What is yhe usage of big primes in Cryptography?
One major use is for RSA. Encrypting with RSA requires finding large random primes and doing modular arithmetic with them, which are easy. Decrypting RSA can be performed by factoring the product of the primes, which is believed to be hard.
As an example: the 663-bit semiprime RSA-200 was factored by a cluster of computers; the lattice sieving alone was the equivalent of 55 years of work on a single processor. I multiplied the factors together on my computer; according to Pari, this took 0 ms.
are there any uses outside cryptography?
Numerical algorithms (e.g. factorial computation), pseudorandom number generation (e.g. Mersenne twister), private information retrieval schemes (see Yekhanin's Ph.D thesis), etc.
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