Addition Chains
Bibliography
- ■ H. Dellac
"Question 49",
L'Intermédiaire des Mathématiciens, V. 1
1894 p. 20.
- ■ E. de Jonquiéres
"Question 49 (H. Dellac)",
L'Intermédiaire des Mathématiciens V. 1
1894 20, pp 162-164.
- ■ A. Goulard
"Question 393",
L'Intermédiaire des Mathématiciens V. 1
1894 p. 234.
- ■ E. de Jonquiéres
"Question 393 (A. Goulard)",
L'Intermédiaire des Mathématiciens V. 2
1895 pp 125-126.
- ■ B. Datta and A.N. Singh
History of Hindu Mathematics V. 1
Bombay 1935, p. 76.
- ■ Arnold Scholz
"Aufgabe 253",
Jahresbericht der deutschen Mathematiker-Vereinigung V. 47, 2. Abteil.
1937 pp 41-42.
- ■ Alfred T. Brauer
"On Addition Chains",
Bulletin of the American Mathematical Society V. 45(10)
1939 pp 736-739.
- ■ W.R. Utz
"A Note on the Scholz-Brauer Problem in Addition Chains",
Proceedings of the American Mathematical Society, V. 4(3)
1953 pp 462-463.
- ■ Samuel M. Hendley
"The Theory and Construction of Addition Chains",
Masters thesis, University of South Carolina, 1954.
- ■ Walter Hansen
"Untersuchungen über die Scholz-Brauerschen Additionsketten und deren Verallgemeinerung",
Göttingen, Wiss. Prüfungsamt, Schriftl. Hausarbeit zum Staatsexamen 1957.
- ■ Walter Hansen
"Zum Scholz-Brauerschen Problem",
Journal für die reine und angewandte Mathematik, V. 202
1959 pp 129-136.
- ■ P.E. Valskii
"On the Lower Bounds of Multiplications in Evaluation of Powers",
Problems of Cybernetics, V. 2
1959 pp 73-74.
- ■ Pál Erdös
"Remarks on Number Theory. III: On Addition Chains",
Acta Arithmetica V. 6(1)
1960 pp 77-81.
- ■ P.E. Valskii,
"The Smallest Number of Multiplications Necessary to Raise a Number to a Given Power",
Problems of Cybernetics V. 2
1961 pp 395-397.
- ■ A.A. Gioia, M.V. Subbarao and M. Sugunamma
"The Scholz-Brauer Problem in Addition Chains",
Duke Mathematical Journal, V. 29
1962 pp 481-487.
- ■ Richard E. Bellman
"Advanced Problem 5125",
Amer. Math. Monthly, V. 70
1963 p. 765.
- ■ Donald E. Knuth
"Addition Chains and the Evaluation of n-th Powers",
Notices of the American Mathematical Society, V. 11
1964 pp 230-231.
- ■ E.G. Strauss
"Solution to Problem 5125",
Am. Math. Monthly, V. 71
1964 pp 806-808.
- ■ C.T. Whyburn
"A Note on Addition Chains",
Proceedings of the Am. Math. Society V. 16(5)
1965 p. 1134.
- ■ A.M. Il'in
"On Additive Number Chains",
Problemy Kibernet, V. 13
1965 pp 245-248.
- ■ Charles A. Demetriou
"Concerning minimal addition chains",
Thesis (Master Science),
University of Mississippi, 1967.
- ■ Evert Wattel, G.A. Jensen
"Efficient calculation of powers in a semigroup",
Stichting Mathematisch Centrum. Zuivere Wiskunde,
Report ZW1968-001 Mathematical Centre Amsterdam, V. 1 pp 1-18.
- ■ Kenneth B. Stolarsky
"A lower bound for the Scholz-Brauer problem",
Canadian Journal of Mathematics, V. 21
1969 pp 675-683.
- ■ Donald E. Knuth
The Art of Computer Programming V. 2
Seminumerical Algorithms, 1st edition,
Addison-Wesly 1969 pp 398-422.
- ■ Donald E. Knuth
"Calculations on Addition chains",
Stanford University, California 1969.
- ■ H. Kato
"On Addition Chains",
Ph.D. thesis,
University of Southern California, 1970.
- ■ Edward G. Thurber
"The Scholz-Brauer Problem for Addition Chains",
Ph.D. thesis,
University of Southern California, 1971.
- ■ Edward G. Thurber
"The Scholz-Brauer Problem on Addition Chains",
Notices of the American Mathematical Society,
Abstract 71T-A274, V. 18
1971 p. 1100.
- ■ P. Gaal
"Additionsketten in einer endlich erzeugten abelschen Gruppe".
Diploma thesis,
Universität Zürich, 1971.
- ■ Robert P. Giese
"A Sequence of Counterexamples of Knuth's Modification of Utz's Conjecture",
Notices of the American Mathematical Society, Abstract 72T-A257, V. 19
1972 p. A-688.
- ■ A. Cottrell
"A Lower Bound for the Scholz-Brauer Problem",
Notices of the American Mathematical Society, V. 20(5)
1973 p. A-476.
- ■ Edward G. Thurber
"The Scholz-Brauer Problem on Addition-Chains",
Pacific Journal of Mathematics, V. 49(1)
1973 pp 229-242.
- ■ Edward G. Thurber
"On addition chains ℓ(mn) <= ℓ(n)-b And lower bounds for c(r)",
Duke Mathematical Journal, V. 40(4)
1973 pp 907-913.
- ■ Edward G. Thurber
"The Scholz-Brauer Problem on Addition Chains",
Notices of the American Mathematical Society,
Abstract 73T-A112, V. 20
1973 p. A-318.
- ■ A. Cottrell
"A Lower Bound for the Scholz-Brauer Problem",
Ph.D. thesis,
University of California, Berkeley, 1974.
- ■ Robert P. Giese
"Selected Topics in Addition Chains",
Ph.D. thesis,
University of Houston, 1974.
- ■ Kevin R. Hebb
"Some Problems on Addition Chains",
Master thesis,
University of Alberta, 1974.
- ■ Kevin R. Hebb
"Some results on addition chains",
Notices of the American Mathematical Society, V. 21(2)
1974 p. A-294.
- ■ Thomas H. Southard
"Addition chains for the first n squares",
Report CNA-84, Center for Numerical Analysis, University of Texas at Austin,
May 1974.
- ■ Thomas H. Southard
"Addition chains for the first n triangular numbers",
Report CNA-85, Center for Numerical Analysis, University of Texas at Austin,
May 1974.
- ■ Emanuel Vegh
"A Note on Addition Chains",
Journal of Combinatorial Theory, Ser. A, V. 19(1)
1975 pp 117-118.
- ■ Emanuel Vegh
"Addition Chains",
Notices of the American Mathematical Society, V. 22(1)
1975 pp A2-A3.
- ■ Arnold Schönhage
"A Lower Bound for the Length of Addition Chains",
Theoretical Computer Science, V. 1(1)
1975 pp 1-12.
- ■ A.A. Gioia, M.V. Subbarao
"The Scholz-Brauer Problem in Addition Chains",
Notices of the American Mathematical Society V. 22
1975 pp A63-A64.
- ■ A.S. Saidan
The Arithmetic of al-Uqlîdisî,
Dordrecht 1975 pp 341-342.
- ■ Nicholas Pippenger
"On the Evaluation of Powers and Related Problems",
Proceedings, 17th Anual IEEE Symposium on the foundations of computer science,
Houston TX. 1976 pp 258-263.
- ■ Edward G. Thurber
"Addition chains and Solutions of ℓ(2n)=ℓ(n) and ℓ(2^n-1)=n+ℓ(n)-1",
Discrete Mathematics, V. 16(3)
1976 pp 279-289.
- ■ Andrew Chi-Chih Yao
"On the Evaluation of Powers",
SIAM Journal of Computing V. 5(1)
1976 pp 100-103.
- ■ Edward G. Belaga
"The additive complexity of a natural number",
Soviet Mathematics Doklady V. 17(1)
1976 pp 5-9.
- ■ Jan van Leeuwen
"An extension of Hansen's theorem for star chains",
J. Reine Angew. Math., V. 295
1977 pp 203-207.
- ■ Daniel P. McCarthy
"The optimal algorithm to evaluate x^n using elementary multiplication methods",
Mathematics of Computation V. 31(137)
1977 pp 251-256.
- ■ Rolf Sonntag
"Theorie der Additionsketten",
Diplomarbeit Technische Universität Hannover, 1978.
- ■ Rolf Sonntag
"On Addition Chains",
Unpublished manuscript 1978.
- ■ A.A. Gioia, M.V. Subbarao
"The Scholz-Brauer Problem in Addition Chains II",
Proc. Eighth Manitoba Conference on Numerical Math. and Computing, XXII V. 22
1978 pp 251-274.
- ■ R.L. Graham and A.C.-C. Yao and F.-F. Yao
"Addition chains with Multiplicative Cost",
Discrete Mathematics, V. 23
1978 pp 115-119.
- ■ David Dobkin, Richard J. Lipton
"Additive Chain Methods for the Evaluation of Specific Polynomials",
Proceedings of a Conference on Theoretical Computer Science (Univ.Waterloo, Waterloo, Ont., 1977),
Comput. Sci. Dept., Univ. Waterloo, Waterloo, Ont., V. 148
1978 pp 146-148.
- ■ Donald E. Knuth
The Art of Computer Programming V. 2
Seminumerical Algorithms, 2nd edition
Addison-Wesly 1981 pp 441-466.
- ■ David Dobkin and Richard J. Lipton
"Addition Chain Methods for the Evaluation of Specific Polynomials",
SIAM Journal on Computing V. 9(1)
1980 pp 121-125.
- ■ Nicholas Pippenger
"On the Evaluation of Powers and Monomials",
SIAM Journal on Computing V. 9(2)
1980 pp 230-250.
- ■ Peter Downey, Benton Leony and Ravi Sethi
"Computing Sequences with Addition Chains",
SIAM Journal of Computing, V. 10(3)
1981 pp 638-646.
- ■ Jorge Olivos
"On Vectorial Addition Chains",
Journal of Algorithms, V. 2(1)
URL https://dblp.uni-trier.de/db/journals/jal/jal2.html#Olivos81
1981 pp 13-21.
- ■ Donald E. Knuth, Christos H. Papadimitriou
"Duality in addition chains",
Bulletin of the European Association for Theoretical Computer Science, V. 13
1981 pp 2-4.
- ■ I. Semba
"Systematic Method for Determining the Number of Multiplications Required to Compute x^m, where m is a Positive Integer",
Journal of Information Processing V. 6(1)
1983 pp 31-33.
- ■ Y.H. Tsai
"A study on some addition chain problems",
Master thesis, National Tsing-Hua University, Hsinchu, Taiwan, 1985.
- ■ Y. H. Chin and Y. H. Tsai
"Algorithms for Finding the Shortest Addition Chain",
Proceedings of National Computer Symposium
Kaoshiung, Taiwan, December 20-22, 1985 pp 1398-1414.
- ■ Hugo Volger
"Some Results on Addition/Subtraction Chains",
Information Processing Letters V. 20(3)
1985 pp 155-160.
- ■ Daniel P. McCarthy
"Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains",
Mathematics of Computation, V. 46(174)
1986 pp 603-608.
- ■ Thomas Lickteig and Hugo Volger
"Some Results on the Complexity of Powers",
Computation Theory and Logic , Editor Egon Börger
Springer-Verlag 1987 pp 249-255.
- ■ Y. H. Tsai, Y. H. Chin
"A study of some addition chain problems",
International Journal of Computer Mathematics, V. 22(2)
1987 pp 117-134.
- ■ W. Orb
"Additionsketten",
Diploma thesis, Mainz, University, 1987.
- ■ M. V. Subbarao
"Addition Chains --- Some Results and Problems",
Number Theory and Applications, Editor R. A. Mollin,
NATO Advanced Science Institutes Series: Series C, V. 265
Kluwer Academic Publisher Group 1989 pp 555-574.
- ■ François Bergeron, Jorge Olivos
"Vectorial addition chains using Euclid's algorithm",
Research Report, Dept. Math., UQAM 105 (1989).
- ■ François Bergeron, Jean Berstel, Srećko Brlek, and Christine Duboc
"Addition chains using continued fractions",
Journal of Algorithms V. 10(3)
Sep. 1989 pp 403-412.
- ■ François Bergeron, Jean Berstel, Srećko Brlek
"A unifying Approach to the generation of Addition Chains",
Proceedings of the XV Latino-American Conference on Informatics,
1989 pp 10-14.
- ■ Jurjen Bos, Matthijs J. Coster
"Addition Chain Heuristics",
Advances in Cryptolog, CRYPTO' 89
Springer-Verlag 1989 pp 400-407.
- ■ M. Elias and F. Neri
"A Note on Addition Chains and Some Related Conjectures",
Sequences, Editor R. M. Capocelli,
Springer-Verlag 1990 pp 166-181.
- ■ Yacov Yacobi
"Exponentiating faster with addition chains",
Advances in Cryptology
EUROCRYPT '90, Springer-Verlag 1990 pp 222-229.
- ■ François Morain, Jorge Olivos
"Speeding up the computations on an elliptic curve using addition-subtraction chains",
R.A.I.R.O. Informatique Théorique et Applications V. 24(6)
1990 pp 531-544.
- ■ Matthijs J. Coster
"Some algorithms on Addition Chains and their Complexity",
Tech. Report CS-R9024, URL https://pure.uvt.nl/ws/portalfiles/portal/5212990/CM5613158.pdf
Centrum voor Wiskunde en Informatica, Amsterdam, 1990 pp 1-69.
- ■ Achim Flammenkamp
"Drei Beiträge zur diskreten Mathematik: Additionsketten, No-Three-in-Line-Problem, Sociable Numbers",
Diplomarbeit an der Fakultät für Mathematik, Universität Bielefeld 1991.
- ■ Hans Zantema
"Minimizing Sums of Addition Chains",
Journal of Algorithms, V. 12(2)
1991 pp 281-307.
- ■ Srećko Brlek, Pierre Castéran, Robert Strandh
"On Addition Schemes",
International Joint Conference on Theory and Practice of Software Development, V. 2
Springer-Verlag 1991 pp 379-393.
- ■ Srećko Brlek, Pierre Castéran, Robert Strandh
"Chaînes d'addition et structures de contrôle",
Journées francophones des langages applicatifs (28-29 Janvier 1991, Gresse-en-Vercors, France) editor C. Queinnec, Bigre 72
1991 pp 54-63.
- ■ Y.H. Chin, Y.H. Tsai
"On Addition Chains",
International Journal Computer Mathematics V. 45(3-4)
1992 pp 145-160.
- ■ Y. H. Tsai and Y. H. Chin
"A Study of Addition Chains",
Proceedings of the National Science Council - Part A: Physical Science and Engineering,
V. 16(6)
Taipei, Taiwan, 1992 pp 506-514.
- ■ Jurjen Norbert E. Bos
"Practical Privacy",
Ph.D. thesis, Eindhoven, University of Technology,
1992, ISBN 9061964059.
- ■ E.F. Brickell, D.M. Gordon, K.S. McCurley, D.B. Wilson
"Fast Exponentiation with Precomputation (Extended Abstract)",
EUROCRYPT 1992 pp 200-207.
- ■ Omprakash K. Gupta
"On Evaluation of Powers",
Journal of Information & Optimization Sciences, V. 13(2)
TARU Publications, India, 1992 pp 331-336.
- ■ Srećko Brlek, Richard Mallette
"Sur le calcul des chaînes d'additions optimales",
dans J. Labelle et J.G. Penaud, éditeurs,
Atelier de combinatoire franco-québecois (6-7 Mai 1991, Bordeaux, France)
Publ. du LACIM 10, 1992 ISBN 2-89276-101-8 pp 71-85.
- ■ Jörg Sauerbrey and Andreas Dietel
"Resource requirements for the application of addition chains in modulo exponentiation",
Workshop on the Theory and Application of of Cryptographic Techniques
Springer-Verlag 1992 pp 174-182.
- ■ Edward G. Thurber
"Addition chains --- an eratic sequence",
Discrete Mathematics, V. 122(1)
1993 pp 287-305.
- ■ Walter Aiello and M. V. Subbarao
"A conjecture in addition chains related to Scholz's conjecture",
Mathematics of Computation, V. 61(203)
1993 pp 17-23.
- ■ Matthijs J. Coster and others
"An Algorithm on Addition Chains with Restricted Memory",
Research memorandum 603
URL https://pure.uvt.nl/ws/portalfiles/portal/5212990/CM5613158.pdf
Tilburg University, Department of Economics, 1993.
- ■ Abdrew G. Dempster, Malcom D. Macleod
"Constant integer multiplication using minimum adders",
IEE Proceedings - Circuits, Devices & Systems, V. 141(5)
1994 pp 407-413.
- ■ François Bergeron, Jean Berstel and Srećko Brlek
"Efficient Computation of Addition Chains",
Journal de Theorie des Nombres de Bordeaux, V. 6(1)
1994 pp 21-38.
- ■ Yuh-Jiun Chen, Chin-Chen Chang and Wei-Pang Yang
"Some Properties of Vectorial Addition Chains",
International Journal of Computer Mathematics, V. 54(3-4)
1994 pp 185-196.
- ■ Y.J. Chen
"Some Properties on Vectorial Addition Chains",
Ph.D. dissertation, Institute of Computer Science and Information Engineering, National Chiao Tung University,
Hsinchu, Taiwan, R.O.C., December 1994.
- ■ Peter J.N de Rooij
"Efficient exponentiation using precomputation and vector addition chains",
Workshop on the Theory and Application of of Cryptographic Techniques
Proceedings EuroCrypt '94, Springer-Verlag 1994 pp 389-399.
- ■ Sergey B. Gashkov and Vadim V. Kochergin
"On addition chains of vectors, gate circuits and the complexity of computations of powers",
Siberian Advances in Mathematics V. 4(4)
1994 pp 1-16.
- ■ Vadim V. Kochergin
"On the computation of powers sets",
Discrete Mathematics and Applications V. 4(2)
1994 pp 119-128.
- ■ O. Kochergin
"The Evaluation of Powers",
Mathematical Problems of Cybernetics, V. 26
1995 pp 76-82.
- ■ Y.J. Chen, C.C Chang, and W.P. Yang
"The Shortest Weighted Length Addition Chains",
Journal of Information Science and Engineering, V. 11(2)
1995 pp 295-305.
- ■ Srećko Brlek, Pierre Castéran, Laurent Habsieger, Richard Mallette
"On-line evaluation of powers using Euclid's algorithm",
R.A.I.R.O. Informatique Théorique et Applications V. 29(5)
1995 pp 431-450.
- ■ I.E. Bocharova, B.D. Kudryashov
"Fast exponentiation in cryptography",
Proceeding of AAECC-11,
Lecture Notes in Computer Science, V. 948
Springer Verlag 1995 pp 146-157.
- ■ Vassil Dimitrov and Todor Cooklev
"Two algorithms for modular exponentiation using nonstandard arithmetics",
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences V. 78(1)
The Institute of Electronics, Information and Communication Engineers
1995 pp 82-87.
- ■ Çetin K. Koc
"Analysis of sliding window techniques for exponentiation",
Computers & Mathematics with Applications V. 30(10)
1995 pp 17-24.
- ■ Youfend Wu
"Strength reduction of multiplications by integer constants",
ACM SIGPLAN Notices V. 30(2)
1995 pp 42-48.
- ■ C.Y. Chen, C.C. Chang, and W.P. Yang
"Hybrid Method for Modular Exponentiation with Precomputation",
Electronics Letters, V. 32(6)
1996 pp 540-541.
- ■ D.C. Lou, and C.C. Chang
"Fast Exponentiation Method Obtained by Folding the Exponent in Half",
Electronics Letters, V. 32(11)
1996 pp 984-985.
- ■ Daniel Bleichenbacher
"Efficiency and Security of Cryptosystems based on Number Theory",
Dissertation to Swiss Federal Institute of Technology Zürich, ETH Zürich 1996.
- ■ Vadim V. Kochergin
"On the complexity of computation of monomials and tuples of powers",
Siberian Advances in Math V. 6(1)
1996 pp 71-86.
- ■ D. Bleichenbacher and A. Flammenkamp
"An Efficient Algorithm for Computing Shortest Addition Chains",
submitted 1997 to SIAM Journal of Discrete Mathematics
The same paper as gziped Postscript included all figures.
- ■ Noboru Kunihiro and Hirosuke Yamamoto
"Optimal addition chain classified by Hamming weight",
IEICE Technical Report, ISEC96-74, 1997.
- ■ Hatem M. Bahig,
"On the algorithms of computational number theory and their applications",
Master thesis, Dept. of Math., Faculty of Science, Ain Shams University, Cairo, Egypt 1998.
- ■ Gérard D. Cohen, Antoine Lobstein, David Naccache, Gilles Zémor
"How to Improve an Exponentiation Black-Box",
EUROCRYPT Springer-Verlag 1998 pp 211-220.
- ■ Daniel M. Gordon
"A survey of fast exponentiation methods",
Journal of Algorithms V. 27(1)
1998 pp 129-146.
- ■ Donald E. Knuth
"The Art of Computer Programming", V. 2
Seminumerical Algorithms, 3rd edition,
Addision Wesley, Reading, MA 1998 pp 461-485.
- ■ Noboru Kunihiro and Hirosuke Yamamoto
"Windows and Extended Window Methods for Addition Chain and Addition-Subtraction Chain",
IEICE Trans. on Fundamentals of Electronics, Communications and C.S. , V. 81(1) E81-A
1998 pp 72-81.
- ■ Colin D. Walter
"Exponentiation Using Division Chains",
IEEE Trans. on Computers V. 47(7)
1998 pp 757-765.
- ■ Der-Chyuan Lou, Chin-Chen Chang
"An adaptive exponentiation method",
Journal of Systems and Software V. 42(1)
1998 pp 59-69.
- ■ Yacov Yacobi
"Fast Exponentiation Using Data Compression",
SIAM J. Comput. V. 28(2)
1998 pp 700-703.
- ■ Richard K. Guy
"Unsolved problems in number theory",
V. 1, 2nd editonSec. C6
Springer Science & Business Media 1998 pp 111-113.
- ■ Achim Flammenkamp
"Integers with a Small Number of Minimal Addition Chains",
Discrete Mathematics V. 205(1)
1999 pp 221-227.
- ■ Edward G. Thurber
"Efficient Generation of Minimal Length Addition Chains",
SIAM Journal of Computing, V. 28(4)
1999 pp 1247-1263.
- ■ Julia Abrahams
"Addition Chains as Test Trees and a Sequential Variant as the Huffman Problem",
DIMACS Technical Report 99-15, February 1999.
- ■ Daniel Bleichenbacher
"Addition chains for large sets",
Unpublished manuscript 1999.
- ■ Joachim von Zur Gathen, Michael Nöcker
"Computing special powers in finite fields",
In Proceedings of the 1999 international Symposium on Symbolic and Algebraic Computation
(Vancouver, British Columbia, Canada, July 28 - 31, 1999). S. Dooley, Ed. ISSAC.
ACM Press,New York, pp 83-90.
- ■ H. Park, K. Park and Y. Cho
"Analysis of the variable length nonzero window method for exponentiation",
Computers & Mathematics with Applications V. 37(7)
1999 pp 21-29.
- ■ Noboru Kunihiro and Hirosuke Yamamoto
"New Methods for Generating Short Addition Chains",
IEICE Transactions on Fundamentals of Electronics, Comm., E83-A, V. 83(1)
2000 pp 60-67.
- ■ Matthijs J. Coster
"The Brun algorithm for Addition Chains",
Unpublished manuscript 2000.
- ■ Matthijs J. Coster
"The Algorithm of Brun and Addition Chains",
Unpublished manuscript 2000.
- ■ Vassil S. Dimitrov, Graham A. Jullien, William C. Miller
"Complexity and Fast Algorithms for Multiexponentiations",
IEEE Trans. on Computers V. 49(2)
2000, pp 141-147.
- ■ Xiao-dong Wang
"An algorithm for generating the shortest addition chains",
Mini-Micro Systems V. 22(10)
Gai-Kan Biajibu, Shenyang, China, 2001 pp 1250-1253.
- ■ Martin Otto
"Brauer Addition-Subtraction Chains",
Diplomarbeit, Universität-Gesamthochschule Paderborn, 2001.
- ■ Hatem M. Bahig, Mohamed H. El-Zahar, Ken Nakamula
"Some Results for Some Conjectures in Addition Chains",
Combinatorics, Computability and Logic Proceedings of the 3rd international conference vol DMTCS '01
Springer-Verlag 2001 pp 47-54.
- ■ C. Clavier and M. Joye
"Universal Exponentiation Algorithm",
In Proceedings of the Third international Workshop on Cryptographic Hardware and Embedded Systems (May 14 - 16, 2001).
Lecture Notes In Computer Science, V. 2162
London, 2001 pp 300-308.
- ■ Hatem M. Bahig, Ken Nakamula
"Some Properties of Nonstar Steps in Addition Chains and New Cases Where the Scholz Conjecture Is True",
Journal of Algorithms V. 42(2)
2002 pp 304-316.
- ■ Nadia Nedjah and Luiza de Macedo Mourelle
"Minimal Addition Chain for Efficient Modular Exponentiation Using Genetic Algorithms",
IEA/AIE '02: Proceedings of the 15th international conference on Industrial and engineering applications of artificial intelligence and expert systems,
Springer-Verlag 2002 pp 88-98.
- ■ Eric A. Lehman
"Approximation algorithms for grammar-based data compression",
Ph.D. thesis, MIT, Dept. of Electrical Engineering and Computer Science, 2002.
- ■ Daniel J. Bernstein
"Pippenger's Exponentiation Algorithm",
URL https://cr.yp.to/papers/pippenger.pdf, 2002.
- ■ Bodo Möller
"Improved techniques for fast exponentiation",
International Conference on Information Security and Cryptology
2002 pp 298-312.
- ■ Hatem M. Bahig, Ken Nakamula
"Erratum to Some properties of nonstar steps in addition chains and new cases where the Scholz conjecture is true",
Journal of Algorithms V. 47(1)
2003 pp 60-61.
- ■ Martijn Stam
"Speeding up Subgroup Cryptosystems",
Ph.D. Technische Universiteit Eindhoven, 2003.
- ■ Peter Tummeltshammer, James C. Hoe, and Markus Püschel
"Multiplexed Multiple Constant Multiplication",
In Proceedings of the 41st Annual Conference on Design Automation
(San Diego, CA, USA, June 07 - 11, 2004). DAC '04
ACM Press, New York, NY, 2004 pp 826-829.
- ■ Richard Parker, Andrew Plater
"Addition Chains with a Bounded Number of Registers",
Information Processing Letters, V. 90(5)
2004 pp 247-252.
- ■ Roberta Maria Avanzi
"A note on the signed sliding window integer recoding and a left-to-right analogue",
Selected Areas in Cryptography, Lecture Notes in Computer Science, ISSUE 3357,
Springer-Verlag 2004 pp 130-143.
- ■ Richard K. Guy
"Unsolved problems in number theory",
V. 1 3rd editon
Springer Science & Business Media
Springer-Verlag & Business Media 2004 pp 169-171.
- ■ Nadia Nedjah and Luiza de Macedo Mourelle
"Finding minimal addition chains using ant colony",
International Conference on Intelligent Data Engineering and Automated Learning
Springer-Verlag 2004 pp 642-647.
- ■ Joachim von Zur Gathen and Michael Nöcher
"Computing special powers in finite fields",
Mathematics of Computation V. 73(247)
2004 pp 1499-1523.
- ■ Nareli Cruz-Cortés, Francisco Rodríguez-Henríquez, Raúl Juárez-Morales and Carlos A. Cello Coello
"Finding Optimal Addition Chains Using a Genetic Algorithm Approach",
in Yue Hao et al. (editors),
International Conference on Computational Intelligence and Security 2005 (CIS'05) Part I,
December 15-19, 2005, Xi'an, China
Lecture Notes in Computer Intelligence,
Springer-Verlag 2005 pp 208-215.
- ■ Vadim Vasil'evich Kochergin
"On the Complexity of Computation of a Pair of Monomials in two Variables",
Discrete Mathematics and Applications V. 15(6)
2005 p 547-572.
- ■ Nadia Nedjah and Luiza de Macedo Mourelle
"Efficient pre-processing for large window-based modular exponentiation using ant colony",
International Conference on Knowledge-Based and Intelligent Information and Engineering Systems
Springer-Verlag 2005 pp 640-646.
- ■ Chi Seong Park and Mun-Kyu Lee and Dong Kyue Kim
"New computation paradigm for modular exponentiation using a graph model",
International Symposium on Stochastic Algorithms
Springer-Verlag 2005 pp 170-179.
- ■ Jan Arpe and Bodo Manthey
"Approximability of minimum and-circuits",
Scandinavian Workshop on Algorithm Theory,
Springer-Verlag 2006 pp 292-303.
- ■ Hatem M. Bahig
"Improved Generation of Minimal Addition Chains",
Computing V. 78(2)
2006 pp 161-172.
- ■ Hatem M. Bahig
"On the Number of Minimal Addition Chains",
Proceedings of the 2006 International Conference on Scientific Computing, CSC 2006, Las Vegas, Nevada, USA, June 26-29, 2006
CSREA Press 2006, ISBN 1-60132-007-8 pp 202-208.
- ■ Hatem M. Bahig, Hazem M. Bahig
"Speeding Up Evaluation of Powers and Monomials",
Proceedings of the 2006 International Conference on Foundations of Computer Science, Las Vegas, Nevada, USA, June 26-29, 2006
CSREA Press 2006, ISBN 1-60132-015-9 pp 149-153.
- ■ Nadia Nedjah and Luiza de Macedo Mourelle
"Towards Minimal Addition Chains Using Ant Colony Optimisation",
Journal of Mathematical Modelling and Algorithms, V. 5(4)
Springer-Verlag 2006 pp 525-543.
- ■ Nadia Nedjah and Luiza de Macedo Mourelle
"Ant colony optimisation for fast modular exponentiation using the sliding window method",
Swarm Intelligent Systems
Springer-Verlag, 2006 pp 133-147.
- ■ Daniel J. Bernstein
"The transposition principle",
URL https://cr.yp.to/transposition.html, 2006.
- ■ Daniel J. Bernstein
"Differential addition chains",
URL https://cr.yp.to/ecdh/diffchain-20060219.pdf, 2006.
- ■ Sergey Borisovich Gashkov and Igor Sergeevich Sergeev
"An application of the method of additive chains to inversion in finite fields",
Discrete Mathematics and Applications, V. 16(6)
2006 pp 601-618.
- ■ Fatih Gelgi and Melih Onus
"Heuristics for minimum brauer chain problem",
International Symposium on Computer and Information Sciences 2006
Springer-Verlag pp 47-54.
- ■ Dong Kyue Kim and Mun-Kyu Lee
"A Computation ,odel for modular Exponentiation using graph representation",
International Conference on Multimedia, Information, Technology and its Application 2006
pp 487-490.
- ■ Takahiro Hayata and Masaya Wagatuma
"On Ternary Addition Chains",
Department of Informatics, Yamagata University, Japan 2006.
- ■ Andrew Byrne, Francis Crowe, William Peter Marnane, Nicolas Meloni, Arnaud Tisserand and Emanuel Popovici
"SPA resistant elliptic curve cryptosystem using addition chains",
International Journal of High Performance Systems Architecture, V. 1(2)
2007 pp 133-142.
- ■ Sander van der Kruijssen
"Addition Chains -- efficient computing of powers",
Bachelor thesis, Amsterdam, 2007.
- ■ Raveen R. Goundar
"A Novel Multi-Exponentiation Method",
IAENG International Journal of Applied Mathematics, V. 37 (2)
,
2007 pp 89-95.
- ■ Peter Tummeltshammer, James C. Hoe and Markus Püschel
"Time-multiplexed multiple-constant multiplication",
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, V. 26(9)
2007 pp 1551-1563.
- ■ Sander Van-Der Kruijssen
"Addition Chains, efficient computing of powers",
Bachelor Project, Amsterdam 2007, pp 13-50.
- ■ Hatem M. Bahig
"On a generalization of addition chains: Addition--multiplication chains",
Discrete Mathematics, V. 308(4)
2008 pp 611-616.
- ■ Raveen R. Goundar
"Addition Chains in Application to Elliptic Curve Cryptosystems",
Ph.D. thesis, Kochi University, Japan, 2008.
- ■ Raveen R. Goundar, Ken-ichi Shiota, and Masahiko Toyonaga
"SPA Resistant Scalar Multiplication using Golden Ratio Addition Chain Method",
IAENG International Journal of Applied Mathematics, V. 38 (2)
2008 pp 83-88.
- ■ Nareli Cruz-Cortés, Francisco Rodríguez-Henríquez and A. Carlos A. Coello Coello
"An Artificial Immune System Heuristic for Generating Short Addition Chains",
IEEE Transactions on Evolutionary Computation, V. 12(1)
2008 pp 1-24.
- ■ Dong Fu-guo and Li Yu-rong
"A novel shortest addition chains algorithm based on Euclid algorithm",
Proceeding of WiCom (2008 4th International Conference on Wireless Communications, Networking and Mobile Computing
2008 pp 4757-4761.
- ■ Raveen R. Goundar, Ken-ichi Shiota and Masahiko Toyonaga
"New Strategy for Doubling-free Short Addition-Subtraction Chain",
Applied Mathematics & Information Sciences -- An International Journal, V. 2
Dixie W Publishing Corporation, China, 2008 pp 123-133.
- ■ Jan Arpe and Bodo Manthey
"Approximability of minimum AND-circuits",
Algorithmica, V. 53(3)
2009 pp 337-357.
- ■ Martin M. Johansen
"Exponentiation with Addition Chains",
undergraduate project paper, 2009.
- ■ Alejandro León-Javier, Nareli Cruz-Cortés, Marco A. Moreno-Armendáriz and Sandra Orantes-Jimènez
"Finding minimal addition chains with a particle swarm optimization algorithm",
Mexican International Conference on Artificial Intelligence
Springer-Verlag 2009 pp 680-691.
- ■ Luis G. Osorio-Hernández, Efrén Mezura-Montes, Nareli Cruz-Cortés and Francisco Rodríguez-Henríquez
"A genetic algorithm with repair and local search mechanisms able to find minimal length addition chains for small exponents",
IEEE Congress on Evolutionary Computation 2009 pp 1422-1429.
- ■ Fu-guo Dong, Yu-rong Li and Jin-jiang Li
"The Consistency Analysis of Addition Chains for Several Fast Algorithm of Modular Exponentiation",
International Journal of Digital Content Technology and its Applications V. 4(5)
2010 pp 82-88.
- ■ Ryan Henry
"Pippenger's Multiproduct and Multiexponentiation Algorithms",
University Waterloo, 2010.
- ■ Fabien Herbaut, Pierre-Yvan Liardet, Nicolas Mèloni, Yannick Teglia and Pascal Vèron
"Random Euclidean addition chain generation and its application to point multiplication",
International Conference on Cryptology in India 2010
Springer-Verlag pp 238--261.
- ■ Fabien Herbaut and Pascal Vèron
"A public key cryptosystem based upon euclidean addition chains",
International Conference on Sequences and Their Applications
Springer-Verlag 2010 pp 284-297.
- ■ Yiling Li and Qingyu Ma
"Design and Implementation of Layer Extended Shortest Addition Chains Database for Fast Modular Exponentiation in RSA",
2010 International Conference on Web Information Systems and Mining (WISM),
IEEE 2010 pp 136-139.
- ■ Anton Mityagin, Ilya Mironov and Yaacov Nissim Kobliner
"Systems and methods for generating random addition chains",
U.S. Patent 7657029B2, 2010.
- ■ Kári Ragnarsson and Bridget Eileen Tenner
"Obtainable sizes of topologies on finite sets",
Journal of Combinatorial Theory, Series A, V. 117(2)
2010 pp 138-151.
- ■ Neill Michael Clift,
"Calculating optimal addition chains",
Computing V. 91(3)
2011 pp 265-284.
- ■ Hatem M. Bahig and Hazem M. Bahig
"A new strategy for generating shortest addition sequences",
Computing V. 91 (3)
2011 pp 285-306.
- ■ Hatem M. Bahig
"Star reduction among minimal length addition chains",
Computing V. 91 (4)
2011 pp 335-352.
- ■ Saúl Domínguez-Isidro and Efrén Mezura-Montes
"An Evolutionary Programming Algorithm to Find Minimal Addition Chains",
I Congreso Internacional de Ingeniería Electrónica, Instrumentación y Computación, de Junio del, Minatitlán Veracruz, Mèxico 2011, 5 pages.
- ■ Saúl Domínguez-Isidro and Efrén Mezura-Montes
"Addition chain length minimization with evolutionary programming",
Proceedings of the 13th annual conference companion on Genetic and evolutionary computation 2011, pp 59-60.
- ■ Yara Elias
"Représentation d'un polynôme par un circuit arithmétique et chaînes additives",
Université de Montréal 2011.
- ■ Dong Fuguo, Wang Pingqin and Feng Lei
"Addition Chains Algorithm Based on Fast Fourier Transform",
International Journal of Digital Content Technology and its Applications V. 5(2),
2011 pp 149-157.
- ■ Adan Jose-Garcia, Hillel Romero-Monsivais, Cindy G. Hernandez-Morales, Arturo Rodriguez-Cristerna, Ivan Rivera-Islas and José Torres-Jiménez
"A simulated annealing algorithm for the problem of minimal addition chains",
Portuguese Conference on Artificial Intelligence
Springer-Verlag 2011 pp 311-325.
- ■ Maurice Mignotte and Amadou Tall
"A note on addition chains",
International Journal of Algebra V. 5(6)
2011 pp 269-274.
- ■ Mohamad Afendee Mohamed
"On the Improvement of Addition Chain in Applications to Elliptic Curve Cryptosystem",
Thesis, Universiti Putra Malaysia 2011.
- ■ M.A. Mohamed, M.R. Md Said, K.A. Mohd Atan and Zuriati Ahmad Zulkarnain
"Shorter addition chain for smooth integers using decomposition method",
International Journal of Computer Mathematics V. 88(11),
2011 pp 2222-2232.
- ■ Arturo Rodriguez-Cristerna,José Torres-Jiménez, Ivan Rivera-Islas, Cindy G. Hernandez-Morales, Hillel Romero-Monsivais and Adan Jose-Garcia
"A mutation-selection algorithm for the problem of minimum brauer chains",
Mexican International Conference on Artificial Intelligence
Springer-Verlag 2011 pp 107-118.
- ■ Peter I. Samovol and Valery M. Zhuravlev
"Faster than the Fastest, or Can the Binary Algorithm be Overhauled"
Journal of the World Federation of National Mathematics V. 24(1)
URL http://www.wfnmc.org/Journal 2011 1.pdf
Australia AMT Publishing, 2011 pp 37-58.
- ■ Sergey Borisovich Gashkov
"The problem of additive chains and its generalization"
Mathematical Education V. 15
https://www.mccme.ru/free-books/matpros/mpf.pdf
Independent University of Moscow, 2011 pp 138-153.
- ■ Muhammad Abbas
"On the implementation of integer and non-integer sampling rate conversion",
Ph.D. thesis, Linköping University, Sweden, 2012.
- ■ Muhammad Abbas and Oscar Gustafsson
"Integer Linear Programming Modeling of Addition Sequences With Additional Constraints for Evaluation of Power Terms",
The Institute of Technology, Linköping University, Sweden, 2012.
- ■ M.A. Mohamed and K.A. Mohd Atan
"Rule Based Representation of Integer for a New Addition Chain Method",
Applied Mathematical Sciences V. 6(30)
2012 pp 1497-1503.
- ■ Colin D. Walter
"A duality in space usage between left-to-right and right-to-left exponentiation",
Cryptographers' Track at the RSA Conference
Springer-Verlag 2012 pp 84-97.
- ■ Mr K. Mani
"Generation of Addition Chain using Deterministic Division Based Method",
International Journal of Computer Science & Engineering Technology V. 1(4)
2013 pp 553-560.
- ■ Arturo Rodriguez-Cristerna,José Torres-Jiménez
"A genetic algorithm for the problem of minimal brauer chains for large exponents",
Soft Computing Applications in Optimization, Control, and Recognition,
Springer-Verlag 2013 pp 27-51.
- ■ Arturo Rodriguez-Cristerna,José Torres-Jiménez
"A genetic algorithm for the problem of minimal brauer chains",
Recent Advances on Hybrid Intelligent Systems, Springer-Verlag 2013 pp 481-500.
- ■ José Maclovio Sautto Vallejo, Agustín Santiago Moreno, Carlos N. Bouza Herrera and Verónica Campos Guzmán
"Scholz's Third Conjecture: A Demonstration for Star Addition Chains",
Applied Mathematics V. 4(10)
2013 pp 1-2.
- ■ Valery M. Zhuravlev, Peter I. Samovol
"Faster than fast, or can one beat the binary algorithm"
Kvant (russian magazin KBAHT) No 2
URL http://kvant.ras.ru/pdf/2013/2013-02.pdf
2013 pp 7-15.
- ■ Xiaodong Wang
"An Efficient Algorithm for Finding Optimal Addition Chains",
Indonesian Journal of Electrical Engineering and Computer Science V. 11(11)
2013 pp 6447-6453.
- ■ Harry J. Altman
"Integer complexity, addition chains, and well-ordering",
Ph.D. thesis University of Michigan 2014.
- ■ Harry J. Altman
"Internal Structure of Addition Chains: Well-Ordering",
arXiv preprint arXiv:1409.1627, 2014.
- ■ Jukka Kohonen and Jukka Corander
"Addition chains meet postage stamps: Reducing the number of multiplications",
Journal of Integer Sequences V. 17(3)
2014, 13 pages.
- ■ Saúl Domínguez-Isidro and Efrén Mezura-Montes and Luis G-Guillermo Osorio-Hernández
"Evolutionary programming for the length minimization of addition chains",
Engineering Applications of Artificial Intelligence V. 37
2015, pp 125-134.
- ■ Vadim Vasil'evich Kochergin
"Improvement of the estimates of the computational complexity for monomials and sets of powers in Bellman's and Knuth's problems",
Journal of Applied and Industrial Mathematics V. 9(1)
2015 pp 68-82.
- ■ Mohamad Afendee Mohamed, Md Said and Mohamad Rushdan
"A hybrid addition chain method for faster scalar multiplication",
WSEAS Transactions on Communications V. 14
2015 pp 144-152.
- ■ Adamu Muhammad Noma, Mohamad Afendee Mohamed, Abdullah Muhammed and Zuriati Ahmad Zulkarnain
"An Initial Solution for Addition Chain Optimization Problem",
Journal of Engineering and Applied Sciences V. 11(3)
2016 pp 640-643.
- ■ Andreas Enge, William Hart and Fredrik Johansson
"Short addition sequences for theta functions",
arXiv preprint arXiv:1608.06810 2016.
- ■ Brian Koziel, Reza Azarderakhsh, David Jao and Mehran Mozaffari-Kermani
"On Fast Calculation of Addition Chains for Isogeny-Based Cryptography",
Information Security and Cryptology:
12th International Conference, Inscrypt 2016, Beijing, China
2016 pp 323-342.
- ■ Nicolas Méloni and M. Anwar Hasan
"Random Digit Representation of Integers",
IEEE 23nd Symposium on Computer Arithmetic
2016 8 pages.
- ■ Stjepan Picek, Carlos A. Coello Coello, Domagoj Jakobovic and Nele Mentens
"Evolutionary Algorithms for Finding Short Addition Chains: Going the Distance",
The 16th European Conference on Evolutionary Computation in Combinatorial Optimization, Porto, Portugal
Springer-Verlag 2016 pp 121-137.
- ■ José M. Sautto, Agustín Santiago, Carlos N. Bouza and Verónica Campos
"Scholz's First Conjecture: A Brief Demonstration",
Applied Mathematics V. 7(01)
2016 pp 70-76.
- ■ Eduardo Vázquez-Fernández, Carlos Cadena and David A. Reyes-Gómez
"A genetic algorithm with a mutation mechanism based on a Gaussian and uniform distribution to minimize addition chains for small exponents",
2016 IEEE Congress on Evolutionary Computation (CEC)
2016 pp 935-940.
- ■ Fernando Aquino and Guillermo Leguizamón
"Optimization of addition chains",
Computer Science & Technology Series, XXII Argentine Congress of Computer Science, 2016 San Luis Argentina
2017 pp 40-54.
- ■ Vadim Vasil'evich Kochergin and D. V. Kochergin
"Improvement of the Lower bound for the Complexity of Exponention",
Prikladnaya Diskretnaya Matematika No. 38
2017 pp 119-132.
- ■ K. Mani and M. Viswambari
"A New Method of Generating Optimal Addition Chain Based on Graph",
International Journal of Mathematical Sciences and Computing V. 3(2)
2017 pp 37-54.
- ■ Josip Užarević
"Postupci konstrukcije i optimizacije adicijskih lanaca",
URL https://www.bib.irb.hr/988402?&rad=988402
Diploma thesis 2017: Faculty of Electrical Engineering and Computing, University of Zagreb 2017.
- ■ Hatem M. Bahig and Khaled A. Abdel Bari
"A fast GPU-based hybrid algorithm for addition chains",
Cluster Computing V. 21(12)
Springer-Verlag 2018 pp 1-11.
- ■ M. F. A. Kadir and Mohamad Afendee Mohamed and Rushdan Mohamad and M. Mamat and Abdullah Muhammed
"Performance Comparison of Some Addition Chain Methods Based on Integer Family",
Information Science and Applications,
Springer Singapore 2018 pp 211-218.
- ■ Mauricio Olguin Carbajal, Juan Carlos Herrera-Lozada, Israel Rivera-Zárate, J. Felix Serrano-Talamantes, Rodrigo Cadena-Martínez and J. Irving Vásquez-Gómez
"Minimum Addition Chains Generation Using Evolutionary Strategies",
Computación y Sistemas V. 22(14)
2018 pp 1463-1472.
- ■ Stjepan Picek, Carlos A. Coello Coello, Domagoj Jakobovic and Nele Mentens
"Finding short and implementation-friendly addition chains with evolutionary algorithms",
Journal of Heuristics V. 24(3)
URL https://doi.org/10.1007/s10732-017-9340-2
Springer Nature, 2018 pp 457-481.
- ■ Hazem Bahig and Yasser Kotb
"An Efficient Multicore Algorithm for Minimal Length Addition Chains",
Computers V. 8(23)
URL https://doi.org/10.3390/computers8010023
MDPI, Basel, Switzerland 2019 23 pages.
- ■ Silvestre Ascencion Garcia Sanchez, Luis Calderon Luis Calderon Osorno and Edmundo Rene Duran Camarillo
"Simulated Annealing Meta-heuristic for Addition Chain Optimization",
European Journal of Electrical Engineering and Computer Science V. 3(6)
2019.
legend:
■ reviewed journal article
■ master, diploma, Ph.D. thesis
■ report, memorandum, arXiv preprint paper
■ in conference proceedings
■ in (conference papers) collection, (non-reviewed) science journal / magazine
■ monography, book
■ unpublished paper
■ private (temporary) manuscript on WWW
■ unknown
■ other
I like to thank Neill Clift, Kirkland, Washington, U.S.A., for almost all references in the preceeding bibliography list.
Corrections and updates are welcome to:
Achim Flammenkamp
Last update: 2020-06-07 13:38:41 UTC+2