@Journal{kiam_9730,
  abstract  = {We consider various generalizations of the effective exponentiation problem, also known as the addition chains problem in additive terminology. We attempt to evaluate and consistently compare the results and the methods that were used to obtain them. The considered generalizations are complexity problems (i.e. finding the minimum number of required operations) for the computation a monomial of many variables, a set of powers, elements of finite Abelian groups, elements of free semigroups, systems of monomials, systems of integer linear forms, systems of elements of free Abelian groups along with some other problems. For all these directions of research we overview both the classical results and the results of the author and his followers.},
  address   = {Moscow},
  annote    = {We consider various generalizations of the effective exponentiation problem, also known as the addition chains problem in additive terminology. We attempt to evaluate and consistently compare the results and the methods that were used to obtain them. The considered generalizations are complexity problems (i.e. finding the minimum number of required operations) for the computation a monomial of many variables, a set of powers, elements of finite Abelian groups, elements of free semigroups, systems of monomials, systems of integer linear forms, systems of elements of free Abelian groups along with some other problems. For all these directions of research we overview both the classical results and the results of the author and his followers.},
  author    = {Kochergin, V.V.},
  booktitle = {Mathematical Problems of Cybernetics},
  crossref  = {https://doi.org/10.20948/mvk-2022-119},
  doi       = {https://doi.org/10.20948/mvk-2022-119},
  eprint    = {https://keldysh.ru/papers/2022/mvk/mvk2022_119.pdf},
  journal   = {Mathematical Problems of Cybernetics},
  pages     = {119--256},
  title     = {Bellman's, Knut's, Lupanov's, Pippenger's problems and their variations as generalizations of the addition chain problem},
  type      = {Mathematical Problems of Cybernetics},
  url       = {https://library.keldysh.ru/mvk.asp?id=2022-119&lg=e},
  volume    = {№20},
  year      = {2022},
}

@Article{dellac1894question,
  author  = {Dellac, H},
  title   = {Question 49},
  journal = {L'Interm\'{e}diaire des Math\'{e}maticiens},
  year    = {1894},
  volume  = {1},
  pages   = {20},
}

@Article{Goulard1895q393,
  author  = {A. Goulard},
  title   = {Question 393},
  journal = {L'Interm\'{e}diaire des Math\'{e}maticiens},
  year    = {1894},
  volume  = {1},
  pages   = {234},
}

@Article{Jonquieres1894question,
  author  = {E. de Jonqui\'{e}res},
  title   = {Question 49 (H. Dellac)},
  journal = {L'Interm\'{e}diaire des Math\'{e}maticiens},
  year    = {1894},
  volume  = {1},
  pages   = {162--164},
}

@Article{Jonquieres1895question393,
  author  = {E. de Jonqui\'{e}res},
  title   = {Question 393 (A. Goulard)},
  pages   = {125--126},
  volume  = {2},
  journal = {L'Interm\'{e}diaire des Math\'{e}maticiens},
  year    = {1895},
}

@Book{data1935history,
  author  = {B. Datta and A.N. Singh},
  title   = {History of Hindu Mathematics},
  year    = {1935},
  pages   = {76},
  address = {Bombay},
}

@Article{scholz1937aufgabe,
  author =  {Scholz, Arnold},
  title =   {Aufgabe 253 (Problem 253)},
  journal = {Jahresbericht der deutschen Mathematiker-Vereinigung},
  year =    {1937},
  volume =  {47},
  number =  {2},
  pages =   {41--42}
}

@Article{brauer1939addition,
  author =  {Brauer, Alfred},
  title =   {On addition chains},
  journal = {Bulletin of the American Mathematical Society},
  year =    {1939},
  volume =  {45},
  number =  {10},
  pages =   {736--739}
}

@Article{utz1953note,
  author =  {Utz, WR},
  title =   {A note on the Scholz-Brauer problem in addition chains},
  journal = {Proceedings of the American Mathematical Society},
  year =    {1953},
  volume =  {4},
  number =  {3},
  pages =   {462--463}
}

@MastersThesis{hendley1954theory,
  author = {Hendley, Samuel Moore},
  title =  {The theory and construction of addition chains},
  year =   {1954}
}

@MastersThesis{hansen1957Thesis,
  author = {Hansen, W},
  title  = {Untersuchungen \"{u}ber die Scholz-Brauerschen Additionsketten und deren Verallgemeinerung},
  school = {G\"{o}ttingen},
  year   = {1957},
}

@Article{hansen1959scholz,
  author  = {Hansen, Walter},
  title   = {Zum Scholz-Brauerschen Problem.},
  pages   = {129--136},
  volume  = {202},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  year    = {1959},
}

@Article{Valskii1959Lower,
  author  = {P.E. Valskii},
  title   = {On the Lower Bounds of Multiplications in Evaluation of Powers},
  note    = {In Russian},
  pages   = {73--74},
  volume  = {2},
  journal = {Problemy Kibernet},
  year    = {1959},
}

@Article{erdos1960remarks,
  author =    {Erd{\"o}s, Paul},
  title =     {Remarks on number theory III. On addition chains},
  journal =   {Acta Arithmetica},
  year =      {1960},
  volume =    {6},
  number =    {1},
  pages =     {77--81},
  publisher = {Institute of Mathematics Polish Academy of Sciences}
}

@Article{Valskii1961Lower,
  author  = {P.E. Valskii},
  title   = {The Smallest Number of Multiplications Necessary to Raise a Number to a Given Power},
  note    = {In English},
  pages   = {395--397},
  volume  = {2},
  journal = {Problems of Cybernetics},
  year    = {1961},
}

@Article{gioia1962scholz,
  author =  {Gioia, AA and Subbarao, MV and Sugunama, M},
  title =   {The Scholz-Brauer problem in addition chains},
  journal = {Duke Math. J},
  year =    {1962},
  volume =  {29},
  pages =   {481--487}
}

@Article{bellman1963advanced,
  author =  {Bellman, Richard},
  title =   {Advanced problem 5125},
  journal = {Amer. Math. Monthly},
  year =    {1963},
  volume =  {70},
  pages =   {765}
}

@Article{knuth1964addition,
  author =  {Knuth, D.E.},
  title =   {Addition chains and the evaluation of n-th powers},
  journal = {Notices Am Math Soc},
  year =    {1964},
  volume =  {11},
  pages =   {230--231}
}

@Article{Straus1964,
  author  = {E. G. Straus},
  title   = {Solution to Problem 5125},
  journal = {Amer. Math. Monthly},
  year    = {1964},
  volume  = {71},
  pages   = {806-808},
  key     = {Straus 1964},
}

@Article{il1965additive,
  author =  {Il'in, AM},
  title =   {On additive number chains},
  journal = {Problemy Kibernet},
  year =    {1965},
  volume =  {13},
  pages =   {245--248}
}

@Article{whyburn1965note,
  author =  {Whyburn, CT},
  title =   {A note on addition chains},
  journal = {Proceedings of the American Mathematical Society},
  year =    {1965},
  volume =  {16},
  number =  {5},
  pages =   {1134}
}

@MastersThesis{demetriou1967concerning,
  author = {Demetriou, Charles A},
  title =  {Concerning minimal addition chains},
  school = {University of Mississippi},
  year =   {1967}
}

@Article{wattel1968efficient,
  author =    {Wattel, Evert and Jensen, GA},
  title =     {Efficient calculation of powers in a semigroup},
  journal =   {Stichting Mathematisch Centrum. Zuivere Wiskunde},
  year =      {1968},
  number =    {ZW 1/68},
  pages =     {1--18},
  publisher = {Stichting Mathematisch Centrum}
}

@Book{knuth1969art,
  author  = {Knuth, Donald E},
  title   = {The art of computer programming, 2: seminumerical algorithms, Addison Wesley},
  pages   = {398--422},
  journal = {Reading, MA},
  year    = {1969},
}

@Unpublished{knuth1969calc,
  author   = {Knuth, Donald Ervin},
  title    = {Calculations on Addition Chains},
  location = {Stanford University, California},
  year     = {1969},
}

@Article{stolarsky1969lower,
  author =  {Stolarsky, Kenneth B},
  title =   {A lower bound for the Scholz-Brauer problem},
  journal = {Canad. J. Math},
  year =    {1969},
  volume =  {21},
  pages =   {675--683}
}

@PhdThesis{kato1970onaddition,
  author = {H. Kato},
  title =  {On Addition Chains},
  school = {University of Southern California},
  year =   {1970}
}

@Thesis{gaal1971additionsketten,
  author      = {Gaal, Peter},
  institution = {University of Z{u}rich},
  title       = {Additionsketten in einer endlich erzeugten abelschen Gruppe},
  year        = {1971},
}

@PhdThesis{Thurber1971,
  author      = {E.G. Thurber},
  date        = {1971},
  institution = {University of Southern California},
  title       = {The Scholz-Brauer Problem on Addition Chains},
}

@Article{thurber1971scholz,
  author  = {Thurber, Edward},
  title   = {The Scholz-Brauer problem on addition chains},
  pages   = {1100},
  series  = {Abstract 71T-A274},
  volume  = {18},
  journal = {Notices of the American Mathematical Society},
  year    = {1971},
}

@InProceedings{giese1972seque,
  author =    {Giese, R.P.},
  title =     {A Sequence of Counterexamples of Knuth's Modification of Utz's Conjecture},
  booktitle = {NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY},
  year =      {1972},
  volume =    {19},
  pages =     {A688}
}

@InProceedings{cottrell1973lower,
  author =       {Cottrell, A},
  title =        {A lower bound for the Scholz-Brauer problem. preliminary report},
  booktitle =    {Notices of the American Mathematical Society},
  year =         {1973},
  volume =       {20},
  number =       {5},
  pages =        {A476},
  organization = {AMER MATHEMATICAL SOC 201 CHARLES ST, PROVIDENCE, RI 02940-2213}
}

@Article{thurber1973scholz,
  author =    {Thurber, Edward},
  title =     {The Scholz-Brauer problem on addition chains},
  journal =   {Pacific Journal of Mathematics},
  year =      {1973},
  volume =    {49},
  number =    {1},
  pages =     {229--242},
  publisher = {Mathematical Sciences Publishers}
}

@Article{thurber1973addition,
  author    = {Thurber, Edward G},
  title     = {On addition chains $l(mn)\leq l(n)-b $ and lower bounds for $ c (r) $},
  number    = {4},
  pages     = {907--913},
  volume    = {40},
  journal   = {Duke Mathematical Journal},
  publisher = {Duke University Press},
  year      = {1973},
}

@PhdThesis{giese1974selected,
  author = {Giese, Robert Paul},
  title =  {Selected topics in addition chains},
  school = {University of Houston},
  year =   {1974}
}

@InProceedings{hebb1974notice,
  author =       {Hebb, KR},
  title =        {Some results on addition chains. Preliminary report.},
  booktitle =    {Notices of the American Mathematical Society},
  year =         {1974},
  volume =       {21},
  number =       {2},
  pages =        {A294--A294},
  organization = {AMER MATHEMATICAL SOC 201 CHARLES ST, PROVIDENCE, RI 02940-2213}
}

@PhdThesis{hebb1974some,
  author = {Hebb, Kevin R},
  title =  {Some problems on addition chains},
  school = {Thesis (M. Sc.)--University of Alberta},
  year =   {1974}
}

@Book{southard1975addition,
  title =     {Addition chains for the first n triangular numbers},
  publisher = {Center for Numerical Analysis, University of Texas at Austin},
  year =      {1974},
  author =    {Southard, Thomas H}
}

@Book{southard1974addition,
  title =     {Addition Chains for the first n squares},
  publisher = {Center for Numerical Analysis, University of Texas at Austin},
  year =      {1974},
  author =    {Southard, Thomas H}
}

@Article{gioia1975scholz,
  author =  {Gioia, AA and Subbarao, MV},
  title =   {The Scholz-Brauer problem in addition chains},
  journal = {Notices of the American Mathematical Society},
  year =    {1975},
  volume =  {22},
  pages =   {A63-A64}
}

@Article{schonhage1975lower,
  author =    {Sch{\"o}nhage, Arnold},
  title =     {A lower bound for the length of addition chains},
  journal =   {Theoretical Computer Science},
  year =      {1975},
  volume =    {1},
  number =    {1},
  pages =     {1--12},
  publisher = {Elsevier}
}

@InProceedings{vegh1975addition,
  author =       {VEGH, E},
  title =        {Addition chains},
  booktitle =    {Notices of the American Mathematical Society},
  year =         {1975},
  volume =       {22},
  number =       {1},
  pages =        {A2--A3},
  organization = {AMER MATHEMATICAL SOC 201 CHARLES ST, PROVIDENCE, RI 02940-2213}
}

@Article{vegh1975note,
  author =    {Vegh, Emanuel},
  title =     {A note on addition chains},
  journal =   {Journal of Combinatorial Theory, Series A},
  year =      {1975},
  volume =    {19},
  number =    {1},
  pages =     {117--118},
  publisher = {Elsevier}
}

@Article{dokl1976additive,
  author    = {Belaga, E.G.},
  title     = {THE ADDITIVE COMPLEXITY OF A NATURAL NUMBER},
  number    = {1},
  pages     = {5--9},
  volume    = {17},
  journal   = {Soviet Mathematics Doklady},
  publisher = {American Mathematical Society.},
  year      = {1976},
}

@InProceedings{pippenger1976evaluation,
  author    = {Pippenger, Nicholas},
  booktitle = {17th Annual Symposium on Foundations of Computer Science (sfcs 1976)},
  title     = {On the evaluation of powers and related problems},
  pages     = {258--263},
  year      = {1976},
}

@Article{thurber1976addition,
  author    = {Thurber, Edward G},
  title     = {Addition chains and solutions of $l(2n) = l(n)$ and $l(2^n-1) = n + l(n) - 1$},
  number    = {3},
  pages     = {279--289},
  volume    = {16},
  journal   = {Discrete Mathematics},
  publisher = {Elsevier},
  year      = {1976},
}

@Article{yao1976evaluation,
  author =    {Yao, Andrew Chi-Chih},
  title =     {On the evaluation of powers},
  journal =   {SIAM Journal on computing},
  year =      {1976},
  volume =    {5},
  number =    {1},
  pages =     {100--103},
  publisher = {SIAM}
}

@Article{van1977extension,
  author =  {van Leeuwen, Jan},
  title =   {An extension of Hansen's theorem for star chains},
  journal = {J. Reine Angew. Math},
  year =    {1977},
  volume =  {295},
  pages =   {203--207}
}

@Article{mccarthy1977optimal,
  author =  {McCarthy, Daniel P},
  title =   {The optimal algorithm to evaluate x^n using elementary multiplication methods},
  journal = {Mathematics of Computation},
  year =    {1977},
  volume =  {31},
  number =  {137},
  pages =   {251--256}
}

@InProceedings{dobkin1980addition1,
  author    = {Dobkin, David and Lipton, Richard J},
  booktitle = {Proceedings of a Conference on Theoretical Computer Science (Univ.Waterloo, Waterloo, Ont., 1977)},
  title     = {Addition chain methods for the evaluation of specific polynomials},
  pages     = {146--148},
  year      = {1978},
}

@InProceedings{gioia1978scholz,
  author =    {Gioia, AA and Subbarao, MV},
  title =     {The Scholz-Brauer problem in addition chains II},
  booktitle = {Proc. eighth Manitoba conference on numerical math. and computing},
  year =      {1978},
  volume =    {22},
  pages =     {251--274}
}

@Thesis{sonntag1978thesis,
  author = {Sonntag R.},
  title  = {Theorie der Additionsketten},
  year   = {1978},
  school = {Technische Universität Hannover},
}

@Unpublished{sonntag1978unpublished,
  author = {Sonntag, R.},
  title  = {On Addition Chains},
  note   = {Unpublished paper giving details of thesis results},
  pages  = {169--171},
  year   = {1978},
}

@Article{dobkin1980addition,
  author    = {Dobkin, David and Lipton, Richard J},
  title     = {Addition chain methods for the evaluation of specific polynomials},
  year      = {1980},
  volume    = {9},
  number    = {1},
  pages     = {121--125},
  journal   = {SIAM Journal on Computing},
  publisher = {SIAM},
}

@Article{pippenger1980evaluation,
  author =    {Pippenger, Nicholas},
  title =     {On the evaluation of powers and monomials},
  journal =   {SIAM Journal on Computing},
  year =      {1980},
  volume =    {9},
  number =    {2},
  pages =     {230--250},
  publisher = {SIAM}
}

@Article{downey1981computing,
  author =    {Downey, Peter and Leong, Benton and Sethi, Ravi},
  title =     {Computing sequences with addition chains},
  journal =   {SIAM Journal on Computing},
  year =      {1981},
  volume =    {10},
  number =    {3},
  pages =     {638--646},
  publisher = {SIAM}
}

@Article{knuth1981duality,
  author =  {Knuth, Donald E and Papadimitriou, Christos H},
  title =   {Duality in addition chains},
  journal = {Bulletin of the European Association for Theoretical Computer Science},
  year =    {1981},
  volume =  {13},
  pages =   {2--4}
}

@Article{Olivos1981OnVector,
  author    = {Olivos, Jorge},
  title     = {On Vectorial Addition Chains.},
  doi       = {10.1016/0196-6774(81)90003-1},
  number    = {1},
  pages     = {13-21},
  url       = {http://dblp.uni-trier.de/db/journals/jal/jal2.html#Olivos81},
  volume    = {2},
  added-at  = {2011-07-01T00:00:00.000+0200},
  biburl    = {http://www.bibsonomy.org/bibtex/272ffd88bc507a7410985a3e418334bd4/dblp},
  interhash = {801a9f3df41bc6162a5998f16765dbd9},
  intrahash = {72ffd88bc507a7410985a3e418334bd4},
  journal   = {J. Algorithms},
  keywords  = {dblp},
  timestamp = {2011-07-02T11:38:13.000+0200},
  year      = {1981},
}

@Article{semba1983systematic,
  author =    {Semba, I},
  title =     {Systematic Method for Determining the Number of Multiplications Required to Compute x\^{} m, Where m is a Positive Integer},
  journal =   {Journal of information processing},
  year =      {1983},
  volume =    {6},
  number =    {1},
  pages =     {31--33},
  publisher = {一般社団法人情報処理学会}
}

@InProceedings{chin1985algorithms,
  author =    {Chin, YH and Tsai, YH},
  title =     {Algorithms for finding the shortest addition chain},
  booktitle = {Proceedings of national computer symposium, Kaoshiung, Taiwan, December},
  year =      {1985},
  pages =     {1398-1414}
}

@MastersThesis{tsai1985study,
  author = {Y.H. Tsai},
  title =  {A study on some addition chain problems},
  school = {National Tsing-Hua University, Hsinchu, Taiwan},
  year =   {1985}
}

@Article{volger1985some,
  author =    {Volger, Hugo},
  title =     {Some results on addition/subtraction chains},
  journal =   {Information Processing Letters},
  year =      {1985},
  volume =    {20},
  number =    {3},
  pages =     {155--160},
  publisher = {Elsevier}
}

@Article{mccarthy1986effect,
  author =  {McCarthy, Daniel P},
  title =   {Effect of improved multiplication efficiency on exponentiation algorithms derived from addition chains},
  journal = {Mathematics of computation},
  year =    {1986},
  volume =  {46},
  number =  {174},
  pages =   {603--608}
}

@InCollection{lickteig1987some,
  author =    {Lickteig, Thomas and Volger, Hugo},
  title =     {Some results on the complexity of powers},
  booktitle = {Computation theory and logic},
  publisher = {Springer},
  year =      {1987},
  pages =     {249--255}
}

@MastersThesis{orb1987,
  author = {Orb, W},
  title  = {Additionsketten},
  school = {Mainz University},
  year   = {1987},
}

@Article{tsai1987study,
  author =    {Tsai, YH and Chin, YH},
  title =     {A study of some addition chain problems},
  journal =   {International Journal of Computer Mathematics},
  year =      {1987},
  volume =    {22},
  number =    {2},
  pages =     {117--134},
  publisher = {Taylor \& Francis}
}

@InProceedings{bergeron1989unifying,
  author    = {Bergeron, F and Berstel, J and Brlek, S},
  booktitle = {Proc. XV Latin American Conf. on Informatics, Santiago, Chile July},
  title     = {A unifying approach to the generation of addition chains},
  note      = {DO NOT HAVE},
  pages     = {10--14},
  year      = {1989},
}

@Article{bergeron1989addition,
  author =    {Bergeron, Fran{\c{c}}ois and Berstel, Jean and Brlek, Srecko and Duboc, Christine},
  title =     {Addition chains using continued fractions},
  journal =   {Journal of Algorithms},
  year =      {1989},
  volume =    {10},
  number =    {3},
  pages =     {403--412},
  publisher = {Elsevier}
}

@TechReport{bergeron1989vectorial,
  author =      {Bergeron, F and Olivos, J},
  title =       {Vectorial addition chains using Euclid's algorithm},
  institution = {Research Report, Dpt. Math., UQAM 105},
  year =        {1989}
}

@InProceedings{bos1989addition,
  author =       {Bos, Jurjen and Coster, Matthijs},
  title =        {Addition chain heuristics},
  booktitle =    {Conference on the Theory and Application of Cryptology},
  year =         {1989},
  pages =        {400--407},
  organization = {Springer}
}

@Article{subbarao1989addition,
  author =    {Subbarao, M},
  title =     {Addition chains-some results and problems},
  journal =   {Number theory and applications},
  year =      {1989},
  pages =     {555--574},
  publisher = {Kluwer Academic Publishers, Dordrecht}
}

@TechReport{coster1990some,
  author      = {Coster, Matthijs J},
  title       = {Some algorithms on addition chains and their complexity},
  institution = {Centrum voor Wiskunde en Informatica, Amsterdam},
  year        = {1990},
  number      = {R 9024},
  pages       = {1--69},
  journal     = {Department of Computer Science [CS]},
  publisher   = {CWI},
}

@InProceedings{elia1990note,
  author =       {Elia, M and Neri, F},
  title =        {A note on addition chains and some related conjectures},
  booktitle =    {Sequences},
  year =         {1990},
  pages =        {166--181},
  organization = {Springer}
}

@Article{morain1990speeding,
  author =  {Morain, Fran{\c{c}}ois and Olivos, Jorge},
  title =   {Speeding up the computations on an elliptic curve using addition-subtraction chains},
  journal = {Informatique th{\'e}orique et Applications},
  year =    {1990},
  volume =  {24},
  number =  {6},
  pages =   {531--543}
}

@InProceedings{yacobi1990exponentiating,
  author =       {Yacobi, Yacov},
  title =        {Exponentiating faster with addition chains},
  booktitle =    {Workshop on the Theory and Application of of Cryptographic Techniques},
  year =         {1990},
  pages =        {222--229},
  organization = {Springer}
}

@InProceedings{brlek1991addition,
  author =       {Brlek, Sre{\'c}ko and Cast{\'e}ran, Pierre and Strandh, Robert},
  title =        {On addition schemes},
  booktitle =    {International Joint Conference on Theory and Practice of Software Development},
  year =         {1991},
  pages =        {379--393},
  organization = {Springer}
}

@Conference{brlek1991jfla91,
  author    = {Brlek S., Castéran P., Strandh R.},
  booktitle = {Journées JFLA91 (Gresse-en-Vercors, France)},
  title     = {Chaînes d'additions et structures de contrôle},
  pages     = {54--63},
  year      = {1991},
}

@Diplomathesis{flammenkamp1991add,
  author = {Flammenkamp, A.},
  school = {Diplomarbeit an der Fakultät für Mathematik, Universität Bielefeld},
  title  = {Drei Beiträge zur diskreten Mathematik: Additionsketten, No-Three-in-Line-Problem, Sociable Numbers},
  year   = {1991},
}

@Article{zantema1991minimizing,
  author =    {Zantema, Hans},
  title =     {Minimizing sums of addition chains},
  journal =   {Journal of Algorithms},
  year =      {1991},
  volume =    {12},
  number =    {2},
  pages =     {281--307},
  publisher = {Elsevier}
}

@PhdThesis{bos1992practical,
  author =    {Bos, Jurjen Norbert Eelco},
  title =     {Practical privacy},
  school =    {Eindhoven University of Technology},
  year =      {1992},
  publisher = {Citeseer}
}

@Article{brlek1992optimales,
  author  = {Brlek, S.;Mallette, R.},
  title   = {Sur le calcul des chaînes d'additions optimales},
  year    = {1992},
  pages   = {71-85},
  note    = {Publ. du LACIM 10, ISBN 2-89276-101-8},
  journal = {Atelier de combinatoire franco-québecois (6-7 Mai 1991, Bordeaux, France)},
}

@Article{gupta1992evaluation,
  author =    {Gupta, Omprakash K},
  title =     {On Evaluation of Powers},
  journal =   {Journal of Information and Optimization Sciences},
  year =      {1992},
  volume =    {13},
  number =    {2},
  pages =     {331--336},
  comment =   {Covering old ground},
  publisher = {Taylor \& Francis}
}

@InProceedings{sauerbrey1992resource,
  author =       {Sauerbrey, J{\"o}rg and Dietel, Andreas},
  title =        {Resource requirements for the application of addition chains in modulo exponentiation},
  booktitle =    {Workshop on the Theory and Application of of Cryptographic Techniques},
  year =         {1992},
  pages =        {174--182},
  organization = {Springer}
}

@InProceedings{Chin1992study,
  author    = {Y.H. Tsai and Y.H. Chin},
  booktitle = {Proceedings of the National Science Council - Part A: Physical Science and Engineering},
  title     = {A Study of Addition Chains},
  number    = {6},
  pages     = {506--514},
  volume    = {16},
  year      = {1992},
}

@Article{Tsai1992OnAddition,
  author   = {Y. H. Tsai and Y. H. Chin},
  title    = {On Addition Chains},
  doi      = {10.1080/00207169208804125},
  eprint   = {http://dx.doi.org/10.1080/00207169208804125},
  number   = {3-4},
  pages    = {145-160},
  volume   = {45},
  abstract = {This paper presents some mathematical properties of the shortest-length addition chain for a certain set of integers whose binary patterns meet a set of special forms; and the correctness of these properties is proved.},
  journal  = {International Journal of Computer Mathematics},
  year     = {1992},
}

@Article{aiello1993conjecture,
  author =  {Aiello, Walter and Subbarao, MV},
  title =   {A conjecture in addition chains related to Scholz’s conjecture},
  journal = {mathematics of computation},
  year =    {1993},
  volume =  {61},
  number =  {203},
  pages =   {17--23}
}

@TechReport{coster1993algorithm,
  author = {Coster, Matthijs Johannes and others},
  title =  {An algorithm on addition chains with restricted memory},
  year =   {1993}
}

@Article{thurber1993addition,
  author =    {Thurber, Edward G},
  title =     {Addition chains—an erratic sequence},
  journal =   {Discrete mathematics},
  year =      {1993},
  volume =    {122},
  number =    {1},
  pages =     {287--305},
  publisher = {Elsevier}
}

@Article{bergeron1994efficient,
  author =  {Bergeron, F and Berstel, Jean and Brlek, S},
  title =   {Efficient computation of addition chains},
  journal = {Journal de th{\'e}orie des nombres de Bordeaux},
  year =    {1994},
  volume =  {6},
  number =  {1},
  pages =   {21--38}
}

@Article{chen1994some,
  author =    {Yuh-Jiun Chen and Chin-Chen Chang and Wei-Pang Yang},
  title =     {Some properties of vectorial addition chains},
  journal =   {International Journal of Computer Mathematics},
  year =      {1994},
  volume =    {54},
  number =    {3-4},
  pages =     {185-196},
  note =      {Don't have this document yet},
  publisher = {Taylor \& Francis}
}

@InProceedings{de1994efficient,
  author =       {De Rooij, Peter},
  title =        {Efficient exponentiation using precomputation and vector addition chains},
  booktitle =    {Workshop on the Theory and Application of of Cryptographic Techniques},
  year =         {1994},
  pages =        {389--399},
  organization = {Springer}
}

@Article{dempster1994constant,
  author =    {Dempster, Andrew G and Macleod, MD},
  title =     {Constant integer multiplication using minimum adders},
  journal =   {IEE Proceedings-Circuits, Devices and Systems},
  year =      {1994},
  volume =    {141},
  number =    {5},
  pages =     {407--413},
  publisher = {IET}
}

@Article{gashkov1994addition,
  author  = {Gashkov, SB and Kochergin, VV},
  title   = {On addition chains of vectors, gate circuits and the complexity of computations of powers},
  year    = {1994},
  volume  = {4},
  number  = {4},
  pages   = {1--16},
  journal = {Syberian Advances in Mathematics},
}

@Article{kochergin1994computation,
  author  = {Kochergin, VV},
  title   = {On the computation of powers sets},
  year    = {1994},
  volume  = {4},
  number  = {2},
  pages   = {119--128},
  journal = {Discrete Mathematics and Applications},
}

@Article{brlek1995line,
  author =  {Brlek, Sre{\'c}ko and Cast{\'e}ran, Pierre and Habsieger, Laurent and Mallette, Richard},
  title =   {On-line evaluation of powers using Euclid's algorithm},
  journal = {Informatique th{\'e}orique et applications},
  year =    {1995},
  volume =  {29},
  number =  {5},
  pages =   {431--450}
}

@Article{dimitrov1995two,
  author    = {Dimitrov, Vassil and Cooklev, Todor},
  title     = {Two algorithms for modular exponentiation using nonstandard arithmetics},
  year      = {1995},
  volume    = {78},
  number    = {1},
  pages     = {82--87},
  journal   = {IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences},
  publisher = {The Institute of Electronics, Information and Communication Engineers},
}

@Article{Bocharova1995fast,
  author =  {I. Bocharova, B. Kudryashow},
  title =   {Fast Exponentiation in cryptography},
  journal = {Lecture Notes in C},
  year =    {1995},
  volume =  {948},
  pages =   {146--157}
}

@Article{kocc1995analysis,
  author    = {Koc, Çetin Kaya},
  title     = {Analysis of sliding window techniques for exponentiation},
  number    = {10},
  pages     = {17--24},
  volume    = {30},
  journal   = {Computers \& Mathematics with Applications},
  publisher = {Elsevier},
  year      = {1995},
}

@Article{wu1995strength,
  author =    {Wu, Youfeng},
  title =     {Strength reduction of multiplications by integer constants},
  journal =   {ACM SIGPLAN Notices},
  year =      {1995},
  volume =    {30},
  number =    {2},
  pages =     {42--48},
  publisher = {ACM}
}

@Thesis{bleichenbacher1996efficiency,
  author      = {Bleichenbacher, Daniel},
  institution = {Swiss Federal Institute of Technology Zürich, ETH Zürich},
  title       = {Efficiency and security of cryptosystems based on number theory},
  publisher   = {Citeseer},
  year        = {1996},
}

@Article{kochergin1996complexity,
  author  = {Kochergin, VV},
  title   = {On the complexity of computation of monomials and tuples of powers},
  year    = {1996},
  volume  = {6},
  number  = {1},
  pages   = {71--86},
  journal = {Siberian Advances in Mathematics},
}

@Article{lou1996fast,
  author    = {Lou, D-C and Chang, C-C},
  title     = {Fast exponentiation method obtained by folding the exponent in half},
  year      = {1996},
  volume    = {32},
  number    = {11},
  pages     = {984--985},
  journal   = {Electronics Letters},
  publisher = {IET},
}

@Unpublished{bleichenbacher1997efficient,
  author = {D. Bleichenbacher; A. Flammenkamp},
  title  = {An Efficient Algorithm for Computing Shortest Addition Chains},
  year   = {1997},
}

@TechReport{kunihiro1997optimal,
  author =      {Kunihiro, Noboru and Yamamoto, Hirosuke},
  title =       {Optimal addition chain classified by Hamming weight},
  institution = {IEICE Technical Report, ISEC96-74},
  year =        {1997}
}

@PhdThesis{bahig1998algorithms,
  author = {Bahig, HM},
  title =  {On the algorithms of computational number theory and their applications},
  school = {Master Thesis, Dept. of Math., Faculty of Science, Ain Shams University, Cairo, Egypt},
  year =   {1998}
}

@InProceedings{cohen1998improve,
  author =       {Cohen, G{\'e}rard and Naccache, David and Lobstein, Antoine and Z{\'e}mor, Gilles},
  title =        {How to improve an exponentiation black-box},
  booktitle =    {International Conference on the Theory and Applications of Cryptographic Techniques},
  year =         {1998},
  pages =        {211--220},
  organization = {Springer}
}

@Article{gordon1998survey,
  author =    {Gordon, Daniel M},
  title =     {A survey of fast exponentiation methods},
  journal =   {Journal of algorithms},
  year =      {1998},
  volume =    {27},
  number =    {1},
  pages =     {129--146},
  publisher = {Elsevier}
}

@Book{knuth1998art,
  author  = {Knuth, Donald E},
  title   = {The art of computer programming, 2: seminumerical algorithms, Addison Wesley},
  pages   = {461--485},
  journal = {Reading, MA},
  year    = {1998},
}

@Article{kunihiro1998window,
  author    = {Kunihiro, Noboru and Yamamoto, Hirosuke},
  title     = {Window and extended window methods for addition chain and addition-subtraction chain},
  number    = {1},
  pages     = {72--81},
  volume    = {E81-A},
  journal   = {IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences},
  publisher = {The Institute of Electronics, Information and Communication Engineers},
  year      = {1998},
}

@Article{lou1998adaptive,
  author =    {Lou, Der-Chyuan and Chang, Chin-Chen},
  title =     {An adaptive exponentiation method},
  journal =   {Journal of Systems and Software},
  year =      {1998},
  volume =    {42},
  number =    {1},
  pages =     {59--69},
  publisher = {Elsevier}
}

@Article{walter1998exponentiation,
  author =    {Walter, Colin D},
  title =     {Exponentiation using division chains},
  journal =   {IEEE Transactions on Computers},
  year =      {1998},
  volume =    {47},
  number =    {7},
  pages =     {757--765},
  publisher = {IEEE}
}

@Article{yacobi1998fast,
  author =    {Yacobi, Yacov},
  title =     {Fast exponentiation using data compression},
  journal =   {SIAM Journal on Computing},
  year =      {1998},
  volume =    {28},
  number =    {2},
  pages =     {700--703},
  publisher = {SIAM}
}

@TechReport{abrahams1999addition,
  author =      {Abrahams, Julia},
  title =       {Addition Chains as Test Trees and a Sequential Variant as the Huffman Problem},
  institution = {The Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)},
  year =        {1999},
  publisher =   {Citeseer}
}

@Article{bleichenbacher1999addition,
  author  = {Bleichenbacher, D},
  title   = {Addition chains for large sets},
  note    = {DONOTHAVE, author does not have it anymore},
  journal = {Unpublished manuscript},
  year    = {1999},
}

@Article{flammenkamp1999integers,
  author =    {Flammenkamp, Achim},
  title =     {Integers with a small number of minimal addition chains},
  journal =   {Discrete mathematics},
  year =      {1999},
  volume =    {205},
  number =    {1},
  pages =     {221--227},
  publisher = {Elsevier}
}

@Article{park1999analysis,
  author =    {Park, H and Park, K and Cho, Y},
  title =     {Analysis of the variable length nonzero window method for exponentiation},
  journal =   {Computers \& Mathematics with applications},
  year =      {1999},
  volume =    {37},
  number =    {7},
  pages =     {21--29},
  publisher = {Elsevier}
}

@Article{thurber1999efficient,
  author =  {Edward G. Thurber},
  title =   {Efficient Generation of Minimal Length Addition Chains},
  journal = {SIAM Journal on Computing},
  year =    {1999},
  volume =  {28},
  number =  {4},
  pages =   {1247--1263}
}

@Unpublished{coster2000brun1,
  author = {Coster, M.J.},
  title  = {The Brun Algorithm for Addition Chains},
  year   = {2000},
}

@Unpublished{coster2000brun,
  author = {Coster, M.J.},
  title  = {The Algorithm of Brun and Addition Chains},
  year   = {2000},
}

@Article{dimitrov2000complexity,
  author =    {Dimitrov, Vassil S and Jullien, Graham A and Miller, William C},
  title =     {Complexity and fast algorithms for multiexponentiations},
  journal =   {IEEE Transactions on Computers},
  year =      {2000},
  volume =    {49},
  number =    {2},
  pages =     {141--147},
  publisher = {Citeseer}
}

@Article{kunihiro2000new,
  author =    {Kunihiro, Noboru and Yamamoto, Hirosuke},
  title =     {New methods for generating short addition chains},
  journal =   {IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences},
  year =      {2000},
  volume =    {83},
  number =    {1},
  pages =     {60--67},
  publisher = {The Institute of Electronics, Information and Communication Engineers}
}

@InCollection{bahig2001some,
  author =    {Bahig, Hatem M and El-Zahar, Mohamed H and Nakamula, Ken},
  title =     {Some Results for Some Conjectures in Addition Chains},
  booktitle = {Combinatorics, Computability and Logic},
  publisher = {Springer},
  year =      {2001},
  pages =     {47--54}
}

@PhdThesis{otto2001brauer,
  author = {Otto, Martin},
  title =  {Brauer addition--subtraction chains},
  school = {Diplomarbeit, Universit{\"a}t--Gesamthochschule Paderborn},
  year =   {2001}
}

@Article{wang2001algorithm,
  author =    {Wang, X-d},
  title =     {An Algorithm for Generating the Shortest Addition Chains},
  journal =   {MINIMICRO SYSTEMS-SHENYANG-},
  year =      {2001},
  volume =    {22},
  number =    {10},
  pages =     {1250--1253},
  publisher = {GAI-KAN BIAJIBU}
}

@Article{bahig2002some,
  author =    {Bahig, Hatem M and Nakamula, Ken},
  title =     {Some properties of nonstar steps in addition chains and new cases where the Scholz conjecture is true},
  journal =   {Journal of Algorithms},
  year =      {2002},
  volume =    {42},
  number =    {2},
  pages =     {304--316},
  publisher = {Elsevier}
}

@Article{bernstein2002pippenger,
  author =    {Bernstein, Daniel J},
  title =     {Pippenger's exponentiation algorithm},
  year =      {2002},
  publisher = {Citeseer}
}

@PhdThesis{lehman2002approximation,
  author = {Lehman, Eric Eric Allen},
  title  = {Approximation algorithms for grammar-based data compression},
  pages  = {30--36},
  url    = {https://dspace.mit.edu/bitstream/handle/1721.1/87172/50504311-MIT.pdf;sequence=2},
  school = {Massachusetts Institute of Technology},
  year   = {2002},
}

@InProceedings{moller2002improved,
  author =       {M{\"o}ller, Bodo},
  title =        {Improved techniques for fast exponentiation},
  booktitle =    {International Conference on Information Security and Cryptology},
  year =         {2002},
  pages =        {298--312},
  organization = {Springer}
}

@InProceedings{nedjah2002minimal,
  author =       {Nedjah, Nadia and de Macedo Mourelle, Luiza},
  title =        {Minimal addition chain for efficient modular exponentiation using genetic algorithms},
  booktitle =    {International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems},
  year =         {2002},
  pages =        {88--98},
  organization = {Springer}
}

@Article{bahig2003erratum,
  author =    {Bahig, Hatem M and Nakamula, Ken},
  title =     {erratum to Some properties of nonstar steps in addition chains and new cases where the Scholz conjecture is true [J. algorithms 42 (2002) 304-316]},
  journal =   {Journal of Algorithms},
  year =      {2003},
  volume =    {47},
  number =    {1},
  pages =     {60--61},
  publisher = {Academic Press, Inc.}
}

@PhdThesis{stam2003speeding,
  author = {Stam, Martijn},
  title =  {Speeding up subgroup cryptosystems},
  school = {Technische Universiteit Eindhoven},
  year =   {2003}
}

@InProceedings{avanzi2004note,
  author =       {Avanzi, Roberto Maria},
  title =        {A note on the signed sliding window integer recoding and a left-to-right analogue},
  booktitle =    {International Workshop on Selected Areas in Cryptography},
  year =         {2004},
  pages =        {130--143},
  organization = {Springer}
}

@Book{guy2013unsolved,
  title =     {Unsolved problems in number theory},
  publisher = {Springer Science \& Business Media},
  year =      {2004},
  author =    {Guy, Richard},
  volume =    {1},
  edition =   {3},
  pages =     {169--171}
}

@InProceedings{nedjah2004finding,
  author       = {Nedjah, Nadia and de Macedo Mourelle, Luiza},
  title        = {Finding minimal addition chains using ant colony},
  booktitle    = {International Conference on Intelligent Data Engineering and Automated Learning},
  year         = {2004},
  organization = {Springer},
  pages        = {642--647},
}

@Article{parker2004addition,
  author =    {Parker, Richard and Plater, Andrew},
  title =     {Addition chains with a bounded number of registers},
  journal =   {Information processing letters},
  year =      {2004},
  volume =    {90},
  number =    {5},
  pages =     {247--252},
  publisher = {Elsevier}
}

@InProceedings{tummeltshammer2004multiple,
  author =       {Tummeltshammer, Peter and Hoe, James C and P{\"u}schel, Markus},
  title =        {Multiple constant multiplication by time-multiplexed mapping of addition chains},
  booktitle =    {Proceedings of the 41st annual Design Automation Conference},
  year =         {2004},
  pages =        {826--829},
  organization = {ACM}
}

@Article{von2004computing,
  author =  {von Zur Gathen, Joachim and N{\"o}cker, Michael},
  title =   {Computing special powers in finite fields},
  journal = {Mathematics of computation},
  year =    {2004},
  volume =  {73},
  number =  {247},
  pages =   {1499--1523}
}

@InProceedings{cruz2005finding,
  author =       {Cruz-Cort{\'e}s, Nareli and Rodr{\'\i}guez-Henr{\'\i}quez, Francisco and Ju{\'a}rez-Morales, Ra{\'u}l and Coello, Carlos A Coello},
  title =        {Finding optimal addition chains using a genetic algorithm approach},
  booktitle =    {International Conference on Computational and Information Science},
  year =         {2005},
  pages =        {208--215},
  organization = {Springer}
}

@Article{kochergin2005complexity,
  author    = {Kochergin, Vadim Vasil'evich},
  title     = {On the complexity of computation of a pair of monomials in two variables},
  number    = {6},
  pages     = {547--572},
  volume    = {15},
  journal   = {Discrete Mathematics and Applications},
  publisher = {Springer},
  year      = {2005},
}

@InProceedings{nedjah2005efficient,
  author =       {Nedjah, Nadia and de Macedo Mourelle, Luiza},
  title =        {Efficient pre-processing for large window-based modular exponentiation using ant colony},
  booktitle =    {International Conference on Knowledge-Based and Intelligent Information and Engineering Systems},
  year =         {2005},
  pages =        {640--646},
  organization = {Springer}
}

@InProceedings{park2005new,
  author =       {Park, Chi Seong and Lee, Mun-Kyu and Kim, Dong Kyue},
  title =        {New computation paradigm for modular exponentiation using a graph model},
  booktitle =    {International Symposium on Stochastic Algorithms},
  year =         {2005},
  pages =        {170--179},
  organization = {Springer}
}

@InProceedings{arpe2006approximability,
  author =       {Arpe, Jan and Manthey, Bodo},
  title =        {Approximability of minimum and-circuits},
  booktitle =    {Scandinavian Workshop on Algorithm Theory},
  year =         {2006},
  pages =        {292--303},
  organization = {Springer}
}

@InProceedings{bahig2006number,
  author =    {Bahig, Hatem M},
  title =     {On the Number of Minimal Addition Chains.},
  booktitle = {CSC},
  year =      {2006},
  pages =     {202--208}
}

@Article{bahig2006improved,
  author =    {Bahig, Hatem M},
  title =     {Improved generation of minimal addition chains},
  journal =   {Computing},
  year =      {2006},
  volume =    {78},
  number =    {2},
  pages =     {161--172},
  publisher = {Springer}
}

@InProceedings{bahig2006speeding,
  author =    {Bahig, Hatem M and Bahig, Hazem M},
  title =     {Speeding Up Evaluation of Powers and Monomials.},
  booktitle = {FCS},
  year =      {2006},
  pages =     {149--153}
}

@WWW{bernstein2006transposition,
  author = {Bernstein, Daniel J},
  title  = {The transposition principle},
  year   = {2006},
  url    = {https://cr.yp.to/transposition.html},
}

@Article{bernstein2006differential,
  author =  {Bernstein, Daniel J},
  title =   {Differential addition chains},
  journal = {URL: http://cr. yp. to/ecdh/diffchain-20060219. pdf},
  year =    {2006}
}

@Article{gashkov2006application,
  author =  {Gashkov, Sergey Borisovich and Sergeev, Igor Sergeevich},
  title =   {An application of the method of additive chains to inversion in finite fields},
  journal = {Discrete Mathematics and Applications dma},
  year =    {2006},
  volume =  {16},
  number =  {6},
  pages =   {601--618}
}

@InProceedings{gelgi2006heuristics,
  author =       {Gelgi, Fatih and Onus, Melih},
  title =        {Heuristics for minimum brauer chain problem},
  booktitle =    {International Symposium on Computer and Information Sciences},
  year =         {2006},
  pages =        {47--54},
  organization = {Springer}
}

@Conference{kim2006computation,
  author  = {Kim, Dong Kyue and Lee, Mun-Kyu},
  title   = {A COMPUTATION MODEL FOR MODULAR EXPONENTIATION USING GRAPH REPRESENTATION},
  pages   = {487--490},
  journal = {International Conference on Multimedia Information Technology and Applications MITA 2006},
  year    = {2006},
}

@Article{nedjah2006towards,
  author    = {Nedjah, Nadia and de Macedo Mourelle, Luiza},
  title     = {Towards minimal addition chains using ant colony optimisation},
  year      = {2006},
  volume    = {5},
  number    = {4},
  pages     = {525--543},
  journal   = {Journal of Mathematical Modelling and Algorithms},
  publisher = {Springer},
}

@InCollection{nedjah2006ant,
  author =    {Nedjah, Nadia and de Macedo Mourelle, Luiza},
  title =     {Ant colony optimisation for fast modular exponentiation using the sliding window method},
  booktitle = {Swarm Intelligent Systems},
  publisher = {Springer},
  year =      {2006},
  pages =     {133--147}
}

@Article{byrne2007spa,
  author =    {Byrne, Andrew and Crowe, Francis and Marnane, William Peter and Meloni, Nicolas and Tisserand, Arnaud and Popovici, Emanuel},
  title =     {SPA resistant elliptic curve cryptosystem using addition chains},
  journal =   {International Journal of High Performance Systems Architecture},
  year =      {2007},
  volume =    {1},
  number =    {2},
  pages =     {133--142},
  publisher = {Inderscience Publishers}
}

@Article{goundar2007novel,
  author    = {Goundar, Raveen R},
  title     = {A Novel Multi Exponentiation Method},
  number    = {2},
  pages     = {89--95},
  volume    = {37},
  journal   = {International Journal of Applied Mathematics},
  publisher = {Citeseer},
  year      = {2007},
}

@Article{tummeltshammer2007time,
  author    = {Tummeltshammer, Peter and Hoe, James C and Puschel, Markus},
  title     = {Time-multiplexed multiple-constant multiplication},
  number    = {9},
  pages     = {1551--1563},
  volume    = {26},
  journal   = {IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems},
  publisher = {IEEE},
  year      = {2007},
}

@Thesis{van2007addition,
  author  = {Van-Der Kruijssen, Sander},
  title   = {Addition Chains, efficient computing of powers},
  pages   = {13--50},
  journal = {Bachelor Project},
  school  = {Amsterdam},
  volume  = {1},
  year    = {2007},
}

@Article{bahig2008generalization,
  author =    {Bahig, Hatem M},
  title =     {On a generalization of addition chains: Addition--multiplication chains},
  journal =   {Discrete Mathematics},
  year =      {2008},
  volume =    {308},
  number =    {4},
  pages =     {611--616},
  publisher = {Elsevier}
}

@Article{cruz2008artificial,
  author =    {Cruz-Cort{\'e}s, Nareli and Rodr{\'\i}guez-Henr{\'\i}quez, Francisco and Coello, Carlos A Coello},
  title =     {An artificial immune system heuristic for generating short addition chains},
  journal =   {IEEE Transactions on Evolutionary Computation},
  year =      {2008},
  volume =    {12},
  number =    {1},
  pages =     {1--24},
  publisher = {IEEE}
}

@InProceedings{fu2008novel,
  author  = {Fu-guo, Dong and Yu-rong, Li},
  title   = {A novel shortest addition chains algorithm based on Euclid algorithm},
  year    = {2008},
  note    = {retracted},
  pages   = {4757--4761},
  journal = {Proceeding of WiCom (2008 4th International Conference on Wireless Communications, Networking and Mobile Computing)},
}

@Article{goundar2008spa,
  author =  {Goundar, R and Shiota, Ken-ichi and Toyonaga, Masahiko},
  title =   {SPA resistant scalar multiplication using golden ratio addition chain method},
  journal = {International Journal of Applied Mathematics},
  year =    {2008},
  volume =  {38},
  number =  {2},
  pages =   {83--88}
}

@Article{arpe2009approximability,
  author =    {Arpe, Jan and Manthey, Bodo},
  title =     {Approximability of minimum AND-circuits},
  journal =   {Algorithmica},
  year =      {2009},
  volume =    {53},
  number =    {3},
  pages =     {337--357},
  publisher = {Springer}
}

@Misc{johansen2009exponentiation,
  author = {Johansen, Martin M},
  title  = {Exponentiation with addition chains},
  year   = {2009},
}

@InProceedings{leon2009finding,
  author       = {Le{\'o}n-Javier, Alejandro and Cruz-Cort{\'e}s, Nareli and Moreno-Armend{\'a}riz, Marco A and Orantes-Jim{\'e}nez, Sandra},
  title        = {Finding minimal addition chains with a particle swarm optimization algorithm},
  booktitle    = {Mexican International Conference on Artificial Intelligence},
  year         = {2009},
  organization = {Springer},
  pages        = {680--691},
}

@InProceedings{osorio2009genetic,
  author =       {Osorio-Hern{\'a}ndez, Luis G and Mezura-Montes, Efr{\'e}n and Cruz-Cort{\'e}s, Nareli and Rodr{\'\i}guez-Henr{\'\i}quez, Francisco},
  title =        {A genetic algorithm with repair and local search mechanisms able to find minimal length addition chains for small exponents},
  booktitle =    {2009 IEEE Congress on Evolutionary Computation},
  year =         {2009},
  pages =        {1422--1429},
  organization = {IEEE}
}

@Article{dong2010consistency,
  author =  {Dong, Fu-guo and Li, Yu-rong and Li, Jin-jiang},
  title =   {The Consistency Analysis of Addition Chains for Several Fast Algorithm of Modular Exponentiation.},
  journal = {JDCTA},
  year =    {2010},
  volume =  {4},
  number =  {5},
  pages =   {82--88}
}

@TechReport{henry2010pippenger,
  author =      {Henry, Ryan},
  title =       {Pippenger’s Multiproduct and Multiexponentiation Algorithms},
  institution = {Citeseer},
  year =        {2010}
}

@InProceedings{herbaut2010random,
  author =       {Herbaut, Fabien and Liardet, Pierre-Yvan and M{\'e}loni, Nicolas and Teglia, Yannick and V{\'e}ron, Pascal},
  title =        {Random Euclidean addition chain generation and its application to point multiplication},
  booktitle =    {International Conference on Cryptology in India},
  year =         {2010},
  pages =        {238--261},
  organization = {Springer}
}

@InProceedings{herbaut2010public,
  author =       {Herbaut, Fabien and V{\'e}ron, Pascal},
  title =        {A public key cryptosystem based upon euclidean addition chains},
  booktitle =    {International Conference on Sequences and Their Applications},
  year =         {2010},
  pages =        {284--297},
  organization = {Springer}
}

@InProceedings{li2010design,
  author =       {Li, Yiling and Ma, Qingyu},
  title =        {Design and Implementation of Layer Extended Shortest Addition Chains Database for Fast Modular Exponentiation in RSA},
  booktitle =    {Web Information Systems and Mining (WISM), 2010 International Conference on},
  year =         {2010},
  volume =       {1},
  pages =        {136--139},
  organization = {IEEE}
}

@Patent{mityagin2010systems,
  year =      {2010},
  author =    {Mityagin, Anton and Mironov, Ilya and Kobliner, Yaacov Nissim},
  title =     {Systems and methods for generating random addition chains},
  month =     feb #{~2},
  note =      {US Patent 7,657,029},
  publisher = {Google Patents}
}

@Article{ragnarsson2010obtainable,
  author    = {Ragnarsson, K{\'a}ri and Tenner, Bridget Eileen},
  title     = {Obtainable sizes of topologies on finite sets},
  year      = {2010},
  volume    = {117},
  number    = {2},
  pages     = {138--151},
  journal   = {Journal of Combinatorial Theory, Series A},
  publisher = {Elsevier},
}

@Article{bahig2011star,
  author =    {Bahig, Hatem M},
  title =     {Star reduction among minimal length addition chains},
  journal =   {Computing},
  year =      {2011},
  volume =    {91},
  number =    {4},
  pages =     {335--352},
  publisher = {Springer}
}

@Article{bahig2011new,
  author    = {Bahig, Hatem M and Bahig, Hazem M},
  title     = {A new strategy for generating shortest addition sequences},
  year      = {2011},
  volume    = {91},
  number    = {3},
  pages     = {285--306},
  journal   = {Computing},
  publisher = {Springer},
}

@Article{Clift:2011:COA:1966629.1966632,
  author     = {Clift, Neill Michael},
  title      = {Calculating Optimal Addition Chains},
  journal    = {Computing},
  year       = {2011},
  volume     = {91},
  number     = {3},
  month      = mar,
  pages      = {265--284},
  issn       = {0010-485X},
  doi        = {10.1007/s00607-010-0118-8},
  acmid      = {1966632},
  address    = {New York, NY, USA},
  issue_date = {March 2011},
  keywords   = {Exponentiation, Optimization, Scholz\&\#x2013;Brauer, Sequences},
  numpages   = {20},
  publisher  = {Springer-Verlag New York, Inc.},
}

@PhdThesis{dominguez2011optimizacion,
  author = {Dom{\'\i}nguez, I},
  title  = {Optimizaci{\'o}n de cadenas de adici{\'o}n en criptograf{\'\i}a utilizando programaci{\'o}n evolutiva},
  school = {Tesis de Maestr{\'\i}a. Laboratorio Nacional de Inform{\'a}tica Avanzada Centro de~…},
  year   = {2011},
}

@InProceedings{dominguez2011evolutionary,
  author =    {Dom{\'\i}nguez-Isidro, S and Mezura-Montes, E},
  title =     {An Evolutionary Programming Algorithm to Find Minimal Addition Chains},
  booktitle = {I Congreso Internacional de Ingenier{\'\i}a Electr{\'o}nica, Instrumentaci{\'o}n y Computaci{\'o}n, de Junio del, Minatitl{\'a}n Veracruz, M{\'e}xico},
  year =      {2011}
}

@InProceedings{dominguez2011addition,
  author =       {Dom{\'\i}nguez-Isidro, Sa{\'u}l and Mezura-Montes, Efr{\'e}n and Osorio-Hern{\'a}ndez, Luis Guillermo},
  title =        {Addition chain length minimization with evolutionary programming},
  booktitle =    {Proceedings of the 13th annual conference companion on Genetic and evolutionary computation},
  year =         {2011},
  pages =        {59--60},
  organization = {ACM}
}

@MastersThesis{elias2011representation,
  author = {Elias, Yara},
  title =  {Repr{\'e}sentation d'un polyn{\^o}me par un circuit arithm{\'e}tique et cha{\^\i}nes additives},
  school = {Université de Montréal},
  year =   {2011}
}

@Article{fuguo2011addition,
  author  = {Fuguo, Dong and Pingqin, Wang and Lei, Feng},
  title   = {Addition Chains Algorithm Based on Fast Fourier Transform},
  year    = {2011},
  volume  = {5},
  number  = {2},
  pages   = {149--157},
  journal = {JDCTA: International Journal of Digital Content Technology and its Applications},
}

@Article{Gashkov2011Problem,
  author    = {Gashkov, Sergey Borisovich},
  title     = {The problem of additive chains and its generalization},
  pages     = {138--153},
  url       = {https://www.mccme.ru/free-books/matpros/mpf.pdf},
  volume    = {15},
  journal   = {Mathematical Education},
  publisher = {Independent University of Moscow - IUM},
  year      = {2011},
}

@InProceedings{jose2011simulated,
  author =       {Jose-Garcia, Adan and Romero-Monsivais, Hillel and Hernandez-Morales, Cindy G and Rodriguez-Cristerna, Arturo and Rivera-Islas, Ivan and Torres-Jimenez, Jose},
  title =        {A simulated annealing algorithm for the problem of minimal addition chains},
  booktitle =    {Portuguese Conference on Artificial Intelligence},
  year =         {2011},
  pages =        {311--325},
  organization = {Springer}
}

@Article{mignotte2011note,
  author =  {Mignotte, Maurice and Tall, Amadou},
  title =   {A note on addition chains},
  journal = {International Journal of Algebra},
  year =    {2011},
  volume =  {5},
  number =  {6},
  pages =   {269--274}
}

@PhdThesis{mohamed2011improvement,
  author = {Mohamed, Mohamad Afendee},
  title =  {On the Improvement of Addition Chain in Applications to Elliptic Curve Cryptosystem Status: Submitted},
  school = {Universiti Putra Malaysia},
  year =   {2011},
  note =   {Don't have this document yet}
}

@Article{mohamed2011shorter,
  author =    {Mohamed, MA and Md Said, MR and Mohd Atan, KA and Ahmad Zulkarnain, Z},
  title =     {Shorter addition chain for smooth integers using decomposition method},
  journal =   {International Journal of Computer Mathematics},
  year =      {2011},
  volume =    {88},
  number =    {11},
  pages =     {2222--2232},
  publisher = {Taylor \& Francis}
}

@InProceedings{rodriguez2011mutation,
  author =       {Rodriguez-Cristerna, Arturo and Torres-Jim{\'e}nez, Jos{\'e} and Rivera-Islas, Ivan and Hernandez-Morales, Cindy G and Romero-Monsivais, Hillel and Jose-Garcia, Adan},
  title =        {A mutation-selection algorithm for the problem of minimum brauer chains},
  booktitle =    {Mexican International Conference on Artificial Intelligence},
  year =         {2011},
  pages =        {107--118},
  organization = {Springer}
}

@InCollection{Samovol2011FasterO,
  author       = {Peter Samovol and Valery Zhuravlev},
  title        = {Faster than the Fastest, or Can the Binary Algorithm be Overhauled},
  number       = {1},
  pages        = {37--58},
  publisher    = {Australia AMT Publishing},
  url          = {http://www.wfnmc.org/Journal 2011 1.pdf},
  volume       = {24},
  journaltitle = {Journal of the World Federation of National Mathematics Competitions},
  year         = {2011},
}

@PhdThesis{abbas2012implementation,
  author =    {Abbas, Muhammad},
  title =     {On the implementation of integer and non-integer sampling rate conversion},
  school =    {Linköping University},
  year =      {2012},
  publisher = {Link{\"o}ping University Electronic Press}
}

@Unpublished{abbas2012integer,
  author = {Abbas, Muhammad and Gustafsson, Oscar},
  title  = {Integer Linear Programming Modeling of Addition Sequences With Additional Constraints for Evaluation of Power Terms},
  year   = {2012},
}

@Article{mohamed2012rule,
  author =  {Mohamed, MA and Atan, KA Mohd},
  title =   {Rule Based Representation of Integer for a New Addition Chain Method},
  journal = {Applied Mathematical Sciences},
  year =    {2012},
  volume =  {6},
  number =  {30},
  pages =   {1497--1503}
}

@InProceedings{walter2012duality,
  author =       {Walter, Colin D},
  title =        {A duality in space usage between left-to-right and right-to-left exponentiation},
  booktitle =    {Cryptographers’ Track at the RSA Conference},
  year =         {2012},
  pages =        {84--97},
  organization = {Springer}
}

@Article{mani2013generation,
  author  = {Mani, Mr K},
  title   = {Generation of Addition Chain using Deterministic Division Based Method},
  number  = {4},
  pages   = {553--560},
  volume  = {1},
  journal = {International Journal of Computer Science \& Engineering Technology},
  year    = {2013},
}

@InCollection{rodriguez2013genetic,
  author =    {Rodriguez-Cristerna, Arturo and Torres-Jimenez, Jose},
  title =     {A genetic algorithm for the problem of minimal brauer chains for large exponents},
  booktitle = {Soft Computing Applications in Optimization, Control, and Recognition},
  publisher = {Springer},
  year =      {2013},
  pages =     {27--51}
}

@InCollection{Rodriguez-CristernaTorres-Jimenez2013,
  author =    {Rodriguez-Cristerna, Arturo and Torres-Jimenez, Jose},
  title =     {A genetic algorithm for the problem of minimal brauer chains},
  booktitle = {Recent Advances on Hybrid Intelligent Systems},
  publisher = {Springer},
  year =      {2013},
  pages =     {481--500}
}

@Article{vallejo2013scholz,
  author    = {Vallejo, Jos{\'e} Maclovio Sautto and Moreno, Agust{\'\i}n Santiago and Herrera, Carlos N Bouza and Guzm{\'a}n, Ver{\'o}nica Campos},
  title     = {Scholz’s Third Conjecture: A Demonstration for Star Addition Chains},
  number    = {10},
  pages     = {1--2},
  volume    = {4},
  comment   = {Covering old ground in a suspect journal},
  journal   = {Applied Mathematics},
  publisher = {Scientific Research Publishing},
  year      = {2013},
}

@Article{wang2013efficient,
  author  = {Wang, Xiaodong},
  title   = {An Efficient Algorithm for Finding Optimal Addition Chains},
  year    = {2013},
  volume  = {11},
  number  = {11},
  pages   = {6447--6453},
  journal = {Indonesian Journal of Electrical Engineering and Computer Science},
}

@Article{Zhuravlev2013Faster,
  author  = {Valery M. Zhuravlev and Peter I. Samovol},
  title   = {Faster than fast, or can one beat the binary algorithm?},
  pages   = {7--15},
  volume  = {2},
  journal = {Kvant},
  year    = {2013},
}

@Article{altman2014internal,
  author =  {Altman, Harry},
  title =   {Internal Structure of Addition Chains: Well-Ordering},
  journal = {arXiv preprint arXiv:1409.1627},
  year =    {2014}
}

@PhdThesis{altman2014integer,
  author = {Altman, Harry J},
  title =  {Integer complexity, addition chains, and well-ordering},
  school = {University of Michigan},
  year =   {2014}
}

@Article{kohonen2014addition,
  author  = {Kohonen, Jukka and Corander, Jukka},
  title   = {Addition chains meet postage stamps: Reducing the number of multiplications},
  number  = {2},
  pages   = {1--13},
  volume  = {17},
  journal = {Journal of Integer Sequences},
  year    = {2014},
}

@Article{кочергин2014уточнение,
  author =    {Кочергин, Вадим Васильевич},
  title =     {Уточнение оценок сложности вычисления одночленов и наборов степеней в задачах Беллмана и Кнута},
  journal =   {Дискретный анализ и исследование операций},
  year =      {2014},
  volume =    {21},
  number =    {6},
  pages =     {51--72},
  publisher = {Институт математики им. СЛ Соболева Сибирского отделения РАН}
}

@Article{dominguez2015evolutionary,
  author =    {Dom{\'\i}nguez-Isidro, Sa{\'u}l and Mezura-Montes, Efr{\'e}n and Osorio-Hern{\'a}ndez, Luis-Guillermo},
  title =     {Evolutionary programming for the length minimization of addition chains},
  journal =   {Engineering Applications of Artificial Intelligence},
  year =      {2015},
  volume =    {37},
  pages =     {125--134},
  publisher = {Elsevier}
}

 
@Article{Kochergin2015,
  author       = {Kochergin, V. V.},
  date         = {2015-01},
  journaltitle = {Journal of Applied and Industrial Mathematics},
  title        = {Improvement of the estimates of the computational complexity for monomials and sets of powers in Bellman’s and Knuth’s problems},
  doi          = {10.1134/s1990478915010081},
  issn         = {1990-4797},
  number       = {1},
  pages        = {68--82},
  volume       = {9},
  publisher    = {Pleiades Publishing Ltd},
}

@Article{mohamed2015hybrid,
  author =    {Mohamed, Mohamad Afendee and Said, Md and Rushdan, Mohamad},
  title =     {A hybrid addition chain method for faster scalar multiplication},
  journal =   {WSEAS Transactions on Communications},
  year =      {2015},
  volume =    {14},
  pages =     {144--152},
  publisher = {World Scientific and Engineering Academy and Society}
}

@Article{enge2016short,
  author =  {Enge, Andreas and Hart, William and Johansson, Fredrik},
  title =   {Short addition sequences for theta functions},
  journal = {arXiv preprint arXiv:1608.06810},
  year =    {2016}
}

@InProceedings{koziel2016fast,
  author       = {Koziel, Brian and Azarderakhsh, Reza and Jao, David and Mozaffari-Kermani, Mehran},
  booktitle    = {International Conference on Information Security and Cryptology: 12th International Conference, Inscrypt 2016, Beijing, China, November 4-6, 2016, Revised Selected Papers},
  title        = {On Fast Calculation of Addition Chains for Isogeny-Based Cryptography},
  organization = {Springer},
  pages        = {323--342},
  year         = {2016},
}

@InProceedings{meloni2016random,
  author       = {M{\'e}loni, Nicolas and Hasan, M Anwar},
  booktitle    = {2016 IEEE 23nd Symposium on Computer Arithmetic (ARITH)},
  title        = {Random Digit Representation of Integers},
  organization = {IEEE},
  pages        = {118--125},
  year         = {2016},
}

@Article{AdamuMuhammadNoma2016,
  author  = {Adamu Muhammad Noma and Mohamad Afendee Mohamed and Abdullah Muhammed and Zuriati Ahmad Zulkarnain},
  title   = {An Initial Solution for Addition Chain Optimization Problem},
  issue   = {3},
  number  = {11},
  pages   = {640-643},
  journal = {Journal of Engineering and Applied Sciences},
  year    = {2016},
}

@InProceedings{picek2016evolutionary,
  author       = {Picek, Stjepan and Coello, Carlos A Coello and Jakobovic, Domagoj and Mentens, Nele},
  booktitle    = {The 16th European Conference on Evolutionary Computation in Combinatorial Optimization, Porto, Portugal},
  title        = {Evolutionary algorithms for finding short addition chains: Going the distance},
  organization = {Springer},
  pages        = {121--137},
  year         = {2016},
}

@Article{sautto2016scholz,
  author    = {Sautto, Jos{\'e} M and Santiago, Agust{\'\i}n and Bouza, Carlos N and Campos, Ver{\'o}nica},
  title     = {Scholz’s First Conjecture: A Brief Demonstration},
  number    = {01},
  pages     = {70--76},
  volume    = {7},
  comment   = {Covering old ground in a suspect journal},
  journal   = {Applied Mathematics},
  publisher = {Scientific Research Publishing},
  year      = {2016},
}

@InProceedings{vazquez2016genetic,
  author       = {V{\'a}zquez-Fern{\'a}ndez, Eduardo and Cadena, Carlos and Reyes-G{\'o}mez, David A},
  title        = {A genetic algorithm with a mutation mechanism based on a Gaussian and uniform distribution to minimize addition chains for small exponents},
  booktitle    = {Evolutionary Computation (CEC), 2016 IEEE Congress on},
  year         = {2016},
  organization = {IEEE},
  pages        = {935--940},
}

@Conference{aquino2017optimization,
  author    = {Fernando Aquino and Guillermo Leguizam{\'o}n},
  booktitle = {Computer Science \& Technology Series},
  title     = {Optimization of addition chains},
  location  = {San Luis, Argentinia},
  subtitle  = {XXII Argentine Congress of Computer Science, Selected Papers},
  year      = {2017},
}

@Article{kochergin2017improvement,
  author    = {Kochergin, Vadim Vasil'evich and Kochergin, Dmitrii Vadimovich},
  title     = {Improvement of the lower bound for the complexity of exponentiation},
  doi       = {10.17223/20710410/38/10},
  number    = {38},
  pages     = {119--132},
  journal   = {Prikladnaya diskretnaya matematika},
  month     = {dec},
  publisher = {Tomsk State University},
  year      = {2017},
}

@Article{mani2017new,
  author  = {Mani, K and Viswambari, M},
  title   = {A New Method of Generating Optimal Addition Chain Based on Graph},
  journal = {International Journal of Mathematical Sciences and Computing},
  year    = {2017},
  volume  = {3},
  number  = {2},
  pages   = {37--54},
  doi     = {10.5815/ijmsc.2017.02.04},
}

@Article{Noma2017,
  author       = {Adamu Muhammad Noma and Abdullah Muhammed and Mohamad Afendee Mohamed and Zuriati Ahmad Zulkarnain},
  date         = {2017-08},
  journaltitle = {Journal of Computer Science},
  title        = {A Review on Heuristics for Addition Chain Problem: Towards Efficient Public Key Cryptosystems},
  doi          = {10.3844/jcssp.2017.275.289},
  number       = {8},
  pages        = {275--289},
  volume       = {13},
  publisher    = {Science Publications},
}

@Thesis{uvzarevic2017postupci,
  author = {U{\v{z}}arevi{\'c}, Josip},
  title  = {Postupci konstrukcije i optimizacije adicijskih lanaca},
  school = {University of Zagreb. Faculty of Electrical Engineering and Computing.},
  year   = {2017},
}

@Article{bahig2018fast,
  author    = {Bahig, Hazem M},
  title     = {A fast optimal parallel algorithm for a short addition chain},
  number    = {1},
  pages     = {324--333},
  volume    = {74},
  journal   = {The Journal of Supercomputing},
  publisher = {Springer},
  year      = {2018},
}

@Article{Bahig2018,
  author    = {Hatem M. Bahig and Khaled A. AbdElbari},
  title     = {A fast {GPU}-based hybrid algorithm for addition chains},
  doi       = {10.1007/s10586-018-2840-5},
  pages     = {1--11},
  volume    = {21(12)},
  journal   = {Cluster Computing},
  month     = {dec},
  publisher = {Springer Nature America, Inc},
  year      = {2018},
}

@InCollection{Kadir2018,
  author    = {M. F. A. Kadir and M. A. Mohamed and R. Mohamad and M. Mamat and A. Muhammed},
  title     = {Performance Comparison of Some Addition Chain Methods Based on Integer Family},
  booktitle = {Information Science and Applications 2018},
  year      = {2018},
  note      = {DO NOT HAVE},
  publisher = {Springer Singapore},
  pages     = {211--218},
  doi       = {10.1007/978-981-13-1056-0_22},
  month     = {jul},
}

@Article{olguin2018minimum,
  author  = {Olguin Carbajal, Mauricio and Herrera-Lozada, Juan Carlos and Rivera-Z{\'a}rate, Israel and Serrano-Talamantes, J Felix and Cadena-Mart{\'\i}nez, Rodrigo and V{\'a}squez-G{\'o}mez, J Irving},
  title   = {Minimum Addition Chains Generation Using Evolutionary Strategies},
  number  = {4},
  pages   = {1463--1472},
  volume  = {22},
  journal = {Computaci{\'o}n y Sistemas},
  year    = {2018},
}

@Article{Picek2018,
  author    = {Stjepan Picek and Carlos A. Coello Coello and Domagoj Jakobovic and Nele Mentens},
  title     = {Finding short and implementation-friendly addition chains with evolutionary algorithms},
  journal   = {Journal of Heuristics},
  year      = {2018},
  volume    = {24},
  issue     = {3},
  month     = {jun},
  pages     = {457-481},
  issn      = {1381-1231},
  doi       = {10.1007/s10732-017-9340-2},
  url       = {https://doi.org/10.1007/s10732-017-9340-2},
  publisher = {Springer Nature},
}

@Article{Bahig2019,
  author    = {Hazem Bahig and Yasser Kotb},
  title     = {An Efficient Multicore Algorithm for Minimal Length Addition Chains},
  doi       = {10.3390/computers8010023},
  number    = {23},
  pages     = {1--12},
  volume    = {8},
  journal   = {Computers},
  month     = {mar},
  publisher = {{MDPI} {AG}},
  year      = {2019},
}

@Article{Bahig_2019,
  author       = {Hazem M Bahig},
  date         = {2019},
  journaltitle = {Journal of Computers},
  title        = {A Multicore Exact Algorithm for Addition Sequence},
  doi          = {10.17706/jcp.14.1.79-87},
  pages        = {79--87},
  publisher    = {International Academy Publishing ({IAP})},
}

@Article{gorodilova2019problems,
  author    = {Gorodilova, Anastasiya and Agievich, Sergey and Carlet, Claude and Gorkunov, Evgeny and Idrisova, Valeriya and Kolomeec, Nikolay and Kutsenko, Alexandr and Nikova, Svetla and Oblaukhov, Alexey and Picek, Stjepan and others},
  title     = {Problems and solutions from the fourth International Students’ Olympiad in Cryptography (NSUCRYPTO)},
  number    = {2},
  pages     = {138--174},
  volume    = {43},
  journal   = {Cryptologia},
  publisher = {Taylor \& Francis},
  year      = {2019},
}

@Article{sanchez2019simulated,
  author  = {Sanchez, Silvestre Ascencion Garcia and Osorno, Luis Calderon Luis Calderon and Camarillo, Edmundo Rene Duran},
  title   = {Simulated Annealing Meta-heuristic for Addition Chain Optimization},
  journal = {European Journal of Electrical Engineering and Computer Science},
  year    = {2019},
  volume  = {3},
  number  = {6},
}

@Article{Bahig2020,
  author       = {Hazem M. Bahig and Khaled A. and Mohammed A. and Amer AlGhadhban and Hatem M.},
  date         = {2020},
  journaltitle = {International Journal of Advanced Computer Science and Applications},
  title        = {An Evolutionary Algorithm for Short Addition Chains},
  doi          = {10.14569/ijacsa.2020.0111258},
  number       = {12},
  volume       = {11},
  publisher    = {The Science and Information Organization},
}

 
@Article{Kotochigov2020,
  author       = {Kotochigov, Aleksandr M. and Suchkov, Andrei I.},
  date         = {2020-03},
  journaltitle = {Computer tools in education},
  title        = {A Method for Reducing Iteration in Algorithms for Building Minimal Additive Chains},
  doi          = {10.32603/2071-2340-2020-1-5-18},
  issn         = {2071-2359},
  number       = {1},
  pages        = {5--18},
  publisher    = {St. Petersburg Electrotechnical University LETI},
}

@Thesis{vanrelative,
  author = {van Buul, Jasper},
  date   = {2021},
  title  = {The relative efficiency of simple addition chain algorithms},
  type   = {Bachelor Thesis},
}

@Article{Mani2021,
  author       = {K, Mani and A, Mullai},
  date         = {2021-02},
  journaltitle = {International Journal of Engineering Trends and Technology},
  title        = {Generation of Addition Chain using Bacteria Foraging Optimization Algorithm},
  doi          = {10.14445/22315381/ijett-v69i2p205},
  issn         = {2231-5381},
  number       = {2},
  pages        = {32--38},
  volume       = {69},
  publisher    = {Seventh Sense Research Group Journals},
}

 
@InProceedings{Smirnov2021,
  author    = {Smirnov, Ivan A. and Cherckesova, Larissa V. and Safaryan, Olga A. and Korochentsev, Denis A. and Chumakov, Vladislav E. and Gavlicky, Alexandr I.},
  booktitle = {2021 IEEE East-West Design & Test Symposium (EWDTS)},
  date      = {2021-09},
  title     = {Development of Fast Exponentiation Algorithm «To Center and Back»},
  doi       = {10.1109/ewdts52692.2021.9580999},
  publisher = {IEEE},
}

@Article{Thurber2021,
  author    = {Edward G. Thurber and Neill M. Clift},
  title     = {Addition chains, vector chains, and efficient computation},
  doi       = {10.1016/j.disc.2020.112200},
  number    = {2},
  pages     = {112200},
  volume    = {344},
  journal   = {Discrete Mathematics},
  month     = {feb},
  publisher = {Elsevier {BV}},
  year      = {2021},
}

@Article{Bahig2022a,
  author       = {Hatem M. Bahig},
  date         = {2022-05},
  journaltitle = {Notes on Number Theory and Discrete Mathematics},
  title        = {A note on the Aiello{\textendash}Subbarao conjecture on addition chains},
  doi          = {10.7546/nntdm.2022.28.2.276-280},
  number       = {2},
  pages        = {276--280},
  volume       = {28},
  publisher    = {Prof. Marin Drinov Publishing House of {BAS} (Bulgarian Academy of Sciences)},
}

@Article{Bahig2022,
  author       = {Bahig, Hatem M and Nassr, Dieaa I},
  date         = {2022},
  journaltitle = {Information Sciences Letters},
  title        = {Generating a Shortest B-Chain using Multi-GPUs},
  number       = {3},
  pages        = {745--750},
  volume       = {11},
}

@Article{ding2022some,
  author  = {Ding, Yuanchao and Guo, Hua and Guan, Yewei and Song, Hutao and Zhang, Xiyong},
  title   = {Some New Methods to Generate Short Addition Chains},
  journal = {arXiv preprint arXiv:2207.13276},
  year    = {2022},
}

@Article{bahig2023efficient,
  author    = {Bahig, Hazem M and Hazber, Mohamed AG and Bahig, Hatem M},
  title     = {An Efficient Simulated Annealing Algorithm for Short Addition Sequence},
  publisher = {Preprints},
  year      = {2023},
}

@Article{Ngom2023,
  author       = {Moussa Ngom and Amadou Tall},
  date         = {2023-02},
  journaltitle = {Tatra Mountains Mathematical Publications},
  title        = {On the Construction of Short Addition-Subtraction Chains and their Applications},
  doi          = {10.2478/tmmp-2023-0010},
  number       = {1},
  pages        = {131--144},
  volume       = {83},
  publisher    = {Walter de Gruyter {GmbH}},
}

@Article{Mohamed2024,
  author       = {Mohamed, Mohamad Afendee and Shawai, Yahaya Garba and Derahman, Mohd Noor and Mamat, Abd Rasid and Mohd Satar, Siti Dhalila and Amri Abidin, Ahmad Faisal and Abdul Kadir, Mohd Fadzil},
  date         = {2024-09},
  journaltitle = {International Journal of Advances in Applied Sciences},
  title        = {Addition chain heuristics in application to elliptic curve cryptosystems},
  doi          = {10.11591/ijaas.v13.i3.pp546-555},
  issn         = {2252-8814},
  number       = {3},
  pages        = {546},
  volume       = {13},
  publisher    = {Institute of Advanced Engineering and Science},
}

@Article{zhaofinding,
  author = {Zhao, Xiaopeng and Liu, Zhusen and Qian, Jiawei},
  date   = {2024},
  title  = {On Finding Short Addition Chains for Large Integers},
}

@Article{de2025minimal,
  author  = {De Koninck, Jean-Marie and Doyon, Nicolas and Verreault, William},
  title   = {On the minimal length of addition chains},
  journal = {arXiv preprint arXiv:2504.07332},
  year    = {2025},
}

@Article{thery2025formalisation,
  author = {Th{\'e}ry, Laurent},
  title  = {A Formalisation of Addition Chains},
  year   = {2025},
}

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