multiscalar

Multiscalar multiplication (MSM) is an operation in elliptic-curve cryptography that computes a linear combination of multiple scalar multiples of points: sum_i a_i P_i, where a_i are scalars and P_i are points on an elliptic curve. MSM is a foundational primitive in contexts where many related EC multiplications must be performed together, such as signature verification, zk-SNARKs, and aggregate signatures, because batching the calculations can significantly reduce the total number of curve operations compared with performing each multiplication independently.

Algorithmic approaches to MSM aim to reuse work across terms. Two principal families are commonly used. Straus-type methods process the scalars bit by bit using a bucketed accumulation, combining many partial results before the next bit is processed. Pippenger's algorithm groups terms into buckets and uses a tree-like accumulation to reduce the total number of elliptic-curve additions and doublings, offering performance gains as the number of terms grows. Variants may use fixed or varying bases, and may emphasize dense or sparse representations of the scalars to balance memory usage and speed. Practical implementations often combine windowing techniques (such as wNAF) with bucketed MSM to optimize both speed and side-channel resistance.

Applications and relevance include batch verification of digital signatures (for example, ECDSA or Schnorr) and other protocols where many public keys and scalars are involved simultaneously. MSM also plays a critical role in privacy-preserving technologies and blockchain systems that rely on efficient verification of large sets of cryptographic statements. The specific performance depends on the curve, the number of terms, the bit-length of the scalars, and available parallelism.