Scalar multiplication
Scalar multiplication applies a scalar to a mathematical object according to the rules of its algebraic structure, such as scaling a vector or repeated addition of a group element.
Real-World Cryptography
Scalar multiplication is the operation of adding an elliptic curve point to itself a specified number of times.
- How it is used
- Deploys repeated point addition as the core operation for elliptic-curve Diffie–Hellman and related cryptographic computations.
- How deep it goes
- Defines repeated addition and notation, then discusses Montgomery’s ladder, timing resistance, and invalid-curve validation without implementation detail.
- Examples
- Uses P = [x]G, geometric point doubling, Montgomery’s ladder, and invalid-curve attack scenarios as concrete examples.
- What you can do after
- Evaluate/understand how elliptic-curve points become cryptographic outputs and why constant-time algorithms and public-key validation matter.
Read how Real-World Cryptography explains scalar multiplication →
Hacking Cryptography
Scalar multiplication is an operation that multiplies a mathematical point or vector by a scalar according to the structure's defining rules.
- How it is used
- Explains elliptic-curve point operations used to advance and output values from DUAL_EC_DRBG.
- How deep it goes
- Defines repeated curve addition and demonstrates package calls, while deferring the internal details of point addition.
- Examples
- Uses points P and Q, Go’s ScalarMult function, and equations for state advancement and truncated output.
- What you can do after
- Apply it via Go’s crypto/elliptic package to multiply curve points and integrate them into a generator.
Read how Hacking Cryptography explains scalar multiplication →
Related concepts
- Montgomery ladderimplemented by · The Montgomery ladder is a regular algorithm for scalar multiplication that performs a consistent sequence of additions and doublings to reduce dependence on secret scalar bits.
- Dual ECused by · Dual EC is an elliptic-curve-based deterministic pseudorandom number generator whose design permits state recovery if its generator points have a hidden mathematical relation.
- matrix factorizationused by · Matrix factorization is the decomposition or approximation of a matrix as a product of simpler, often lower-dimensional or structured matrices.
- Gaussian eliminationused by · Gaussian elimination is a procedure that applies elementary row operations to transform a linear system or matrix into an equivalent triangular or row-reduced form.
- scalaruses · A scalar is a single numerical quantity, as opposed to a vector or multidimensional collection of values.
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