On the History of the Square and Multiply Algorithm

2026-05-30

AI Summary

This study investigates the historical origins and formalization of the square-and-multiply algorithm, clarifying its cross-cultural evolution spanning over two millennia. Through rigorous historical documentation, textual analysis, and comparative historiography of mathematics, the paper systematically traces the algorithm’s development from its embryonic binary notions in Pingala’s Sanskrit prosody in ancient India, through refinements by Arabic scholars al-Uqlidisi and al-Biruni, to its first explicit articulation as a general-purpose algorithm by al-Kashi. The research affirms al-Kashi’s originality in formulating the algorithm in its universal form, elucidates the profound mathematical significance underlying Pingala’s early work, and fills a critical gap in scholarly understanding of the origins of this foundational computational method.

Abstract

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a technique for fast exponentiation commonly used in modern cryptography and computational number theory. Despite its prominence, the historical origins of the algorithm are not known with certainty. This paper critically examines the origins and formalization of the algorithm through primary source analysis. We focus on Jamshid al-Kashi's fifteenth-century Miftah al-Hisab where the algorithm is articulated explicitly as a general method and claimed by al-Kashi as his own innovation. To contextualize this, we trace earlier instances of successive squaring in the works of al-Uqlidisi and al-Biruni, who applied these techniques for specific calculations, but did not formalize them into a general procedure. The earliest known work on this method of computation is found in Pingala's prosodic studies in ancient India (c. 200 BCE). Even though it was not fully developed as a general technique, Pingala's work seems to contain the conceptual foundation of the algorithm which is to employ the binary representation of a positive integer. By mapping this intellectual progression, the paper illustrates the historical background of an algorithm that is prominent in modern computation.

Problem

• square-and-multiply algorithm
• binary exponentiation
• historical origins
• computational number theory
• algorithm history

Innovation

• square-and-multiply algorithm
• binary exponentiation
• history of algorithms
• al-Kashi
• Pingala