Elliptic Curve Cryptography (ECC) is a public-key cryptography system that relies on the algebraic structure of elliptic curves over finite fields. The curve is defined by an equation such as:
where ( a ) and ( b ) are coefficients, and ( p ) is a large prime number defining the finite field. The set of all points ( (x, y) ) satisfying the equation, along with a special “point at infinity,” forms a group under defined addition operations.
Group Operations on an Elliptic Curve
The addition of two points ( P ) and ( Q ) involves drawing a line through ( P ) and ( Q ) to find the intersection with the curve. Reflecting this intersection across the x-axis yields the resulting point ( P + Q ). Doubling a point ( P ) involves taking the tangent at ( P ), finding its intersection with the curve, and reflecting it. Scalar multiplication, used extensively in ECC, is the operation of adding a point to itself multiple times. For instance, ( nP ) means adding ( P ) to itself ( n ) times.
These operations are computationally efficient but mathematically hard to invert, making ECC secure. For example, given ( P ) and ( Q = nP ), determining ( n ) is infeasible due to the “elliptic curve discrete logarithm problem.”
ECC vs. RSA
ECC provides equivalent security to RSA with much smaller key sizes. A 256-bit ECC key offers the same level of security as a 3072-bit RSA key. This efficiency reduces computational overhead, memory usage, and power consumption, making ECC particularly suitable for constrained environments like IoT devices.
Curve 25519 Cryptography
Curve25519 is a widely-used elliptic curve optimized for cryptographic performance and security. It is defined by:
where ( p = 2^{255} - 19 ). Curve25519 supports high-speed cryptographic operations and mitigates side-channel attacks through careful parameter selection.
x25519 Key Exchange
x25519 implements the Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol. It allows two parties to securely compute a shared secret over an untrusted channel.
- Each party generates a private scalar ( a ) or ( b ), which is a large random number.
- The private scalar is multiplied with the curve’s base point ( G ), producing the public key ( A = a ⋅ G ) or ( B = b ⋅ G ).
- These public keys are exchanged between the parties.
- To compute the shared secret, each party multiplies their private scalar with the received public key:
This shared secret is identical for both parties due to the commutative property of scalar multiplication. The shared secret is then passed through a key derivation function (KDF) to produce encryption keys.
ed25519 Digital Signatures
ed25519 is a digital signature algorithm based on Curve25519, providing fast and secure signatures.
Key Generation
The private key is a random scalar ( a ). The public key is derived as ( A = a ⋅ G ), where ( G ) is the curve’s base point.
Signature Generation
To sign a message ( M ), the private key and ( M ) are hashed to generate a deterministic nonce ( r ). The ephemeral public key is computed as ( R = r ⋅ G ). The signature ( (R, S) ) is formed, where:
Here, l is the curve’s order.
Signature Verification
The verifier checks the equation:
If this holds, the signature is valid. This process ensures authenticity and integrity of the message.
Applications of Curve25519
Curve25519 is widely used in secure communication protocols such as TLS 1.3, Signal, and SSH. x25519 ensures efficient and secure key exchanges, while ed25519 is favored for compact and reliable digital signatures, particularly in environments demanding both high performance and strong security guarantees.