Prime Numbers in Cryptography: Uses & Benefits

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Large prime numbers play a crucial role in cryptography, particularly in RSA encryption, which relies on modular arithmetic with these primes. The security of RSA hinges on the difficulty of factoring the product of large primes, making decryption challenging. For instance, the RSA-200 semiprime was factored using extensive computational resources, illustrating the complexity involved. Beyond cryptography, large primes are utilized in numerical algorithms, pseudorandom number generation, and private information retrieval schemes. Their applications extend into various fields, showcasing their versatility and importance.
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What is yhe usage of big primes in Cryptography?
 
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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?
 
soandos said:
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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