Brute-force Attack - Theoretical Limits

Theoretical Limits

The resources required for a brute-force attack grow exponentially with increasing key size, not linearly. Although US export regulations historically restricted key lengths to 56-bit symmetric keys (e.g. Data Encryption Standard), these restrictions are no longer in place, so modern symmetric algorithms typically use computationally stronger 128- to 256-bit keys.

There is a physical argument that a 128-bit symmetric key is computationally secure against brute-force attack. The so-called Landauer limit implied by the laws of physics sets a lower limit on the energy required to perform a computation of kT · ln 2 per bit erased in a computation, where T is the temperature of the computing device in kelvins, k is the Boltzmann constant, and the natural logarithm of 2 is about 0.693. No irreversible computing device can use less energy than this, even in principle. Thus, in order to simply flip through the possible values for a 128-bit symmetric key (ignoring doing the actual computing to check it) would theoretically require 2128 − 1 bit flips on a conventional processor. If it is assumed that the calculation occurs near room temperature (~300 K) the Von Neumann-Landauer Limit can be applied to estimate the energy required as ~1018 joules, which is equivalent to consuming 30 gigawatts of power for one year. This is equal to 30×109 W×365×24×3600 s = 9.46×1017 J or 262.7 TWh (more than 1/100th of the world energy production). The full actual computation—checking each key to see if you have found a solution—would consume many times this amount.

However, this argument assumes that the register values are changed using conventional set and clear operations which inevitably generate entropy. It has been shown that computational hardware can be designed not to encounter this theoretical obstruction (see reversible computing), though no such computers are known to have been constructed.

As commercial available successors of governmental ASICs Solution also known as custom hardware attack, today two emerging technologies have proven their capability in the brute-force attack of certain ciphers. One is modern graphics processing unit (GPU) technology, the other is the field-programmable gate array (FPGA) technology. GPUs benefit from their wide availability and price-performance benefit, FPGAs from their energy efficiency per cryptographic operation. Both technologies try to transport the benefits of parallel processing to brute-force attacks. In case of GPUs some hundreds, in the case of FPGA some thousand processing units making them much better suited to cracking passwords than conventional processors. Various publications in the fields of cryptographic analysis have proved the energy efficiency of today’s FPGA technology, for example, the COPACOBANA FPGA Cluster computer consumes the same energy as a single PC (600 W), but performs like 2,500 PCs for certain algorithms. A number of firms provide hardware-based FPGA cryptographic analysis solutions from a single FPGA PCI Express card up to dedicated FPGA computers. WPA and WPA2 encryption have successfully been brute-force attacked by reducing the workload by a factor of 50 in comparison to conventional CPUs and some hundred in case of FPGAs.

AES permits the use of 256-bit keys. Breaking a symmetric 256-bit key by brute force requires 2128 times more computational power than a 128-bit key. A device that could check a billion billion (1018) AES keys per second (if such a device could ever be made - as of 2012, supercomputers have computing capacities of 20 Peta-FLOPS, see Titan. So 50 supercomputers would be required to process (1018) operations per second) would in theory require about 3×1051 years to exhaust the 256-bit key space.

An underlying assumption of a brute-force attack is that the complete keyspace was used to generate keys, something that relies on an effective random number generator, and that there are no defects in the algorithm or its implementation. For example, a number of systems that were originally thought to be impossible to crack by brute force have nevertheless been cracked because the key space to search through was found to be much smaller than originally thought, because of a lack of entropy in their pseudorandom number generators. These include Netscape's implementation of SSL (famously cracked by Ian Goldberg and David Wagner in 1995) and a Debian/Ubuntu edition of OpenSSL discovered in 2008 to be flawed. A similar lack of implemented entropy lead to the breaking of Enigma's code.