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Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys to provide equivalent security, compared to cryptosystems based on modular exponentiation in Galois fields , such as the RSA cryptosystem and ElGamal cryptosystem .
As with elliptic-curve cryptography in general, the bit size of the private key believed to be needed for ECDSA is about twice the size of the security level, in bits. [1] For example, at a security level of 80 bits—meaning an attacker requires a maximum of about 2 80 {\displaystyle 2^{80}} operations to find the private key—the size of an ...
As of October 2012, CNSSP-15 [4] stated that the 256-bit elliptic curve (specified in FIPS 186-2), SHA-256, and AES with 128-bit keys are sufficient for protecting classified information up to the Secret level, while the 384-bit elliptic curve (specified in FIPS 186-2), SHA-384, and AES with 256-bit keys are necessary for the protection of Top ...
Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K. If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form
The curve set is named after mathematician Harold M. Edwards. Elliptic curves are important in public key cryptography and twisted Edwards curves are at the heart of an electronic signature scheme called EdDSA that offers high performance while avoiding security problems that have surfaced in other digital signature schemes.
To send an encrypted message to Bob using ECIES, Alice needs the following information: The cryptography suite to be used, including a key derivation function (e.g., ANSI-X9.63-KDF with SHA-1 option), a message authentication code system (e.g., HMAC-SHA-1-160 with 160-bit keys or HMAC-SHA-1-80 with 80-bit keys) and a symmetric encryption scheme (e.g., TDEA in CBC mode or XOR encryption scheme ...
Elliptic-curve Diffie–Hellman (ECDH) is a key agreement protocol that allows two parties, each having an elliptic-curve public–private key pair, to establish a shared secret over an insecure channel. [1] [2] [3] This shared secret may be directly used as a key, or to derive another key.
A Montgomery curve of equation = + +. A Montgomery curve over a field K is defined by the equation,: = + + for certain A, B ∈ K and with B(A 2 − 4) ≠ 0.. Generally this curve is considered over a finite field K (for example, over a finite field of q elements, K = F q) with characteristic different from 2 and with A ≠ ±2 and B ≠ 0, but they are also considered over the rationals with ...