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Exact Short Products From Truncated Multipliers

Daniel Lemire

TL;DR

This work addresses computing the $d$ most significant digits of a product when the multiplier is large or irrational and only its leading digits are accessible. It derives an exact-ness criterion for short multipliers: the discarded remainder must satisfy $( w z) \% M < M - w + 1$ to ensure exact MSDs for the range of interest. The authors develop a logarithmic-time algorithm that enumerates extrema of the remainders $(w z) \% M$ and extends it to offsets, enabling efficient computation of the valid $w$-range via a final find_range procedure. The methods are implemented in an open-source Python library, enabling precise, storage-efficient digit extraction with potential impact on parsing, numeric representations, and cryptography.

Abstract

We sometimes need to compute the most significant digits of the product of small integers with a multiplier requiring much storage: e.g., a large integer (e.g., $5^{100}$) or an irrational number ($π$). We only need to access the most significant digits of the multiplier-as long as the integers are sufficiently small. We provide an efficient algorithm to compute the range of integers given a truncated multiplier and a desired number of digits.

Exact Short Products From Truncated Multipliers

TL;DR

This work addresses computing the most significant digits of a product when the multiplier is large or irrational and only its leading digits are accessible. It derives an exact-ness criterion for short multipliers: the discarded remainder must satisfy to ensure exact MSDs for the range of interest. The authors develop a logarithmic-time algorithm that enumerates extrema of the remainders and extends it to offsets, enabling efficient computation of the valid -range via a final find_range procedure. The methods are implemented in an open-source Python library, enabling precise, storage-efficient digit extraction with potential impact on parsing, numeric representations, and cryptography.

Abstract

We sometimes need to compute the most significant digits of the product of small integers with a multiplier requiring much storage: e.g., a large integer (e.g., ) or an irrational number (). We only need to access the most significant digits of the multiplier-as long as the integers are sufficiently small. We provide an efficient algorithm to compute the range of integers given a truncated multiplier and a desired number of digits.
Paper Structure (13 sections, 3 theorems, 1 equation, 4 figures, 3 tables)

This paper contains 13 sections, 3 theorems, 1 equation, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Mathematical Preliminaries
  4. Plan
  5. Most Significant Digits
  6. Short Multipliers
  7. Finding the Valid Range
  8. Enumerating the Extrema of Remainders
  9. Bounding Remainders with an Offset
  10. Computing the Range
  11. Conclusion and Future Work
  12. Funding
  13. Data Availability

Key Result

lemma 1

Let $z$ be a truncated integer multiplier and $z'=z + \epsilon$ be the exact multiplier with $\epsilon \in [0,1)$. For integers $w$ and $M$, we have that $( w \times z) \div M = ( w \times z') \div M$ for all $\epsilon$ if and only if $(w\times z) \mathop{\mathrm{\mathbin{\%}}}\nolimits M < M - w +

Figures (4)

  • Figure 1: Illustration of how the next extrema is computed from the last minimum and the last maximum
  • Figure 2: Python code to compute all of the gaps between the extrema of $(w \times z)\mathop{\mathrm{\mathbin{\%}}}\nolimits M$ for $w=1,2,\ldots, \frac{M}{\gcd(z,M)} - 1$. $M$ and $z$ should be positive integers, and $z$ should not be a multiple of $M$.
  • Figure 3: Python code to enumerate all extrema of $(w\times z) \mathop{\mathrm{\mathbin{\%}}}\nolimits M$ for $w = A,\ldots, B$. $M$ and $z$ should be positive integers, and $z$ should not be a multiple of $M$.
  • Figure 4: Python code to compute the range of values of $w$ for which the most significant digits of $w\times z$ are exact even if $z$ is truncated. The function allows the user to specify the number of exact most-significant digits desired (digits) as well as the basis (base).

Theorems & Definitions (6)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof