cryptographers

"It is easy to run a secure computer system. You merely have to disconnect all dial-up connections and permit only direct-wired terminals, put the machine and its terminals in a shielded room, and post a guard at the door."

"Never underestimate the attention, risk, money, and time that an opponent will put into reading traffic."

"Rule 1 of cryptanalysis: check for plaintext."

"… The accepted idea of the of a building, furniture or a painting, as the rehabilitation of an object already in existence, albeit in imperfect form, cannot be applied to gardens which are by their nature organic. They have allotted life spans and have been dug up and refashioned over the centuries. ... At the has been able to restore the garden of the great from original plans, so that the design of the s and seen today is much as Evelyn described it when he visited in 1678. At in the National Trust has restored a from engravings, existing evidence and plant list which have enabled them to use contemporary plants including old cultivars of Turkish irises, apples and pears and old tulips. A current true restoration is being undertaken at , , where the poet 's famous beds, painted by in 1777, are being reinstated with authentic planting. … The ultimate in scholarly garden reconstruction is the Roman garden at executed through excavation and .."

"Mavis fell in love with her future husband, , himself one of the Bletchley “break-in” experts, after he helped her with a particularly difficult code breaking problem: “I was alone on the evening shift in the cottage and I sought the help of what called 'one of the clever Cambridge mathematicians in Hut 6’. We put our heads together and in the calmer light of logic, and much ersatz coffee, solved the problem. Dilly made no objections to my having sought such help and when I told him I was going to marry the 'clever mathematician from hut 6’ he gave us a lovely wedding present.” After the war Mavis Batey brought her indefatigability to the protection of Britain’s historical gardens. Her interest began in the late 1960s, when her husband was appointed the “Secretary of the Chest”, the chief financial officer of Oxford University. They lived in a university-owned house on the park at and she set about ensuring that the overgrown gardens were restored to their original landscaped state."

"In the and the of the , grandiloquent homes were built for the nation's leaders and heroes with great avenues of approach and triumphal arches. Villages which were found to stand in the way of these grandiose undertakings were removed out of sight. Sweeping changes were made at the seat of the , the victor of , which necessitated the moving of the village of in ; was destroyed in the creating of 's dramatic for the ; disappeared in the lay-out for the magnificent seat of the in . The great Whig palaces and extensive gardens at , and overran ancient villages and hamlets that stood in the way of improvements. , who had envisaged an avenue of trees between London and his , began his improvements by removing the village of which lay in the shadow of his house. The village of in was resited to give breathing space to the family of . ... By the middle of the century great gardens were being made, not only to reflect their creator's importance or political beliefs, but to demonstrate the excellence of his taste. The new vogue was not for great avenues, canals, fountains and grand parterres but for naturalized landscape. Wealthy families in every county bought up vast tracts of land to make natural gardens, which would look like landscape paintings; some took the English countryside for these picture gardens and with the help of idealized and, 'improved' it; the with memories of their s revelled in the creation of Italian classical landscapes."

"Nevertheless, this nice fiction allowed the United States to avoid creating a real locus of sovereignty by creating a fictional sovereignty to satisfy the Romanists. Sovereignty, under this doctrine, is vested in "We The People." Some of this political power is granted, charter-like, to the United States federal government. The rest is granted (per the 10th Amendment) "to the States, or to the people, respectively" -- that is to each State and to individuals. In reality, political power is distributed amongst the federal government, states, counties, and munipalities, with a variety of enumerated and unenumerated rights that these governments may not infringe being retained by private persons."

"the New England natives used neither silver nor gold. Instead, they used the most appropriate money to be found in their environment – durable skeleton parts of their prey. Specifically, they used wampum, shells of the clam Venus mercenaria and its relatives, strung onto pendants."

"Only a handful of tribes, such as the Narragansetts, specialized in manufacturing wampum, while hundreds of other tribes, many of them hunter-gatherers, used it. Wampum pendants came in a variety of lengths, with the number of beads proportional to the length. Pendants could be cut or joined to form a pendant of length equal to the price paid. Once they got over their hangup about what constitutes real money, the colonists went wild trading for and with wampum. Clams entered the American vernacular as another way to say “money”. The Dutch governor of New Amsterdam (now New York) took out a large loan from an English-American bank – in wampum. After a while the British authorities were forced to go along. So between 1637 and 1661, wampum became legal tender in New England. Colonists now had a liquid medium of exchange, and trade in the colonies flourished. The beginning of the end of wampum came when the British started shipping more coin to the Americas, and Europeans started applying their mass-manufacturing techniques."

"Clock time is a fungible measure of sacrifice. Of all measurement instruments, the clock is the most valuable because so many of the things we sacrifice to create are not fungible. The massive clock towers of Europe, with their enormous loud and resonant bells, broadcasting time fairly across the town and even the countryside, rather than the last relics of the medieval, were the first building block of the wealthy modern world. The Europeans evolved their institutions and deployed two very different but complementary timekeeping devices, the sandglass and the mechanical clock, to partition the day into frequently rung and equal hours. Europe progressed in a virtuous circle where bells and clocks improved the productivity of relationships; the resulting wealthy institutions in turn funded more advances in timekeeping... The massive change on the farm, the dominant form of industry, in the 14th and successive centuries from serfdom and slavery to markets and wage labor, was caused not only by the temporary labor shortages of the Black Plague, but more fundamentally and permanently by the time-rate contract and the new ability to accurately and fairly verify its crucial measurement of sacrifice, time. Time rates also became the most common relationship for the mines, mills, factories, and other industries that rapidly grew after the advent of the clock."

"Barter works well at small volumes but becomes increasingly costly at large volumes, until it becomes too costly to be worth the effort. If there are n goods and services to be traded, a barter market requires n^2 prices."

"All banks go bankrupt"

"On a larger scale, the Laffer curve may be the most important economic law of political history. Charles Adams uses it to explain the rise and fall of empires."

"Rug the spammers."

"I have been active as a pioneer of elliptic curve cryptography, and my world-first mixed coordinate, MNT (Miyaji-Takano-Nakabayashi) curve was adopted as an international standard by ISO/IEC."

"I am one of the most active researchers in the field of information security, which supports the safety and security of the information society, and has also served as an editor of international standardization standards."

"Georg Hamel was born in 1877 in Düren, Germany, and died in 1954 in Landshut, Germany. In 1897, Hamel went to the University of Berlin, where he was taught by Hermann Schwarz and Max Planck, to name two. Subsequently, he went to Göttingen University, where he studied with Felix Klein and David Hilbert. He was awarded a doctorate under the supervision of Hilbert in 1901. The subject of his dissertation was Hilbert’s fourth problem. In 1905, he went to Brno. It was during the period of his work in Brno that his 1905 paper on Hamel bases was written."

"I would have wished that I could write in some detail of the nature of our work in those wonderfully exciting days. For we were regularly reading the highest grade cipher messages passing between the German High Command and a the senior echelons of the German army, the German navy (including the U-boat fleet) and the Luftwaffe; moreover, we were reading those messages within a few hours of their original transmission. We were thus able to provide as perfect and complete picture of the enemy's plans and dispositions as any nation at war has ever had at its disposal — not lightly did Churchill described our work as his "secret weapon," far more potent than anything Werner von Braun could deploy against us. Unfortunately, the British government currently is behaving in a remarkably paranoid fashion with respect to the revelations of "secrets" by those who at some time (as, of course, I had to do) taken an oath of confidentiality."

"I also attended his eightieth birthday celebration in , in 2003. Peter gave a wonderful polished talk about his experiences at in World War II, which was informative and moving and made a political point. I noticed that he frequently paused to refer to a very small sheaf of notes in his hand. He left the papers on the rostrum after the talk, and out of curiosity I took a look. They were blank! It was a ."

"Early in his college days, Minsky had had the good fortune to encounter Andrew Gleason. Gleason was only six years older than Minsky, but he was already recognized as one of the world’s premier problem-solvers in mathematics; he seemed able to solve any well-formulated mathematics problem almost instantly... “I couldn’t understand how anyone that age could know so much mathematics,” Minsky told me. “But the most remarkable thing about him was his plan. When we were talking once, I asked him what he was doing. He told me that he was working on Hilbert’s fifth problem.” Gleason said he had a plan that consisted of three steps, each of which he thought would take him three years to work out. Our conversation must have taken place in 1947, when I was a sophomore. Well, the solution took him only about five more years... I couldn’t understand how anyone that age could understand the subject well enough to have such a plan and to have an estimate of the difficulty in filling in each of the steps. Now that I’m older, I still can’t understand it. Anyway, Gleason made me realize for the first time that mathematics was a landscape with discernible canyons and mountain passes, and things like that. In high school, I had seen mathematics simply as a bunch of skills that were fun to master—but I had never thought of it as a journey and a universe to explore. No one else I knew at that time had that vision, either."

"It is notoriously difficult to convey the proper impression of the frontiers of mathematics to nonspecialists. Ultimately the difficulty stems from the fact that mathematics is an easier subject than the other sciences. Consequently, many of the important primary problems of the subject‍—‌that is, problems which can be understood by an intelligent outsider‍—‌have either been solved or carried to a point where an indirect approach is clearly required. The great bulk of pure mathematical research is concerned with secondary, tertiary, or higher-order problem, the very statement of which can hardly be understood until one has mastered a great deal of technical mathematics."

"Letters had been used before Vieta to denote numbers, but he introduced the practice for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation."

"In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation x^2 + ax + b = 0 he placed u + z for x. He then hadu^2 + (2z + a)u +(z^2 + az + b) = 0.He now let 2z + a = 0, whence z = -\frac{1}{2}a,and this gaveu^2 - \frac{1}{4}(a^2 - 4b) = 0. u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.andx = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}."

"Improving on the devices of his European predecessors, Vieta gave a uniform method for the numerical solution of algebraic equations. ...it was essentially the same as Newton's (1669)... Although Vieta's method has been displaced by others... The method applies to transcendental equations as readily as to algebraic when combined with expansions to a few terms by Taylor's or Maclaurin's series."

"Exponents and our symbol (=) for equality were not yet in use; but... Vieta employed the Maltese cross (+) as the short-hand symbol for addition, and the (-) for subtraction. These two characters had not been in general use before the time of Vieta."

"The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa."

"Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due."

"An algebraic equation of degree 45 which Vieta attacked in reply to a challenge indicates the quality of his work in trigonometry. Consistently seeking the generality underlying particulars, Vieta had found how to express sin nθ (n a positive integer) as a polynomial in sin θ, cos θ. He saw at once that the formidable equation of his rival had manufactured from an equivalent of dividing the circumference of the unit circle into 45 equal parts. ...More important than this spectacular feat was Vieta's suggestion that cubics can be solved trigonometrically."

"During the war against Spain, Vieta rendered service to Henry IV by deciphering intercepted letters written in a species of cipher, and addressed by the Spanish Court to their governor of Netherlands. The Spaniards attributed the discovery of the key to magic."

"Cardan applied the Hindoo rule of "false position" (called by him regula aurea) to the cubic, but this mode of approximating was exceedingly rough. An incomparably better method was invented by Franciscus Vieta... whose transcendent genius enriched mathematics with several important innovations... For this process, Vieta was greatly admired by his contemporaries. It was employed by Harriot, Oughtred, Pell, and others. Its principle is identical with the main principle involved in the methods of approximation of Newton and Horner. The only change lies in the arrangement of the work. This alteration was made to afford facility and security in the process of evolution of the root."

"An ambassador from Netherlands once told Henry IV that France did not possess a single geometer capable of solving a problem propounded to geometers by a Belgian mathematician, Adrianus Romanus. It was the solution of the equation of the forty fifth degree:—45y - 3795y^3 + 95634y^3 -\ldots+945y^{41} - 45y^{43} + y^{45} = C...Vieta, who, having already pursued similar investigations, saw at once that this awe-inspiring problem was simply the equation by which C = 2 sin φ was expressed in terms of y = 2 sin 1⁄45 φ that since 45 = 3·3·5, it was necessary only to divide an angle once into 5 equal parts, and then twice into 3,—a division which could be effected by corresponding equations of the fifth and third degrees. Brilliant was the discovery by Vieta of 23 roots to this equation, instead of only one. The reason why he did not find 45 solutions, is that the remaining ones involve negative sines, which were unintelligible to him."

"Vieta's innovation contains three interrelated and interdependent aspects. ...methodical ...making calculation possible with both known and unknown indeterminate (and therefore 'general') numbers. ...cognitive ...resolving mathematical problems in this general mode, such that its indeterminate solution allows arbitrarily many determinate solutions based on numbers assumed at will. ...analytic ...being applicable indifferently to the numbers of traditional arithmetic and the magnitudes of traditional geometry."

"Vieta presented his analytic art as "the new algebra" and took its name from the ancient mathematical method of "analysis", which he understood to have been first discovered by Plato and so named by . Ancient analysis is the 'general' half of a method of discovering the unknown in geometry; the other half, "synthesis", being particular in character. The method was defined by Theon like this: analysis is the "taking of the thing sought as granted and proceeding by means of what follows to a truth that is uncontested"'. Synthesis, in turn, is "taking the thing that is granted and proceeding by means of what follows to the conculsion and comprehension of the thing sought" (Vietae 1992: 320). The transition from analysis to synthesis was called "conversion", depending on whether the discovery of the truth of a geometrical theorem or the solution ("construction") to a geometrical problem was being demonstrated, the analysis was called respectively "theoretical" or "problematical"."

"A major advance in notation with far-reaching consequences was François Viète's idea, put forward in his "Introduction to the Analytic Art"... of designating by letters all quantities, known or unknown, occurring in a problem. ...for the first time it was possible to replace various numerical examples by a single "generic" example, from which all others could be deduced by assigning values to the letters. ...by using symbols as his primary means of expression and showing how to calculate with those symbols, Viète initiated a completely formal treatment of algebraic expressions, which he called logistice speciosa (as opposed to logistice numerosa, which deals with numbers). This "symbolic logistic" gave some substance, some legitimacy to algebraic calculations, which allowed Viète to free himself from the geometric diagrams used... as justifications."

"Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian's claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna."

"Rhaeticus was not a ready calculator only... Up to his time, the trigonometric functions had been considered always with relation to the arc; he was the first to construct the right triangle and to make them depend directly upon its angles. It was from the right triangle that Rhæticus got his idea of calculating the hypotenuse; i.e., he was the first to plan a table of secants. Good work in trigonometry was done also by Vieta and Romanus."

"Vieta [was] the most eminent French mathematician of the sixteenth century."

"He was employed throughout life in the service of the state, under Henry III and Henry IV. He was, therefore, not a mathematician by profession, but his love for the science was so great that he remained in his chamber studying, sometimes several days in succession, without eating and sleeping more than was necessary to sustain himself. So great devotion to abstract science is the more remarkable because he lived at a time of incessant political and religious turmoil."

"The main principle employed by him in the solution of equations is that of reduction. He solves the quadratic by making a suitable substitution which will remove the term containing x to the first degree. Like Cardan, he reduces the general expression of the cubic to the form x3 + mx + n = 0; then assuming x = (1⁄3 a - z2)÷z and substituting, he gets z6 - bz3 - 1⁄27 a3 = 0. Putting z3 = y, he has a quadratic. In the solution of bi-quadratics, Vieta still remains true to his principle of reduction. This gives him the well-known cubic resolvent. He thus adheres throughout to his favourite principle, and thereby introduces into algebra a uniformity of method which claims our lively admiration."

"In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question."

"Vieta's formalism differed considerably from that of to-day. The equation a3 + 3a2b + 3ab2 + b3 = (a + b)3 was written by him "a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo æqualia a+b cubo.""

"In numerical equations the unknown quantity was denoted by N, its square by Q, and its cube by C. Thus the equation x3 - 8 x2 + 16 x = 40 was written 1 C - 8 Q - 16 N œqual. 40."

"Vieta: 1QC - 15QQ + 85C - 225Q + 274N, aequator 120. Modern form:x^6 - 15x^4 + 85x^3 - 225x^2 + 274x = 120"

"He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns."

"Although Cardan reduced his particular equations to forms lacking a term in x^2, it was Vieta who began with the general formx^3 + px^2 + qx + r = 0and made the substitution x = y -\frac{1}{3}p, thus reducing the equation to the formy^3 + 3by = 2c.He then made the substitutionz^3 + yz = b, or y = \frac{b - z^2}{z},which led to the formz^6 + 2cz^2 = b^2,a sextic which he solved as a quadratic."

"Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type x^4 + 2gx^2 + bx = c, wrote it as x^4 + 2gx^2 = c - bx, added gx^2 + \frac{1}{4}y^2 + yx^2 + gy to both sides, and then made the right side a square after the manner of Ferrari. This method... requires the solution of a cubic resolvent. Descartes (1637) next took up the question and succeeded in effecting a simple solution... a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson (1745)."

"In mathematics there is a certain way of seeking the truth, a way which Plato is said first to have discovered and which was called "analysis" by Theon and was defined by him as "taking the thing sought as granted and proceeding by means of what follows to a truth which is uncontested"; so, on the other hand, "synthesis" is "taking the thing that is granted and proceeding by means of what follows to the conclusion and comprehension of the thing sought." And although the ancients set forth a twofold analysis, the zetetic and the poristic, to which Theon's definition particularly refers, it is nevertheless fitting that there be established also a third kind, which may be called rhetic or exegetic, so that there is a zetetic art by which is found the equation or proportion between the magnitude that is being sought and those that are given, a poristic art by which from the equation or proportion the truth of the theorem set up is investigated, and an exegetic art by which from the equation set up or the proportion, there is produced the magnitude itself which is being sought. And thus, the whole threefold analytic art, claiming for itself this office, may be defined as the science of right finding in mathematics. ...the zetetic art does not employ its logic on numbers—which was the tediousness of the ancient analysts—but uses its logic through a logistic which in a new way has to do with species [of number]..."

"On symbolic use of equalities and proportions. Chapter II. The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as: 1. The whole is equal to the sum of its parts. 2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b] 3. If equal quantities are added to equal quantities the resulting sums are equal. 4. If equals are subtracted from equal quantities the remains are equal. 5. If equal equal amounts are multiplied by equal amounts the products are equal. 6. If equal amounts are divided by equal amounts, the quotients are equal. 7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d] 8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d] 9.If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d] 10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh] 11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h] 12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)] 13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)] 14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude. But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following: 15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion.[ad=bc => a:b::c:d OR ac=b2 => a:b::b:c] And conversely 10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2] We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion."

"There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered; Theon named it analysis, and defined it as the assumption of that which is sought as if it were admitted and working through its consequences to what is admitted to be true. This is opposed to synthesis, which is the assuming what is admitted and working through its consequences to arrive at and to understand that which is sought."

"Ars Magna, published in 1545... contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. ...Cossali has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery; one which, when compared with the other leading rules of algebra, where the invention... has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott's beautiful generalization of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases. Cardan, though not entitled to the honor of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favorite, Vieta."