you would genuinely need to know that. So if there’s something being displayed on the screen that you, a malicious program, are not allowed to know what it is, but you can influence how it is being displayed, you might be able to make the information you’re not allowed to know affect the rendering time, and thus extract the information—slowly, but not too slowly to be practical. There is also a scenario involving differentially private databases, where you’re allowed to know an approximation to an aggregate value but not see individual table rows; the aggregate is computed with floating point, and, again, computation time can reveal the secret values.

In both cases, the floating-point computations are uniform, with no data-dependent branches or anything like that, so how does timing variation sneak in? It turns out that on all tested CPUs and GPUs, primitive floating-point arithmetic operations—add, multiply, divide—don’t always take the same amount of time to execute. Certain combinations of input values are slower than others, and predictably so. As the authors point out, this is a well-known problem for numerical programmers. It has to do with a feature of IEEE floating point known as subnormal numbers. These allow IEEE floating point to represent numbers that are very close to zero, so close that they don’t fit into the bit representation without bending the rules. This is mathematically desirable because it means that if two floating-point values are unequal, subtracting one from the other will never produce zero. However, subnormals are awkward to implement in hardware; so awkward that CPUs in the 1990s were notorious for suffering a slowdown of 100x or more for basic arithmetic on subnormals. Nowadays it’s not so bad; if I’m reading the (poorly designed) charts in this paper correctly, it’s only a 2–4x slowdown on modern hardware. But that’s still enough to detect and build a timing channel out of.

Timing channels are a perennial problem because they tend to be side-effects of something desirable. Algorithms are often easier to understand if you dispose of special cases up-front—this paper also talks about how division by zero might be much faster than division by a normal value, presumably because the CPU doesn’t bother running through the divide circuit in that case. Running fast most of the time, slow in unusual cases, is often an excellent algorithmic choice for overall performance: hash tables, quicksort, etc. The papers that defined the concept of covert channels [1] [2] discuss timing channels introduced by data caching, without which Moore’s Law would have been hamstrung by fundamental physics decades ago.

However, I don’t think there’s any excuse for variable-time arithmetic or logic operations, even in floating point. (I might be persuaded to allow it for division, which is genuinely algorithmically hard.) In particular I have never understood why subnormal numbers are a problem for hardware. Now I don’t know beans about hardware design, but I have written floating-point emulation code, in assembly language, and here’s the thing: in software, subnormals are straightforward to implement. You need larger internal exponent precision, an additional test in both decode and encode, a count-leading-zeroes operation on the way in and an extra bit-shift on the way out. I didn’t try to make my emulator constant-time, but I can imagine doing it without too much trouble, at least for addition and multiplication. In hardware, that seems like it ought to translate to a wider internal data bus, a couple of muxes, a count-leading-zeroes widget, and a barrel shifter that you probably need anyway, and it should be easy to make it constant-time across both subnormals and normals, without sacrificing speed at all.

Constant time floating point operations for all inputs and outputs does strike me as harder, but only because IEEE floating point is specified to generate hardware faults and/or out-of-band exception bits under some circumstances. This was a mistake. A floating-point standard that has neither control nor status registers, and generates exclusively in-band results, is decades overdue. It would require sacrificing a bit of mantissa for a sticky this number is inexact flag, but I think that should be acceptable.

As a final headache, how long do you suppose it’ll be before someone comes up with an exploit that can only be fixed by making all of the transcendental function library (sin, cos, log, exp, …) constant-time? Never mind performance, I doubt this can be done without severely compromising accuracy.