Page 193 - Cyber Defense eMagazine Annual RSA Edition for 2024

Here’s a basic example of elliptic curve operations over a finite field F p where p is a prime number:

1. Point Addition: Given two points ( P ) and ( Q ) on the curve, the sum ( P + Q ) is the point ( R )
that lies on the line intersecting both ( P ) and ( Q ) but reflected over the x-axis.
2. Point Doubling: When ( P = Q ), the line tangent to the point on the curve will intersect the curve
at another point, which when reflected over the x-axis gives the result of ( P + P ) or ( 2P ).
3. Scalar Multiplication: This involves adding a point to itself repeatedly, which is computationally
intensive and forms the basis for the security of ECC. For example, computing ( kP ) means
adding ( P ) to itself ( k ) times.

Elliptic Curve Cryptography (ECC) is a type of cryptography that involves using the mathematics of elliptic
curves to secure data. To understand it from a beginner’s perspective, let’s break it down:

1. Cryptography Basics: At its core, cryptography is about securing communication so that only the
intended recipient can understand the message. Traditional methods use complex algorithms to
scramble data into unreadable formats that can only be deciphered with a specific key.
2. What Are Elliptic Curves?: Imagine drawing a smooth, symmetrical curve on a graph, similar to a
sideways “S” shape. This is a simplified view of an elliptic curve. It’s a type of mathematical curve
that has some fascinating properties, which make it useful for cryptography.
3. Why Use Elliptic Curves?: Elliptic curves are used in cryptography because they offer high
security with smaller key sizes. This means that to achieve the same level of security, ECC can
use a smaller key than other types of cryptography, like RSA. This results in faster computations
and less storage space needed, which is especially beneficial for devices with limited resources,
like smartphones.

What common cryptography algorithms implement ECC?

Several cryptographic algorithms leverage the properties of Elliptic Curve Cryptography (ECC) to provide
security for digital communications and data. Some of the most common ones include:

1. Elliptic Curve Digital Signature Algorithm (ECDSA): This is used for digital signatures, similar to
the way RSA is used but with the benefits of smaller key sizes and faster computation that ECC
provides. ECDSA is widely used in various security protocols and applications, including SSL/TLS
certificates and cryptocurrency wallets.
2. Elliptic Curve Diffie-Hellman (ECDH): An algorithm for key agreement that allows two parties,
each having an elliptic curve public-private key pair, to establish a shared secret over an insecure
channel. This shared secret can then be used to encrypt subsequent communications. ECDH is
used in many secure communication protocols, including HTTPS and VPNs.
3. Elliptic Curve Integrated Encryption Scheme (ECIES): This is a hybrid encryption scheme which
combines the benefits of ECC for key exchange with the performance of symmetric key
cryptography for encrypting data. ECIES is used in scenarios where both secure key exchange
and data encryption are required.

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