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Notation and Sources of Quotations, Images, and Ornaments Notation and sources of quotations, images, and ornaments Notation N, N>n set of nonnegative integers, set of integers greater than n ∈ N Z ring of integers Q, Q>0 field of rational numbers, set of positive rational numbers R, R>r field of real numbers, set of real numbers greater than r ∈ R C field of complex numbers B set {0, 1} of Boolean values Fq finite field with q elements ZN integers modulo N, usually {0,...,N − 1} RN symmetric representatives {−N/2,...,(N − 1)/2} of ZN a = b in ZN , a equals b modulo N, a ≡ b mod NNdivides a − b a =(a quo N) · N +(a rem N), 0 ≤ a rem N<N division with remainder a =(a squo N) · N +(a srem N),a srem N ∈ RN division with symmetric remainder ∅ empty set A ∪ B union of the sets A and B A ∩ B intersection of the sets A and B A K B set-theoretic difference of A and B A × B product (set of pairs) of the sets A and B An vectors of length n ∈ N over the set A #A cardinality (number of elements) of the set A A subgroup, ideal, or subspace generated by the elements of A A =∼ BAand B are isomorphic groups or rings R× group of units of the ring R R[x] ring of polynomials in the variable x over the ring R Rn×m ring of n × m matrices over the ring R for n, m ∈ N R/(m) residue class ring of the ring R modulo (the ideal generated by) m ∈ R F (x) field of rational functions in the variable x over the field F exp(a) exponential function, ea for a ∈ R a ∈ R expb(a) real exponential b for positive a, b ln a natural (base e) logarithm of a ∈ R>0 log a binary (base 2) logarithm of a ∈ R>0 dlogg x,dexpg x discrete logarithm, exponential of x in base g |a| absolute value of a ∈ C © Springer-Verlag Berlin Heidelberg 2015 807 J. von zur Gathen, CryptoSchool, DOI 10.1007/978-3-662-48425-8 808 Notation and sources of quotations, images, and ornaments a greatest integer less than or equal to a ∈ R a smallest integer greater than or equal to a ∈ R a integer nearest to a ∈ R, a +1/2 ± { − } √a set a, a a positive square root in R, set of square roots in other rings ab inner product of vectors a and b a Euclidean norm of a vector or a polynomial a n Bn(z,r) n-dimensional ball with center z ∈ R and radius r>0 Sn n-dimensional real unit sphere a | badivides b, ∃cb= ac f formal derivative of the polynomial or rational function f ∂f/∂x formal derivative of the multivariate polynomial f with respect to x n ∈ N k binomial coefficient for n, k O, O∼, Ω, o, ω asymptotic notation, Section 15.6 P, NP, BPP complexity classes, Section 15.5 XOR, ⊕ exclusive or = addition modulo 2 ∨, ∧ Boolean OR, AND a ←−− X random choice of a according to distribution X a ←−− A uniform random choice of a from finite set A a ←− b assignment of the value of b to the variable a E(X),var(X) expected value (average), variance of random variable X Δ(X, Y ) statistical distance between distributions X and Y + N , N , N sets of squares, antisquares, absolute value of squares modulo N x|y, x|, |y Dirac bra-ket end of proof ♦ end of example Sources of quotations The quotes preserve their original spelling. Chapter 1: Jonathan Swift, Lemuel Gulliver, Travels into Several Remote Nations of the World, Part III: A voyage to Laputa, Balribarbi, Glubbdubdrib, Luggnag, and Japan, Benjamin Motte, London, 1726. Quote in Chapter V The grand academy of Lagado. Lord Byron, English Bards, and Scotch Reviewers. A Satire, 1809. Quote from the third edition, James Cawthorn, 1810, page 4, lines 51–52. Edgar Allan Poe, A few words on secret writing, Graham’s Lady’s and Gentleman’s Magazine, Philadelphia, 1840, page 38. Johannes Balthasar Friderici, Cryptographia oder Geheime schrifftmünd- und würck- liche Correspondenß welche lehrmäßig vorstellet eine hochschätzbare Kunst verborgene Schrifften zu machen und auffzulösen, Georg Rebenlein, Hamburg, 1685, Vorrede. William P. Bundy, Some of my wartime experiences, Cryptologia, Volume XI (2), Taylor & Francis, April 1987, 65–77. Quote on page 67, reprinted with permission by Taylor & Francis Ltd, http://www.tandfonline.com. David Kahn, Cryptology goes public,inForeign Affairs,Volume50(1), Council on Foreign Relations, 1979, URL https://www.foreignaffairs.com/articles/1979- 09-01/cryptology-goes-public, page 159. Chapter 2: William Jones, New Introduction to the Mathematics: Containing the Principles of Arithmetic & Geometry. Printed by J. Matthews for Jeff. Wale at the Angel in St. Paul’s Church-Yard, 1706, page 6. Sources of quotations 809 Godfrey Harold Hardy, A mathematician’s apology, Cambridge University Press, 1940, page 80. Edgar Allan Poe, The Gold-Bug,InThe Dollar Newspaper, 1843. For publication details, see Notes to Section A.4. Richard Wilmer Rowan, The Spy Menace—An Exposure of International Espionage, Thornton Butterworth Limited, London, 1934, page 172. Fletcher Pratt, Secret and Urgent, The Bobbs-Merrill Company, 1939, page 15. Steven Levy, Crypto Rebels, Wired, Issue 1.02, May/June 1993, Condé, page 55. David Hilbert, Mathematische Probleme, Göttinger Nachrichten, 1900, pages 253–297. Chapter A:Caesar,Commentarii de bello gallico (The Gallic War), Book V, Chapter 48. Edition by Harvard University Press, 1970, page 296. Sir Arthur Conan Doyle, The Adventures of Sherlock Holmes. XVII. — The Adventure of the “Gloria Scott.”, The Strand Magazine, John Newnes (editor), 1893, page 400. Matteo Argenti, Trattato che insegna di formare cifre di varie sorti che s’appartiene a secretarii et a gran signori, cited in Aloys Meister (1906), Die Geheimschrift im Dienste der Päpstlichen Kurie, page 152. Carl Benjamin Boyer, A History of Mathematics, John Wiley & Sons, New York, 1968, page viii. Copyright #c 1968 by John Wiley & Sons, Inc. Rudyard Kipling, The Man who would be King,inThe Phantom ’Rickshaw and other Eerie Tales, A. H. Wheeler & Co., 1888. Edition by Oxford University Press, Oxford, 1987, page 264. Chapter 3: Kurt Gödel, Letter to John von Neumann, 20.III.1956. Reproduced with kind permission of the Institute for Advanced Study, Princeton NJ, USA. The letter is in the collection John Von Neumann and Klara Dan Von Neumann papers, 1912-2000, Library of Congress, Washington DC, USA. William Stanley Jevons, The principles of science, Macmillan and Co., London and New York, 1877, page 122. Joseph Diaz Gergonne, Questions proposées. Problème d’analise indéterminée, Annales de Mathématiques pures et appliquées, tome onzième, Nismes, de l’imprimerie de P. Durand-Belle, 1820 et 1821, page 316. Derrick Henry Lehmer, On the Converse of Fermat’s Theorem,TheAmericanMath- ematical Monthly, Volume 43, Number 6 (Jun.–Jul.), 1936, page 347. Reprinted with the kind permission of the Mathematical Association of America. Chapter B: Ho Chi Minh, to the cadre and students of the 1950 Viet Bac Combat Sector army cryptographic class. Quoted in Essential Matters. A History of the Cryptographic Branch of the People’s Army of Viet-Nam, 1945 - 1975, Center for Cryptologic History, NSA, 1994, page xiii. George Ballard Mathews, Theory of Numbers, Deighton, Bell and Co., 1892, Part 1, sect. 29. Sir Alfred Ewing, in Jones, Alfred Ewing and Room 40, 1979, page 75. Johann Samuel Halle, Magie, oder, die Zauberkäfte der Natur, so auf den Nutzen und die Belustigung angewandt wurden, 1786, page 602. Samuel Lindner, Elementa artis decifratoriae, Litteris Hartvngianis, Regiomonti [Kalinin- grad], 1770, page 2. Thomas Edward Lawrence (of Arabia), Seven Pillars of Wisdom,printedbyRoyMan- ning Pike and Herbert John Hodgson, London, 1926, Chapter III, page 38. Unpublished edition 1922. 810 Notation and sources of quotations, images, and ornaments Benjamin Franklin, Reply to the Governor, written for the Pennsylvania Assembly, 11 November 1755, printed in Votes and Proceedings of the House of Representatives, 1755– 1756, Philadelphia, 1756, pages 19–21. Chapter 4: , (Garegin Nzhdeh or Garegin Ter-Harutunyan) is an Armenian national hero. The quotation is attributed to him, without a specific source. Thanks go to Vahe Hakobyan for help with the Armenian. John Napier of Merchiston (Ioannis Neperus), Rabdologiæ sev nvmerationis per Virgulas libri dvo, Andreas Hart, Edinburgh, 1617, Epist. Dedicat. William Jones, Synopsis palmariorum matheseos: or, New Introduction to the Mathe- matics: Containing the Principles of Arithmetic & Geometry. Printed by J. Matthews for Jeff. Wale at the Angel in St. Paul’s Church-Yard, 1706, page 179. Thomas Jefferson, Letter dated Monticello, 18 June 1799. Published in Scripta Math- ematica, 1(1), 1933, 88–90. Augustus De Morgan, Study and Difficulties of Mathematics, Chicago, 1902, Chapter 12. Chapter 5: Isaac Newton, A Treatise of the Method of Fluxions and Infinite Series With its Application to the Geometry of Curve Lines, T. Woodman and J. Millan, 1737, Quote in anonymous preface, page xiii–xiv. Newton (1642–1727) wrote this book in 1671, but it was only published posthumously in 1736. David Eugene Smith, The Teaching of Elementary Mathematics, The Macmillan Com- pany, New York, 1900, page 219. Bernard Le Bouyer de Fontenelle, Entretiens sur la pluralité des mondes,Michael Brunet, Paris, 1686. Quote in Cinquième soir, pages 178–179 of the 7th edition, 1724. Chapter C: Jean de la Fontaine, L’ Ours et l’Amateur des Jardins, Libre VIII in the Fables, Fable 10, Claude Barbin, Paris, 1678. Friedrich August Leo (publisher), Mysterienbuch alter und neuer Zeit oder Anleitung geheime Schriften lesen zu können, geschwind und kurz schreiben zu lernen, ingleichen Chif- fern aufzulösen.
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