Abstract
Modern security considerations make it desirable for us to have new types of encryption schemes. It is no longer enough to render a message so that only the intended recipient can read it (and outsiders cannot). In today’s complex world, and with the advent of high-speed digital computers, there are new demands on the technology of cryptography. We would now like to have secure messages that anyone can encode, but only select people can read. We would like cryptography systems that are protected against various types of eavesdropping and fraud. The important system that we shall describe here, developed during the 1980s by Ron Rivest, Adi Shamir, and Leonard Adleman, enables many such technologies.
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Why is that? Suppose that there were only finitely many primes \(p_{1},p_{2},\ldots,p_{k}\). Define \(P = (p_{1} \cdot p_{2} \cdot \cdots \cdot p_{k}) + 1\). If we divide P by any p j we get the remainder 1. So P is not composite. It must be prime. But P is greater than all the primes in our exhaustive list \(p_{1},p_{2},\ldots,p_{k}\). That is a contradiction. Hence there are infinitely many primes.
References and Further Reading
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Blum, M., Goldwasser, S.: An efficient probabilistic public-key encryption scheme which hides all partial information. Advances in Cryptology (Santa Barbara, 1984). Lecture Notes in Computer Science, vol. 196, pp. 289–299, Springer, Berlin (1985)
Blum, M., De Santis, A., Micali, S., Persiano, G.: Noninteractive zero-knowledge. SIAM Journal on Computing 20, 1084–1118 (1991)
Feige, U., Fiat, A., Shamir, A.: Zero-knowledge proofs of identity. Journal of Cryptology 1, 77–94 (1988)
Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the Association for Computing Machinery 21, 120–126 (1978)
De Santis, A., Micali, S., Persiano, G.: Noninteractive zero-knowledge proof systems. Advances in Cryptology—CRYPTO ‘87 (Santa Barbara, 1987). Lecture Notes in Computer Science, vol. 293, pp. 52–72, Springer, Berlin (1988)
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Krantz, S.G., Parks, H.R. (2014). RSA Encryption. In: A Mathematical Odyssey. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8939-9_9
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