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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.
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. [1] [2] Key pairs are generated with cryptographic algorithms based on mathematical problems termed one-way functions.
The PKCS #1 standard defines the mathematical definitions and properties that RSA public and private keys must have. The traditional key pair is based on a modulus, n, that is the product of two distinct large prime numbers, p and q, such that =.
To create signature keys, generate an RSA key pair containing a modulus, N, that is the product of two random secret distinct large primes, along with integers, e and d, such that e d ≡ 1 (mod φ(N)), where φ is Euler's totient function. The signer's public key consists of N and e, and the signer's secret key contains d.
A cryptographic key is categorized according to how it will be used and what properties it has. For example, a key might have one of the following properties: Symmetric, Public or Private. Keys may also be grouped into pairs that have one private and one public key, which is referred to as an Asymmetric key pair.
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.
Symmetric-key algorithms use a single shared key; keeping data secret requires keeping this key secret. Public-key algorithms use a public key and a private key. The public key is made available to anyone (often by means of a digital certificate). A sender encrypts data with the receiver's public key; only the holder of the private key can ...
Key exchange (also key establishment) is a method in cryptography by which cryptographic keys are exchanged between two parties, allowing use of a cryptographic algorithm. In the Diffie–Hellman key exchange scheme, each party generates a public/private key pair and distributes the public key. After obtaining an authentic copy of each other's ...