research-article

On the Evaluation of Powers and Monomials

Author:

Nicholas PippengerAuthors Info & Claims

SIAM Journal on Computing, Volume 9, Issue 2

Pages 230 - 250

https://doi.org/10.1137/0209022

Published: 01 May 1980 Publication History

Abstract

Let $y_1, \cdots,y_p $ be monomials over the indeterminates $x_1, \cdots,x_q $. For every $y = (y_1, \cdots,y_p )$ there is some minimum number $L(y)$ of multiplications sufficient to compute $y_1, \cdots,y_p $ from $x_1, \cdots,x_q $ and the identity 1. Let $L(p,q,N)$ denote the maximum of $L(y)$ over all y for which the exponent of any indeterminate in any monomial is at most N. We show that if $p = (N + 1^{o(q)} )$ and $q = (N + 1^{o(p)} )$, then $L(p,q,N) = \min \{ p,q\} \log N + H/\log H + o(H /\log H)$, where $H = pq\log (N + 1)$ and all logarithms have base 2.

References

[1]

R. E. Bellman, Addition chains of vectors, Amer. Math. Monthly, 70 (1963), 765–

Google Scholar

[2]

Alfred Brauer, On addition chains, Bull. Amer. Math. Soc., 45 (1939), 736–739

Crossref

Google Scholar

[3]

P. Erdös, Remarks on number theory. III. On addition chains, Acta Arith., 6 (1960), 77–81

Crossref

Google Scholar

[4]

Donald E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969xi+624

Google Scholar

[5]

Nicholas Pippenger, On the evaluation of powers and related problems (preliminary version), 17th Annual Symposium on Foundations of Computer Science (Houston, Tex., 1976), IEEE Comput. Soc., Long Beach, Calif., 1976, 258–263, 25-27 Oct.

Google Scholar

[6]

Nicholas Pippenger, The minimum number of edges in graphs with prescribed paths, Math. Systems Theory, 12 (1979), 325–346

Crossref

Google Scholar

[7]

A. Scholz, Jahresbericht der Deutschen Mathematiker-Vereinigung (II), 47 (1937), 41–42

Google Scholar

[8]

E. G. Straus, Addition chains of vectors, Amer. Math. Monthly, 71 (1964), 806–808

Google Scholar

[9]

A. C.-C. Yao, On the evaluation of powers, SIAM J. Comput., 5 (1976), 100–103

Digital Library

Google Scholar

Cited By

View all

Zhu YWu YLuo ZOoi BXiao X(2024)Secure and Verifiable Data Collaboration with Low-Cost Zero-Knowledge ProofsProceedings of the VLDB Endowment10.14778/3665844.366586017:9(2321-2334)Online publication date: 6-Aug-2024
https://doi.org/10.14778/3665844.3665860
Regev O(2024)An Efficient Quantum Factoring AlgorithmJournal of the ACM10.1145/370847172:1(1-13)Online publication date: 12-Dec-2024
https://dl.acm.org/doi/10.1145/3708471
Ray ADevlin BQuah FYesantharao RZhang ZPutnam A(2024)Hardcaml MSM: A High-Performance Split CPU-FPGA Multi-Scalar Multiplication EngineProceedings of the 2024 ACM/SIGDA International Symposium on Field Programmable Gate Arrays10.1145/3626202.3637577(33-39)Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1145/3626202.3637577
Show More Cited By

Index Terms

On the Evaluation of Powers and Monomials

Index terms have been assigned to the content through auto-classification.

Recommendations

Detecting monomials with k distinct variables

We study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We ...
On the Evaluation of Powers

It is shown that for any set of positive integers $\{ n_1 ,n_2 , \cdots ,n_p \} $, there exists a procedure which computes $\{ x^{n_1 } ,x^{n_2 } , \cdots ,x^{n_p } \} $ for any input x in less than $\lg N + c\sum_{i = 1}^P [\lg n_i /\lg \lg (n_i + 2)]$ ...
Sums of Two Exact Powers

An asymptotic formula is obtained for the number of representations of an element of a finite field as a weighted sum of two prescribed powers of primitive elements. This generalises previous work on sums of primitive elements, including that relating ...

Comments

Information & Contributors

Information

Published In

SIAM Journal on Computing Volume 9, Issue 2

May 1980

221 pages

ISSN:0097-5397

DOI:10.1137/smjcat.1980.9.issue-2

Issue’s Table of Contents

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 May 1980

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

24
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 29 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

View all

Zhu YWu YLuo ZOoi BXiao X(2024)Secure and Verifiable Data Collaboration with Low-Cost Zero-Knowledge ProofsProceedings of the VLDB Endowment10.14778/3665844.366586017:9(2321-2334)Online publication date: 6-Aug-2024
https://doi.org/10.14778/3665844.3665860
Regev O(2024)An Efficient Quantum Factoring AlgorithmJournal of the ACM10.1145/370847172:1(1-13)Online publication date: 12-Dec-2024
https://dl.acm.org/doi/10.1145/3708471
Ray ADevlin BQuah FYesantharao RZhang ZPutnam A(2024)Hardcaml MSM: A High-Performance Split CPU-FPGA Multi-Scalar Multiplication EngineProceedings of the 2024 ACM/SIGDA International Symposium on Field Programmable Gate Arrays10.1145/3626202.3637577(33-39)Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1145/3626202.3637577
Deng JWu B(2024)A Practical Data Trading Protocol for Sudoku SolutionsIEEE Transactions on Information Forensics and Security10.1109/TIFS.2024.341970219(6935-6948)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1109/TIFS.2024.3419702
Masson SSanso AZhang Z(2024)Bandersnatch: a fast elliptic curve built over the BLS12-381 scalar fieldDesigns, Codes and Cryptography10.1007/s10623-024-01472-092:12(4131-4143)Online publication date: 1-Dec-2024
https://dl.acm.org/doi/10.1007/s10623-024-01472-0
Abdolmaleki BFauzi PKrips TSiim J(2024)Shuffle Arguments Based on Subset-CheckingSecurity and Cryptography for Networks10.1007/978-3-031-71070-4_16(345-366)Online publication date: 11-Sep-2024
https://dl.acm.org/doi/10.1007/978-3-031-71070-4_16
Nguyen WDatta TChen BTyagi NBoneh D(2024)Mangrove: A Scalable Framework for Folding-Based SNARKsAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68403-6_10(308-344)Online publication date: 18-Aug-2024
https://dl.acm.org/doi/10.1007/978-3-031-68403-6_10
Garg SKolonelos DPolicharla GWang M(2024)Threshold Encryption with Silent SetupAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68394-7_12(352-386)Online publication date: 18-Aug-2024
https://dl.acm.org/doi/10.1007/978-3-031-68394-7_12
Hhan MYamakawa TYun A(2024)Quantum Complexity for Discrete Logarithms and Related ProblemsAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68391-6_1(3-36)Online publication date: 18-Aug-2024
https://dl.acm.org/doi/10.1007/978-3-031-68391-6_1
Bell JGascón ALepoint TLi BMeiklejohn SRaykova MYun CCalandrino JTroncoso C(2023)ACORNProceedings of the 32nd USENIX Conference on Security Symposium10.5555/3620237.3620506(4805-4822)Online publication date: 9-Aug-2023
https://dl.acm.org/doi/10.5555/3620237.3620506
Show More Cited By

Abstract

References

Cited By

Index Terms

Recommendations

Detecting monomials with k distinct variables