RSA Problem

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In cryptography, the **RSA problem** summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a *message* to an *exponent*, modulo a composite number *N* whose factors are not known. As such, the task can be neatly described as finding the *e*<sup>th</sup> roots of an arbitrary number, modulo N. For large RSA key sizes (in excess of 1024 bits), no efficient method for solving this problem is known; if an efficient method is ever developed, it would threaten the current or eventual security of RSA-based cryptosystems -- both for public-key encryption and digital signatures.

More specifically, the RSA problem is to efficiently compute*P* given an RSA public key (*N*, *e*) and a ciphertext *C* ≡ *P*<sup>*e*</sup> (**mod** *N*). The structure of the RSA public key requires that *N* be a large semiprime (i.e., a product of two large prime numbers), that 2 < *e* < *N*, that *e* be coprime to φ(*N*), and that 0 ≤ *C* < *N*. *C* is chosen randomly within that range; to specify the problem with complete precision, one must also specify how *N* and *e* are generated, which will depend on the precise means of RSA random keypair generation in use.

, the most efficient means known to solve the RSA problem is to first factor the modulus*N*, which is believed to be impractical if *N* is sufficiently large (see integer factorization). The RSA key setup...

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More specifically, the RSA problem is to efficiently compute

, the most efficient means known to solve the RSA problem is to first factor the modulus

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