Bremermann's limit, named after

Hans-Joachim Bremermann Hans-Joachim Bremermann (1926 – 1996) was a German-American mathematician and biophysicist. He worked on computer science and evolution, introducing ideas of how mating generates new gene combinations. Bremermann's limit, named after him, is th ...

, is a theoretical limit on the maximum rate of

computation A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms. Mechanical or electronic devices (or, hist ...

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

Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

's mass–energy equivalency and the

Heisenberg uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...

, and is ''c''²/ ''h'' ≈ 1.3563925 × 10⁵⁰

bits per second In telecommunications and computing, bit rate (bitrate or as a variable ''R'') is the number of bits that are conveyed or processed per unit of time. The bit rate is expressed in the unit bit per second (symbol: bit/s), often in conjunction ...

per kilogram. This value establishes an asymptotic bound on adversarial resources when designing

cryptographic Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More gen ...

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 Iteration#Computing, systematically checking all possible candida ...

. 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 Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

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,

\textstyle \Delta t = \frac .

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

\textstyle \Delta t = \frac

to evolve into an orthogonal state. This is one of the quantum speed limit theorems. However, it has been shown that chaining multiple computations or access to

quantum memory In quantum computing, quantum memory is the Quantum mechanics, quantum-mechanical version of ordinary computer memory. Whereas ordinary memory stores information as Binary number, binary states (represented by "1"s and "0"s), quantum memory s ...

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

References

{{reflist

External links

* Gorelik, G. (2010
Bremermann's Limit in cGh-physics // arXiv:0910.3424v4
* 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