Bremermann's limit, named after Hans-Joachim Bremermann, is a limit on the maximum rate of

computation Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm). Mechanical or electronic devices (or, historically, people) that perform computations are known as '' computers''. An esp ...

that can be achieved in a self-contained system in the material universe. It is derived from

Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

's mass-energy equivalency and the

Heisenberg uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physi ...

, and is '' c''²/'' h'' ≈ 1.36 × 10⁵⁰ bits per second per kilogram. This value is important when designing

cryptographic Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...

algorithms, as it can be used to determine the minimum size of

encryption key A key in cryptography is a piece of information, usually a string of numbers or letters that are stored in a file, which, when processed through a cryptographic algorithm, can encode or decode cryptographic data. Based on the used method, the key ...

s or hash values required to create an algorithm that could never be cracked by a

brute-force search In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the soluti ...

. For example, a computer with the mass of the entire

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...

operating at Bremermann's limit could perform approximately 10⁷⁵ mathematical computations per second. If one assumes that a cryptographic key can be tested with only one operation, then a typical 128-bit key could be cracked in under 10⁻³⁶ seconds. However, a 256-bit key (which is already in use in some systems) would take about two minutes to crack. Using a 512-bit key would increase the cracking time to approaching 10⁷² years, without increasing the time for encryption by more than a constant factor (depending on the encryption algorithms used). The limit has been further analysed in later literature as the maximum rate at which a system with energy spread

\Delta E

can evolve into an orthogonal and hence distinguishable state to another,

\Delta t = \frac .

In particular, Margolus and Levitin have shown that a quantum system with average energy ''E'' takes at least time

\Delta t = \frac

to evolve into an orthogonal state. However, it has been shown that access to

quantum memory In quantum computing, quantum memory is the quantum-mechanical version of ordinary computer memory. Whereas ordinary memory stores information as binary states (represented by "1"s and "0"s), quantum memory stores a quantum state for later ...

in principle allows computational algorithms that require arbitrarily small amount of energy/time per one elementary computation step.

References

{{reflist

External links

* Gorelik, G. (2003)
Bremermann's Limit and cGh-physics
* Lokshin, A (2017)
Arbitrary choice, ‘understanding’ and Gorelik-Bremermann limit. ''Far East Journal of Mathematical Sciences'', V. 102, Issue 1, P. 215–222
Cybernetics Theory of computation Limits of computation

See also

References

External links