Elliptic curve cryptography (ECC)
is a type of public-key cryptography that uses elliptic curves over finite
fields to secure communication. The security of the system is based on the
difficulty of computing discrete logarithms in a finite field, which is a
mathematical problem that is believed to be hard to solve efficiently.
An elliptic curve is a curve
defined by an equation in the form y^2 = x^3 + ax + b, where a and b are constants. The curve also has a special
point called the "point at infinity," which is often denoted as O.
The curve and the point O together form an additive group, which means that you
can add two points on the curve to get a third point that also lies on the
curve. The addition operation is defined using a line that intersects the curve
in exactly three points.
In ECC, each user has a public
key and a private key. The public key is a point on the elliptic curve, and the
private key is a random number. To generate a public key, the user selects a
starting point on the curve and then adds it to itself a certain number of
times, where the number of times is determined by the private key. For example,
if the private key is 5, the user would add the starting point to itself 5
times to get the public key.
To encrypt a message, the sender
uses the recipient's public key to compute a shared secret. The shared secret
is a point on the curve that is calculated by multiplying the recipient's
public key by the sender's private key. Once the shared secret is calculated,
the sender uses it to encrypt the message.
To decrypt the message, the
recipient uses their private key to compute the shared secret. The shared
secret is calculated by multiplying the sender's public key by the recipient's
private key. Once the shared secret is calculated, the recipient uses it to
decrypt the message.
ECC is very secure, and it is
often used in applications that require a high level of security, such as
online banking, e-commerce, and digital signatures. It is also used in
situations where there are limitations on computational power or storage space,
such as in mobile devices or smart cards. However, it is important to use
proper key management practices to ensure the security of the system, such as
using strong random number generators and protecting private keys from
unauthorized access.
HOW THE ELLIPTICAL CRYPTOGRAPHY WORK:
Key Generation: A user
generates a pair of related keys, a public key and a private key. The public
key is derived from a point on an elliptic curve, and the private key is a
random number that is kept secret. The user first chooses an elliptic curve,
which is a set of points that satisfy a specific mathematical equation. The
user also chooses a base point on the curve, which is a fixed point used as a
reference for all other points on the curve. The user then generates a private
key, which is a random number between 1 and the order of the base point. The
public key is then derived by multiplying the base point by the private key
using elliptic curve point multiplication. The result is a new point on the
elliptic curve that represents the public key. It also work on luna crypto. It also tell the luna crypto price.
Encryption: To encrypt a message,
the sender uses the recipient's public key to generate a shared secret. This is
done by multiplying the recipient's public key with the sender's private key
using elliptic curve point multiplication. The resulting point on the elliptic
curve is then used as the encryption key to encrypt the message using a
symmetric encryption algorithm like AES. This process is known as Elliptic
Curve Diffie-Hellman (ECDH) key exchange.
Decryption: To decrypt the
message, the recipient uses their private key to compute the shared secret.
This is done by multiplying the encrypted point on the elliptic curve with the
recipient's private key using elliptic curve point multiplication. The
resulting point is then used as the decryption key to decrypt the message.
Digital Signatures: ECC can also
be used for digital signatures, which are used to authenticate the identity of
the sender and ensure the integrity of the message. To sign a message, the
sender uses their private key to compute a digital signature, which is a
mathematical function of the message. The recipient can then use the sender's
public key to verify the digital signature and ensure that the message has not
been tampered with. It store in crypto wallet.
The security of ECC is based on the elliptic curve discrete logarithm problem, which is believed to be hard to solve efficiently. This problem involves finding the private key from the public key, which is computationally difficult for large key sizes. Additionally, the use of elliptic curves over finite fields allows for smaller key sizes and faster operations than other public-key cryptography systems like RSA. Overall, ECC provides strong security and efficient performance, making it a popular choice for secure communication protocols and applications. The crypto trading in terra usd and top cryptocurrency.
BENEFIT OF ELLIPTICAL CRYPTOGRAPHY:
Smaller key sizes: ECC can
achieve the same level of security as other public-key cryptography systems,
such as RSA, with much smaller key sizes. For example, a 256-bit ECC key
provides the same level of security as a 3072-bit RSA key. This is because the
underlying mathematical problem in ECC, the elliptic curve discrete logarithm
problem, is harder to solve than the corresponding problem in RSA, the integer factorization
problem. This means that ECC can save storage space and reduce computational
costs, especially in resource-constrained environments like mobile devices or
smart cards.
Faster operations: ECC
operations like encryption, decryption, and key generation are faster than
those in RSA. This is because the mathematical operations involved in ECC are
simpler and more efficient than those in RSA. For example, the multiplication
of two points on an elliptic curve can be computed faster than the multiplication
of two large prime numbers in RSA. Voyager crypto is very fast operation.
Stronger security: ECC is very secure, and it is resistant to many common attacks like brute-force attacks and factorization attacks. This is because the security of ECC is based on the difficulty of solving the elliptic curve discrete logarithm problem, which is believed to be hard to solve efficiently. In contrast, RSA is vulnerable to attacks based on the ability to factor large composite numbers. It also provide strong security in reddit crypto. It also provide strong security in celsius crypto.
Flexibility: ECC can be
used in a variety of applications, including digital signatures, key agreement,
and encryption. It can also be used with different elliptic curves and finite
fields, which provides more flexibility in designing secure systems. Different
curves and finite fields can be chosen based on their security properties,
performance characteristics, and suitability for specific applications.