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In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. [1] For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. [2] [4] Euclid proved c. 300 BCE that every prime expressed as M p = 2 p − 1 has a corresponding perfect number M p × (M p +1)/2 = 2 p − 1 × (2 p − 1). For example, the Mersenne prime 2 2 − 1 = 3 leads to the corresponding perfect number 2 2 ...
List of recreational number theory topics; Topics in cryptography; Divisibility. Composite number. ... Almost perfect number; Amicable number; Betrothed numbers;
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The development of cryptography has been paralleled by the development of cryptanalysis — the "breaking" of codes and ciphers. The discovery and application, early on, of frequency analysis to the reading of encrypted communications has, on occasion, altered the course of history.
Kerckhoffs viewed cryptography as a rival to, and a better alternative than, steganographic encoding, which was common in the nineteenth century for hiding the meaning of military messages. One problem with encoding schemes is that they rely on humanly-held secrets such as "dictionaries" which disclose for example, the secret meaning of words.
"Communication Theory of Secrecy Systems" is a paper published in 1949 by Claude Shannon discussing cryptography from the viewpoint of information theory. [1] It is one of the foundational treatments (arguably the foundational treatment) of modern cryptography. [ 2 ]
An example of an information-theoretically hiding commitment scheme is the Pedersen commitment scheme, [18] which is computationally binding under the discrete logarithm assumption. [19] Additionally to the scheme above, it uses another generator h of the prime group and a random number r. The commitment is set =. [20]