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The RSA problem is defined as the task of taking e th roots modulo a composite n: recovering a value m such that c ≡ m e (mod n), where (n, e) is an RSA public key, and c is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus n.
More specifically, the RSA problem is to efficiently compute P given an RSA public key (N, e) and a ciphertext C ≡ P e (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.
The public key in the RSA system is a tuple of integers (,), where N is the product of two primes p and q.The secret key is given by an integer d satisfying (() ()); equivalently, the secret key may be given by () and () if the Chinese remainder theorem is used to improve the speed of decryption, see CRT-RSA.
PKCS Standards Summary; Version Name Comments PKCS #1: 2.2: RSA Cryptography Standard [1]: See RFC 8017. Defines the mathematical properties and format of RSA public and private keys (ASN.1-encoded in clear-text), and the basic algorithms and encoding/padding schemes for performing RSA encryption, decryption, and producing and verifying signatures.
It provides the basic definitions of and recommendations for implementing the RSA algorithm for public-key cryptography. It defines the mathematical properties of public and private keys, primitive operations for encryption and signatures, secure cryptographic schemes, and related ASN.1 syntax representations. The current version is 2.2 (2012 ...
RSA uses exponentiation modulo a product of two very large primes, to encrypt and decrypt, performing both public key encryption and public key digital signatures. Its security is connected to the extreme difficulty of factoring large integers , a problem for which there is no known efficient general technique.
A key encapsulation mechanism, to securely transport a secret key from a sender to a receiver, consists of three algorithms: Gen, Encap, and Decap. Circles shaded blue—the receiver's public key and the encapsulation —can be safely revealed to an adversary, while boxes shaded red—the receiver's private key and the encapsulated secret key —must be kept secret.
In the RSA cryptosystem, Bob might tend to use a small value of d, rather than a large random number to improve the RSA decryption performance. However, Wiener's attack shows that choosing a small value for d will result in an insecure system in which an attacker can recover all secret information, i.e., break the RSA system.