Addisjonkjede

Addisjonkjede, also known as the "addition chain," is a mathematical sequence used in cryptography and computer science to efficiently compute large exponents through a series of additions. The concept is particularly relevant in modular exponentiation, where calculating large powers modulo a number is a common requirement in algorithms like RSA encryption.

An addition chain for an integer *n* is a sequence of numbers starting with 1, where each subsequent number is the sum of two previous numbers in the sequence, and the last number in the sequence equals *n*. The goal is to minimize the total number of additions required to reach *n*. For example, the addition chain for 5 could be 1, 2 (1+1), 3 (1+2), 5 (2+3), which uses three additions.

Addition chains are crucial in optimizing performance in cryptographic operations, as they reduce the computational overhead of repeated squaring and multiplication. They are also used in hardware implementations of modular exponentiation, where minimizing the number of operations can significantly improve efficiency. The length of an addition chain for a given *n* is known as the "addition chain length," and finding the shortest possible chain for a number is a well-studied problem in mathematics.

Research in this area has led to the development of algorithms and heuristics for generating efficient addition chains. Some numbers have known optimal chains, while others remain unsolved, making them subjects of ongoing mathematical exploration. The study of addition chains intersects with number theory, combinatorics, and computer science, contributing to both theoretical insights and practical applications in secure computing.