Interactive visualization of elliptic curves showing point addition, scalar multiplication, and ECDH key exchange.
Move from a curve equation to the group operations that make elliptic-curve cryptography possible. The visualizer shows chosen points, their sum, multiples of a base point, and the two sides of an ECDH exchange. It is best for tracing how coordinates change and why both ECDH participants reach the same shared result. The common trap is reading the picture as ordinary real-number geometry: for a finite-field curve, the points and operations are governed by modular arithmetic and include a special point at infinity.
The point-addition view identifies two curve points and the resulting sum. For distinct points, the usual construction follows the line through them; doubling uses the tangent at one point. The third intersection is reflected to obtain the sum. If the selected curve is over a finite field, the coordinate arithmetic is modular rather than ordinary screen geometry.
The scalar-multiplication view computes kP, meaning the group sum of P with itself k times, while showing intermediate points or the final point. In the ECDH view, private scalars a and b produce public points aG and bG. Each side multiplies the other public point; by associativity of the group law, a(bG) and b(aG) both equal (ab)G, so the two sides always reach the same point.
The point at infinity is the identity element of the elliptic-curve group, not a coordinate located somewhere beyond the plot. It appears, for example, when adding a point to its inverse. It can also arise during scalar multiplication according to the point's order. Seeing it is therefore a valid group result, although it may indicate an unsuitable input for a cryptographic protocol.
No. ECDH produces a shared elliptic-curve result; it does not by itself encrypt, authenticate, frame, or integrity-protect a message. Practical protocols derive symmetric key material from the shared secret and add authenticated encryption, validation, and protocol context. The visualization explains the agreement step, not a complete secure messaging design.
The behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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