Dear Maged Z. Youssef ,
To go deep into the topic, one needs to read and learn many courses in abstract algebra and elliptic curves. But one can start reading elementary facts about prime numbers and cryptography, for example, the following simple, readable article
https://math.berkeley.edu/~kpmann/encryption.pdf
Best regards
Dear Abdellah LAMNII ,
Prime numbers are the best for cryptographic purposes to protect our communications. Not for little secrets, but also for the sensitive and essential ones, as well as military secrets. Prime numbers are the hardest to decipher. As an example bitcoin protocols used the large prime numbers generated by special elliptic curves tools to protect their clients. Our ID, credit cards, passports, all are ciphered using prime numbers.
Best regards
Dear Dr. Issam, would you please refer us by some materials about applications of prime numbers. Thank you
Dear Maged Z. Youssef , RSA system is an important application.
https://en.wikipedia.org/wiki/RSA_(cryptosystem)
Dear Maged Z. Youssef ,
To go deep into the topic, one needs to read and learn many courses in abstract algebra and elliptic curves. But one can start reading elementary facts about prime numbers and cryptography, for example, the following simple, readable article
https://math.berkeley.edu/~kpmann/encryption.pdf
Best regards
Notice that Carmichael numbers are used for Public Key Encription systems and RSA (cryptosystem) (see the link given in the answer by Dr. Tapas Chatterjee and the paper: R.G.E. Pinch, "On Using Carmichael Numbers for Public Key Encryption Systems", published in IMA International Conference on Cryptography and Coding
1997, pp 265-269, https://link.springer.com/chapter/10.1007%2FBFb0024472.
In the Abstact of this paper, the author wrote: “We show that the inadvertent use of a Carmichael number instead of a prime factor in the modulus of an RSA cryptosystem is likely to make the system fatally vulnerable, but that such numbers may be detected.”
On Carmichael numbers and their generalizations (the so called “weak Carmichael numbers”), see my article: “Generalizations of Carmichael numbers I”, May 2013, 46 pages, preprint https://arxiv.org/pdf/1305.1867.pdf (also available at https://www.researchgate.net/publication/236651729_Generalizations_of_Carmichael_numbers_I).
The very large phenomenology of the cellular automata model (see https://en.wikipedia.org/wiki/Cellular_automaton) and its apparently big complexity offer a good basis for applications in cryptography. Notice that the famous Lucas’ theorem (Lucas’ congruence) concerning congruences of binomial coefficients modulo prime numbers, also can be interpreted as a result about cellular automata. Namely, Lucas theorem can be interpreted as a two-dimensional p-automaton; see Remark 2 on page 7 of my survey article: Romeo Meštrović, Lucas’ theorem, its generalizations, extensions and applications (1878-2014); available at https://arxiv.org/pdf/1409.3820.pdf
(preprint arXiv, September 2014, 51 pages; also available at https://www.researchgate.net/publication/265644350_Lucas'_theorem_its_generalizations_extensions_and_applications_1878--2014).
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