Abstract
The transmission of information in a communication process faces various threats. These threats arise if during the transmission, the messages are at the mercy of unauthorized actions of an adversary, that is, if the channel used for the communication is insecure. Basically there are three attacks the communicants have to be aware of when using an information transmission system. An adversary might observe the communication and gain information about it, he might insert false messages or he might replace legally sent messages by false messages. The protection against the first attack is a question of secrecy and the protection against the latter two attacks is a question of authenticity.
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Notes
- 1.
First enunciated by A. Kerckhoffs (1835–1903) ([15], pp. 235).
- 2.$$\frac{{\left( {\begin{array}{c}K-a\\ a-i\end{array}}\right) }}{{\left( {\begin{array}{c}K\\ a\end{array}}\right) }}=\underbrace{\frac{K-a}{K}\cdots \frac{K-2a+i+1}{K-a+i+1}}_{a-i\;\text {factors}}\underbrace{\frac{a}{K-a+i}\cdots \frac{a-i+1}{K-a+1}}_{i\; \text {factors}}\le \left( \frac{K-a}{K}\right) ^{a-i}\left( \frac{a}{K-a}\right) ^{i}. $$
- 3.
\({\left( {\begin{array}{c}n\\ k\end{array}}\right) }\le \left( \frac{n}{k}\right) ^{k}(1+\frac{k}{n-k})^{n-k}e^{\frac{1}{12n}-\frac{1}{12k+1}-\frac{1}{12(n-k)+1}+\frac{1}{2}\ln (\frac{n}{2\pi k(n-k)})}\)
\(\le \left( \frac{ne}{k}\right) ^{k}e^{\frac{1}{2n}-\frac{1}{6n+1}+\frac{1}{2}\ln (\frac{n}{2\pi (n-1)})}\le \left( \frac{ne}{k}\right) ^{k}\).
- 4.
Remark by the editors: This statement is not up to date, because in the paper “M. Agrawal, N. Kayal, and N. Saxena, “PRIMES is in P”, Annals of Mathematics, Vol. 160, No. 2, 781–793, 2004, the authors proved the asymptotic time complexity of the algorithm to be \(\tilde{O}(\log ^{12}(n))\). In other words, the algorithm takes less time than the twelfth power of the number of digits in n times a polylogarithmic (in the number of digits) factor. However, the upper bound proved in the paper was rather loose; indeed, a widely held conjecture about the distribution of the Sophie Germain primes would, if true, immediately cut the worst case down to \(\tilde{O}(\log ^6(n))\).
- 5.
See the Remark in the previous footnote.
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Ahlswede, R. (2016). Authentication and Secret-Key Cryptology. In: Ahlswede, A., Althöfer, I., Deppe, C., Tamm, U. (eds) Hiding Data - Selected Topics. Foundations in Signal Processing, Communications and Networking, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-31515-7_2
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