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Constant-Time Crypto

Why Constant-Time Crypto?

In 1996, Paul Kocher published a novel attack on RSA, specifically on RSA implementations , that extracted information on the private key by simply measuring the time taken by the private key operation on various inputs. It took a few years for people to accept the idea that such attacks were practical and could be enacted remotely on, for instance, an SSL server; see this article from Boneh and Brumley in 2003, who conclude that:

Our results demonstrate that timing attacks against network servers are practical and therefore all security systems should defend against them.

Since then, many timing attacks have been demonstrated in lab conditions, against both symmetric and asymmetric cryptographic systems.

This requires a few comments. First, while timing attacks work well in research conditions, they are extremely rarely spotted in the wild (I am not aware of a single case). Timing attacks usually require many attempts to gather enough samples for statistics to reveal the sought timing difference; as such, they tend to be somewhat slow and not very discreet. This does not mean that timing attacks are not real or do not apply, only that the state of the security of many systems is such that typical attackers have easier, faster ways in.

Another important point is that when timing attacks apply, they are all-encompassing: if the context is such that secret information held in a system may leak through external timing measures, then everything the system does may be subject to such leaking. This is not limited to cryptographic algorithms. Research on timing attacks tends to focus on secret keys because keys are high-value targets (a key concentrates a lot of secrecy), and cryptographers talk mostly about cryptography; however, even if all cryptographic algorithms in your system are protected against timing attacks, you are not necessarily out of trouble in that respect. In BearSSL I am doing my part, by providing constant-time implementations for all operations that are relevant to SSL; but slapping a constant-time SSL implementation over existing software is not sufficient to achieve general timing immunity. This is only a good start.

Timing attacks are a subset of a more general class of attacks known as side-channel attacks . A computer system runs operations in a conceptual abstract machine, that takes some inputs and provides some outputs; side-channel attacks are all about exploiting the difference between that abstract model and the real thing. In the context of smart card security, for instance, power analysis attacks (in particular Differential Power Analysis , that compares power usage between successive runs) have proven to be a great threat. Timing attacks still have a special place in that they can be applied remotely, through a network, while all other side-channel leakages require the attacker to be physically close to its target.

Constant-time implementationsare pieces of code that do not leak secret information through timing analysis. This is one of the two main ways to defeat timing attacks: since such attacks exploit differences in execution time that depend on secret elements, make it so that execution time does not depend on secret elements. Or, more precisely, that variations in execution time are not correlated with secret elements: execution time may still vary, but not in a way that can be traced back to any kind of value that you wish to keep secret, in particular (but not only) cryptographic keys.

The other main method is masking . For instance, in the case of RSA, the core operation is computing m d mod n , for a messsage m (encrypted message to decrypt, or hashed and padded data to sign), modulus n (a public value) and private exponent d . Masking entails generating a random integer r modulo n , and really computing r -1 ( mr e ) d mod n . Thus, the private exponent is applied on a value that the attacker does not know (the product of m with r e ), depriving him of any chance of correlating the execution time with the actual message or private key data.

Masking has some drawbacks, in particular:

It works only in algorithms that have an algebraic structure that is amenable to such manipulations.

Whether the masking is really efficient at thwarting attacks depends on a mathematical analysis which rarely leads to a conclusive proof (i.e.it more often is: “This breaks the attack as described, and nobody found a way to fix it. Yet.”).

Masking with a random value requires a source of randomness, which can be a hard requirement, especially in embedded systems. Even if a strong RNG is available at some level (a running SSL client or server mostly needs one anyway), bringing it to the implementation of a nominally deterministic algorithm (e.g.RSA decryption) can be troublesome in terms of internal API.

For these reasons, BearSSL aims for constant-time implementations for all algorithms, or at least all implementations used by default. For instance, for AES, version 0.2 contains four implementations, two of them being constant-time (and used by default), the two others meant only for special purposes.

Generic Tools Execution Model

Most elementary operations (opcodes in the machine language) are constant-time. The potentially non-constant-time operations to look for are the following:

Memory accesses. Whenever a table element is read from or written to, the address of the element may leak. In general, in the presence of caches, out-of-cache accesses take longer time; moreover, the new access will evict from cache some older data, and the eviction victim depends on the target address, so this can be detected afterwards. Cache-based leaks have been successfully leveraged to retrieve secret keys (for both RSA and AES implementations) by remote attackers, or by attackers running their own code on the same hardware (from another unprivileged process, or even from another virtual machine co-hosted on the same CPU). All levels of cache could be exploited that way.

Even in cache-less architectures, memory accesses may leak some data; for instance, many Flash controllers will take an extra delay to serve read requests that occur at the very start of a block.

Conditional jumps. Since executing code involves reading the opcodes from RAM or ROM, a conditional jump necessarily involves reading bytes from distinct addresses, so the points about memory accesses apply. But there is also an additional effect in that conditional jumps are subject to jump prediction: modern CPU try to work out in advance how the jump will be taken, and fetch the corresponding instructions. A bad prediction entails a detectable delay. Jump prediction uses both static rules, and a dedicated cache system, which can lead to the same kind of attack as table lookups.

Note that conditional jumps may leak information on the condition. This is a problem only if the condition is secret. For instance, when implementing AES-128, there are ten rounds, so an implementation may use a loop with a conditional jump that will exit after the tenth round. That AES-128 includes ten rounds is not secret, so that specific conditional jump is not problematic.

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