Retrieving "Rsa Cryptosystem" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Modular Arithmetic

   Linked via "RSA"

   Applications in Cryptography
   Modular arithmetic is indispensable in modern Public Key Cryptography, particularly systems like RSA. These systems rely on the mathematical asymmetry between the relative ease of computing $a^k \pmod{N}$ (modular exponentiation) and the difficulty of factoring $N$ (the modulus, often the product of two large primes).
   To decrypt a message encrypted with a public exponent $e$, one must calculate the [modular multiplicative inverse](/entries/modular-multiplica…

2. Shors Algorithm

   Linked via "RSA cryptosystem"

   Shor's Algorithm, developed by Peter Shor in 1994, is a quantum algorithm designed to solve the integer factorization problem in polynomial time, offering an exponential speedup over the best-known classical algorithms for this task [2]. The core mathematical problem it addresses is finding the prime factors of a composite number $N$. Classically, the best algorithms are sub-exponential, making large factorizations computationally infeasible and forming the basis for modern public-key cryptography, such as the […

3. Shors Algorithm

   Linked via "RSA"

   Cryptographic Implications and Obsolescence
   The ability of Shor's algorithm to factor large numbers efficiently renders the security foundation of widely used public-key cryptography schemes obsolete. The security of RSA relies directly on the assumption that factorization is computationally hard for classical computers [2].
   The successful implementation of Shor's algorithm on a sufficiently large-scale, fault-tolerant quantum computer would immediately break most extant secure communications protected by these systems. Th…