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 &equiv; P<sup>e</sup> (mod&nbsp;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&nbsp;<&nbsp;e&nbsp;<&nbsp;N, that e be coprime to &phi;(N), and that 0&nbsp;≤&nbsp;C&nbsp;<&nbsp;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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