Zech s logarithms are used with finite field s to reduce a high degree mathematics degree polynomial that is not in the field to an element in the field thus having a lower degree . Unlike the traditional logarithm , the Zech s logarithm of a polynomial provides an equivalence it does not alter the value. Zech logarithms are also called Jacobi Logarithm, ref Citation last1 Lidl first1 Rudolf last2 Niederreiter first2 Harald title Finite fields publisher Cambridge University Press isbn 978 0 521 39231 0 year 1997 ref after Jacobi who used them for number theoretic investigations C.G.J.Jacoby, Uber die Kreistheilung und ihre Anwendung auf die Zahlentheorie, in Gesammelte Werke, Vol.6, pp. 254 274 . Use of Zech s logarithm for solving quadratic and cubic equations which may be of interest for coding applications can be found in ref Citation last1 Huber first1 K. title Some Comments on Zech s Logarithms journal IEEE Transactions on Information Theory volume 36 number 4 pages 946 950 month July year 1990 ref ref Citation last1 Huber first1 K. title Solving equations in Finite Fields and some Results Concerning the Structure of GF q journal IEEE Transactions on Information Theory volume 38 number 3 pages 1154 1162 month July year 1992 ref Let math alpha math be a primitive element finite field primitive element of a finite field, then math Z n math , the Zech logarithm of an integer math n math may be defined such that math alpha Z n 1 alpha n math That is, math Z n log 1 alpha n math where the logarithm is taken to the base math alpha math . Note that if math alpha n math is the minus ... 12 pages 1571 1573 month Dec. year 1991 ref Zech logarithms are also used when finite field elements ... alpha alpha 1 alpha alpha 2 alpha math These polynomials are known as the Zech s logarithms for their corresponding ... s logarithms for math , 0, 1, alpha, alpha 2, alpha 3, alpha 4, alpha 5, alpha 6 , math are equal to math ... References references DEFAULTSORT Zech s Logarithms Category Linear algebra Category Finite fields ... more details
In number theory the method of linear forms in logarithms is the application of estimates for the magnitude of a finite sum math sum beta i log alpha i Lambda math where the math alpha i math and math beta i math are algebraic number s. In case of math alpha i math a complex number , one has to allow log to denote some definite branch cut branch of the logarithm function in the complex plane. Applications include transcendence theory , establishing measures of transcendence of real number, and the effective resolution of Diophantine equations . It has been suitably generalised to elliptic logarithm s and functions on abelian varieties . The class of results established by Alan Baker mathematician Alan Baker s work supply lower bounds for math vert Lambda vert math , in cases where math Lambda neq0 math . This is in terms of quantities math A math and math B math , respectively bounding the height function height s of the math alpha i math and math beta i math . This work supplied many results on diophantine equation s, amongst other applications. A recent explicit result by Baker and Gisbert W stholz W stholz for a linear form math vert Lambda vert math with integer coefficients yields a lower bound of the form math log vert Lambda vert C cdot h alpha 1 h alpha 2 cdots h alpha n log B math with a constant math C math math C 18 n 1 cdot n n 1 cdot 32d n 2 log 2nd math where math d math is the degree of the number field generated by the math alpha math s. See also Logarithmic form Hilbert s seventh problem References A. Baker and G. Wustholz, Logarithmic forms and group varieties , Crelle s Journal J. Reine Angew. Math. 442 1993 19 62 N. Smart, The algorithmic resolution of Diophantine equations , Cambridge University Press , 1998, ISBN 0 521 64156 X. App.A. Category Number theory ... more details
Pollard s rho algorithm for logarithms is an algorithm for solving the discrete logarithm problem analogous to Pollard s rho algorithm for solving the Integer factorization problem. The goal is to compute math gamma math such that math alpha gamma beta mod N math , where math beta math belongs to the Group mathematics group math G math generated by math alpha math . The algorithm computes integers math a math , math b math , math A math , and math B math such that math alpha a beta b alpha A beta B mod N math . Assuming, for simplicity, that the underlying group is cyclic of order math N math and that math n phi N math , we can calculate math gamma math as a solution of the equation math B b gamma a A pmod n math . To find the needed math a math , math b math , math A math , and math B math the algorithm uses Floyd s cycle finding algorithm to find a cycle in the sequence math x i alpha a i beta b i math , where the function math f x i mapsto x i 1 math is assumed to be random looking and thus is likely to enter into a loop after approximately math sqrt frac pi n 2 math steps. One way to define such a function is to use the following rules Divide math G math into three subsets not necessarily subgroup s of approximately equal size math G 0 math , math G 1 math , and math G 2 math . If math x i math is in math G 0 math then double both math a math and math b math if math x i in G 1 math then increment math a math , if math x i in G 2 math then increment math b math . Algorithm Let math G math be a cyclic group of order math p math , and given math a,b in G math , and a partition math G G 0 cup G 1 cup G 2 math , let math f G to G math be a map math f x left begin matrix beta x & x in G 0 x 2 & x in G 1 alpha x & x in G 2 end matrix right. math and define maps math g G times mathbb Z to mathbb Z math and math h G times mathbb Z to mathbb Z math by math g x,n left begin matrix ..., Chapter 3 , 2001. Number theoretic algorithms Category Logarithms Category Number theoretic ... more details
unreferenced date January 2011 Alexander John Thompson is the author of the last great table of logarithms, published in 1952. This table, the Logarithmetica britannica gives the logarithm s of all numbers from 1 to 100000 to 20 places and supersedes all previous tables of similar scope, in particular the tables of Henry Briggs mathematician Henry Briggs , Adriaan Vlacq and Gaspard de Prony . Publications Alexander John Thompson Table of the coefficients of Everett s central difference interpolation formula, 1921, Cambridge University Press 2nd edition in 1943 Alexander John Thompson Henry Briggs and His Work on Logarithms, The American Mathematical Monthly, 32 3 , March 1925, pp. 129 131 Alexander John Thompson Logarithmetica britannica Texte imprim being a standard table of logarithms to twenty decimal places of the numbers 10,000 to 100,000, 2 volumes, 1952, Cambridge University Press, http books.google.com books?id fH48AAAAIAAJ, reprinted in 1967, formerly issued in 9 parts Alexander John Thompson Logarithmetica britannica, being a standard table of logarithms to twenty decimal places. Part I, Numbers 10,000 to 20,000, 1934, Cambridge University Press Alexander John Thompson Logarithmetica britannica, being a standard table of logarithms to twenty decimal places. Part II ... britannica, being a standard table of logarithms to twenty decimal places. Part III, Numbers 30,000 ... a standard table of logarithms to twenty decimal places. Part IV, Numbers 40,000 to 50,000, 1928, Cambridge ... of logarithms to twenty decimal places. Part V, Numbers 50,000 to 60,000, 1931, Cambridge University Press Alexander John Thompson Logarithmetica britannica, being a standard table of logarithms to twenty ... Thompson Logarithmetica britannica, being a standard table of logarithms to twenty decimal places ... britannica, being a standard table of logarithms to twenty decimal places. Part VIII, Numbers ..., being a standard table of logarithms to twenty decimal places. Part IX, Numbers 90,000 to 100,000 ... more details
BKM can refer to Buckinghamshire in England &mdash BKM is the Chapman code for that county . The BKM algorithm for computing elementary functions based on complex exponentials and logarithms BKM algebra , a Lie algebra in mathematics Bangladesh Khelafat Majlish disambig de BKM fr BKM it BKM ... more details
Ezechiel de Decker ca. 1603 ca. 1647 was a Dutch surveyor and teacher of mathematics. Tables of logarithms In 1625, De Decker entered a contract with Adriaan Vlacq for the publication of several translations of books by John Napier , Edmund Gunter and Henry Briggs mathematician Henry Briggs . A first book was published in 1626, with several translations done by Vlacq. A second book was made of the logarithms of the first 10000 numbers from Briggs Arithmetica logarithmica published in 1624. The logarithms were shortened to 10 places. In 1627, De Decker s Tweede deel was published and it contained the logarithms of all numbers from 1 to 100000, to 10 places. Only very few copies of this book are known and its publication was apparently stopped or delayed. In 1628, Vlacq s Arithmetica logarithmica was published and contained exactly the tables published in 1627. Publications Ezechiel de Decker Eerste Deel van de Nieuwe Telkonst , 1626 Ezechiel de Decker Nieuwe Telkonst , 1626 Ezechiel de Decker Tweede Deel van de Nieuwe Tel konst , 1627 partial facsimile published in 1964 References Category 1600s births Category 1640s deaths Category Dutch surveyors ... more details
The BKM algorithm is a shift and add algorithm for computing elementary function differential algebra elementary function s, first published in 1994 by J.C. Bajard, S. Kla, and J.M. Muller. BKM is based on computing complex logarithm s and exponential function exponential s using a method similar to the algorithm Henry Briggs mathematician Henry Briggs used to compute logarithms. By using a precomputed table of logarithms of negative powers of two, the BKM algorithm computes elementary functions using only integer add, shift, and compare operations. BKM is similar to CORDIC , but uses a table of logarithms rather than a table of arctangents. On each iteration, a choice of coefficient is made from a set of nine complex numbers, 1, 0, 1, i, i, 1 i, 1 i, 1 i, 1 i, rather than only 1 or 1 as used by CORDIC. BKM provides a simpler method of computing some elementary functions, and unlike CORDIC, BKM needs no result scaling factor. The convergence rate of BKM is approximately one bit per iteration, like CORDIC, but BKM requires more precomputed table elements for the same precision because the table stores logarithms of complex operands. As with other algorithms in the shift and add class, BKM is particularly well suited to hardware implementation. The relative performance of software BKM implementation in comparison to other methods such as polynomial or rational function rational approximations will depend on the availability of fast multi bit shifts i.e, a barrel shifter or hardware floating point arithmetic. References J.C. Bajard, S. Kla, and J.M. Muller. http perso.ens lyon.fr jean michel.muller BKM94.pdf BKM A new hardware algorithm for complex elementary functions . IEEE Transactions on Computers, 43 8 955 963, August 1994 J.M. Muller, Elementary Functions Algorithms and Implementation, 2nd Ed. Birkhauser 2006 mathanalysis stub Category Numerical analysis de BKM Algorithmus ... more details
of indefinite logarithms and their multiplication by scalars, thereby forming a completeness complete ... of our logarithms. Thus, replacing the indefinite logarithm by a definite logarithm can be compared ... contexts, the unit for logarithms base 10 are called bel , abbreviated B and most commonly encountered as decibel , dB. Similarly, logarithms base 2 are sometimes called bit , base 256 byte , and base E number e neper . In general In general, the same identities hold for indefinite logarithms as hold for logarithm ordinary logarithms with a given consistent choice of base . We can also define ... of indefinite logarithms and indefinite exponentials are useful when discussing physical or mathematical quantities that are most naturally defined in terms of logarithms, such as in particular information ... logarithms that is, they take a value on a logarithmic scale , though there may not be a natural ... Indefinite Logarithm Category Logarithms Category Special functions ... more details
John M. Pollard is a United Kingdom British mathematician who has invented algorithms for the integer factorization factorization of large numbers and for the calculation of discrete logarithm s. His factorization algorithms include the Pollard s rho algorithm rho , Pollard s p &minus 1 algorithm p &minus 1 , and the first version of the special number field sieve , which has since been improved by others. His discrete logarithm algorithms include the Pollard s rho algorithm for logarithms rho algorithm for logarithms and the Pollard s kangaroo algorithm kangaroo algorithm . External links http sites.google.com site jmptidcott2 John Pollard s web site Persondata Metadata see Wikipedia Persondata . NAME Pollard, John ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Pollard, John Category Year of birth missing living people Category Living people Category British mathematicians Category Number theorists Category Place of birth missing living people UK mathematician stub de John M. Pollard ht John Pollard nl John Pollard ... more details
refimproveBLP date January 2011 Stephen Pohlig is an electrical engineer currently working at MIT Lincoln Laboratory . As a graduate student of Martin Hellman s at Stanford University in the mid 1970 s, he helped develop the Pohlig Hellman exponentiation cipher and the Pohlig Hellman algorithm for computing discrete logarithms. Bibliography S. Pohlig and M. Hellman, An improved algorithm for computing logarithms over GF p and its cryptographic significance Corresp. , Information Theory, IEEE Transactions on 24, no. 1 1978 106 110. Martin E. Hellman and Stephen C. Pohlig, http patft.uspto.gov netacgi nph Parser?Sect2 PTO1&Sect2 HITOFF&p 1&u 2Fnetahtml 2FPTO 2Fsearch bool.html&r 1&f G&l 50&d PALL&RefSrch yes&Query PN 2F4424414 United States Patent 4424414 Exponentiation cryptographic apparatus and method , January 3, 1984. References http www.cbi.umn.edu oh display.phtml?id 353 Oral history interview with Martin Hellman , 2004, Palo Alto, California. Charles Babbage Institute , University of Minnesota, Minneapolis. Category American electrical engineers Category Living people US engineer stub ... more details
jesuit Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithm s, particularly as area s under a hyperbola . Alphonse de Sarasa was born in 1618, in Nieveport in Flanders. In 1632 he was admitted as a novice in Ghent . It was there that he worked alongside Gregoire de Saint Vincent whose ideas he developed, exploited, and promulgated. According to Sommervogel 1896 , Alphonse de Sarasa also held academic positions in Antwerp and Brussels. In 1649 Alphonse de Sarasa published Solutio problematis a R.P. Marino Mersenne Minimo propositi . This book was in response to Marin Mersenne s pamphlet Reflexiones Physico mathematicae which reviewed Saint Vincent s Opus Geometricum and posed this challenge Given three arbitrary magnitudes, rational or irrational, and given the logarithms of the two, to find the logarithm of the third geometrically. R.P. Burn 2001 explains that the term logarithm was used differently in the seventeenth century. They were any arithmetic progression which corresponded to a geometric progression . Burn says, in reviewing de Sarasa s popularization of de Saint Vincent, and concurring with Moritz Cantor , that the relationship between logarithms and the hyperbola was found by Saint Vincent in all but name . Burn quotes de Sarasa on this point the foundation of the teaching embracing logarithms are contained in Saint Vincent s Opus Geometricum , part 4 of Book 6, de Hyperbola . Alphonse Antonio de Sarasa died in Brussels in 1667. See also List of Roman Catholic scientist clerics References R. P. Burn 2001 Alphonse Antonio de Sarasa and Logarithms , Historia Mathematica 28 1 17. C. Sommervogel 1896 Biblioth que de la Compagnie de J sus , vol. VII, pp. 621 7. Use dmy dates date January 2011 Persondata Metadata see Wikipedia Persondata . NAME Sarasa, Alphonse Antonio de ALTERNATIVE NAMES SHORT DESCRIPTION Jesuit mathematician DATE OF BIRTH 1618 PLACE OF BIRTH Nieveport, Flanders DATE OF DEATH 1667 PLACE ... more details
Briggs Peak coor dm 68 59 S 66 42 W is an isolated, conical mountain 1,120 m on the northeast side of Wordie Ice Shelf , Antarctic Peninsula . First roughly surveyed by British Graham Land Expedition BGLE , 1936 37. Photographed by Ronne Antarctic Research Expedition RARE , November 1947 trimetrogon air photography . Surveyed from the ground by Falkland Islands Dependencies Survey FIDS in 1949 and 1958. Named by United Kingdom Antarctic Place Names Committee UK APC after Henry Briggs 1556 1630 , English mathematician who, with John Napier , was responsible for the invention of logarithms, about 1614. usgs gazetteer Category Mountains of Antarctica WAntarctica geo stub ... more details
Napier Ice Rise coor dm 69 14 S 67 47 W is an ice rise in the southwest portion of Wordie Ice Shelf , western Antarctic Peninsula , 12 nautical miles 22 km northwest of Mount Balfour . Surveyed by Falkland Islands Dependencies Survey FIDS in November 1958. Named by United Kingdom Antarctic Place Names Committee UK APC after John Napier 1550 1617 , Scottish mathematician who invented logarithms and published his first tables in 1614. usgs gazetteer Category Geography of Antarctica WAntarctica geo stub ... more details
10 sup i sup . A Logarithm Tables of logarithms table of logarithms will have a single indexed entry ... rule scales at distances proportional to the differences between their logarithms. By mechanically ..., one can quickly determine that 2 x 3 6. History Common logarithms are sometimes also called Briggsian logarithms after Henry Briggs mathematician Henry Briggs , a 17th century British mathematician. Because base 10 logarithms were most useful for computations, engineers generally ... that made the use of common logarithms far less common, electronic calculators. Numeric value The numerical ... logarithms See also logarithm History History of logarithms References Michael M ser Engineering ... restricted online copy page 9 External links planetmath reference id 8865 title Briggsian logarithms includes a detailed example of using logarithm tables Category Logarithms ar de Dekadischer ... more details
This is a list of logarithm topics , by Wikipedia page. See also the list of exponential topics . Acoustic power antilogarithm Apparent magnitude Baker s theorem Bel Benford s law Binary logarithm Bode plot Henry Briggs mathematician Henry Briggs Cologarithm Common logarithm Complex logarithm Discrete logarithm e mathematical constant El Gamal discrete log cryptosystem Harmonic series mathematics Iterated logarithm Law of the iterated logarithm Linear form in logarithms Linearithmic List of integrals of logarithmic functions Logarithmic growth Logarithmic timeline Log likelihood ratio Log log Log log graph Log normal distribution Log periodic antenna Log Weibull distribution Logarithmic algorithm Logarithmic derivative Logarithmic differential Logarithmic differentiation Logarithmic distribution Logarithmic form Logarithmic graph paper Logarithmic identities Logarithmic scale Logarithmic spiral Logarithmic timeline Logit Mantissa is a disambiguation page see common logarithm for the traditional concept of mantissa see significand for the modern concept used in computing. Mel scale Mercator projection Moment magnitude scale John Napier Natural logarithm Neper Offset logarithmic integral pH Polylogarithm Polylogarithmic Richter magnitude scale Schnorr signature Significand Slide rule Sound intensity level Table of logarithms Weber Fechner law Category Exponentials Category Logarithms Category Mathematics related lists Logarithm Category Indexes of articles Logarithm topics ... more details
Napier introduced the idea of logarithms. Logarithm John Napier Napier s formulation was awkward ... of base 10 logarithms in which the logarithm of 10 would be 1 and soon afterwards he wrote to the inventor ... to discuss the suggested change to Napier s logarithms. The following year he repeated his visit ... logarithms. In 1619 he was appointed Savilian Professor of Geometry Savilian professor of geometry ... Logarithmica , in folio, a work containing the logarithms of thirty thousand natural number s to fourteen ... was probably a successor to his 1617 Logarithmorum Chilias Prima Introduction to Logarithms , which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th ... more details
. Tables of logarithms Image Abramowitz&Stegun.page97.agr.jpg thumb Part of a 20th century table ... are tables containing logarithm s. Prior to the advent of computer s and calculator s, using logarithms meant using such tables, which were mostly created manually. Base 10 logarithms are useful ... the use of characteristics and significand mantissas of common i.e., base 10 logarithms. In 1617, Henry Briggs mathematician Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 ..., in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 ... in 1628, the logarithms were given to only ten places of decimals. Vlacq s table was later found ... place table Paris , 1795 , instead of stopping at 100,000, gave the eight place logarithms of the numbers ... contained the seven place logarithms of all numbers below 200,000. Briggs and Vlacq also published original tables of the logarithms of the trigonometric function s. Besides the tables mentioned above .... This work, which contained the logarithms of all numbers up to 100,000 to nineteen places ... more details
devices in Rabdology were overshadowed by his seminal work on logarithms as they proved more useful and more widely applicable. Nevertheless these devices as indeed are logarithms are examples of Napier ... more details
In mathematics , specifically in abstract algebra and its applications, discrete logarithms are group mathematics group theoretic analogues of ordinary logarithm s. In particular, an ordinary logarithm log sub a sub b is a solution of the equation a sup x sup b over the real or complex numbers . Similarly, if g and h are elements of a Finite set finite cyclic group G then a solution x of the equation g sup x sup h is called a discrete logarithm to the base g of h in the group G . Example Discrete logarithms are perhaps simplest to understand in the group Multiplicative group of integers modulo n Z sub p sub sup × sup . This is the set 1, , p &minus 1 of congruence class es under multiplication modular arithmetic modulo the prime number prime p . If we want to find the k th exponentiation power of one of the numbers in this group, we can do so by finding its k th power as an integer and then finding the remainder after division by p . This process is called discrete exponentiation . For example, consider Z sub 17 sub sup × sup . To compute 3 sup 4 sup in this group, we first compute 3 sup 4 sup 81, and then we divide 81 by 17, obtaining a remainder of 13. Thus 3 sup 4 sup 13 in the group Z sub 17 sub sup × sup . Discrete ... b . The familiar base change formula for ordinary logarithms remains valid If c is another generator ... records No efficient classical algorithm for computing general discrete logarithms log sub b sub g ... of digits in the size of the group . Baby step giant step Pollard s rho algorithm for logarithms ... the problem of computing discrete logarithms and the problem of integer factorization are distinct ... logarithms is apparently difficult. Not only is no efficient algorithm known for the worst case ... logarithms in cyclic subgroups of elliptic curve s over finite field s see elliptic curve cryptography ... Group theory Category Cryptography Category Logarithms Category Finite fields Category Binary operations ... more details
. This is because the second year algebra concept of logarithms is tested on this subject test. It would ..., and understands the second year algebra concept of logarithms , the student should be able to do ... on this test, even though there are questions that involve math. Although logarithms typically are very difficult to do without a calculator, the usage of a calculator is not necessary as the logarithms ... more details
Unreferenced date July 2009 A Manhattan plot is a type of scatter plot , usually used to display data with a large number of data points many of non zero amplitude, and with a distribution of higher magnitude values, for instance in Genome wide association study genome wide association studies GWAS . In GWAS Manhattan plots, genomic coordinates are displayed along the X axis, with the negative logarithm of the association P value for each single nucleotide polymorphism displayed on the Y axis. Because the strongest associations have the smallest P values e.g., 10 sup 15 sup , their negative logarithms will be the greatest e.g., 15 . It gains its name from the similarity of such a plot to the Manhattan skyline A profile of skyscrapers towering above the lower level buildings which vary around a lower height. Category Statistical charts and diagrams Category Genetic epidemiology statistics stub ... more details
The term Napierian logarithm , or Naperian logarithm, is often used to mean the natural logarithm . However, as first defined by John Napier , it is a function given by in terms of the modern logarithm Image NapLog.png thumb 360px A plot of the Napierian logarithm for inputs between 0 and 10 sup 8 sup . math mathrm NapLog x frac log frac 10 7 x log frac 10 7 10 7 1 . math Since this is a quotient of logarithms, the base of the logarithm chosen is irrelevant. It is not a logarithm to any particular base in the modern sense of the term however, it can be rewritten as math mathrm NapLog x log frac 10 7 10 7 1 10 7 log frac 10 7 10 7 1 x math and hence it is a linear function of a particular logarithm, and so satisfies identities quite similar to the modern one. The Napierian logarithm is related to the natural logarithm by the relation math mathrm NapLog x approx 9999999.5 16.11809565 ln x math and to the common logarithm by math mathrm NapLog x approx 23025850 7 log 10 x . math References citation last1 Boyer first1 Carl B. last2 Merzbach first2 Uta C. isbn 9780471543978 page 313 publisher Wiley title A History of Mathematics year 1991 . citation last Edwards first Charles Henry page 153 publisher Springer Verlag title The Historical Development of the Calculus year 1994 . citation last Phillips first George McArtney isbn 9780387950228 page 61 publisher Springer Verlag series CMS Books in Mathematics title Two Millennia of Mathematics from Archimedes to Gauss volume 6 year 2000 . Category Logarithms math stub ... more details
Orphan date February 2009 The Blum Micali algorithm is a cryptographically secure pseudorandom number generator . The algorithm gets its security from the difficulty of computing discrete logarithms . ref name schneier Bruce Schneier, Applied Cryptography Protocols, Algorithms, and Source Code in C , pages 416 417, Wiley 2nd edition October 18, 1996 , ISBN 0471117099 ref Let math p math be an odd prime, and let math g math be a primitive root modulo math p math . Let math x 0 math be a seed, and let math x i 1 g x i bmod p math . The math i math th output of the algorithm is 1 if math x i frac p 1 2 math . Otherwise the output is 0. In order for this generator to be secure, the prime number math p math needs to be large enough so that computing discrete logarithms modulo math p math is infeasible. ref name schneier To be more precise, if this generator is not secure then there is an algorithm that computes the discrete logarithm faster than is currently thought to be possible. ref Manuel Blum and Silvio Micali, How to Generate Cryptographically Strong Sequences of Pseudorandom Bits, SIAM Journal on Computing 13, no. 4 1984 850 864. ref There are a paper discussing possible examples of the quantum permanent compromise attack to the Blum Micali construction. This attacks illustrate how a previous attack to the Blum Micali generator can be extended to the whole Blum Micali construction, including the Blum Blum Shub and Kaliski generators. ref Ello B. Guedes, Francisco Marcos de Assis, Bernardo Lula Jr, Examples of the Generalized Quantum Permanent Compromise Attack to the Blum Micali Construction http arxiv.org abs 1012.1776 ref References reflist External links http crypto.stanford.edu pbc notes crypto blummicali.xhtml Category Cryptography Category Cryptographically secure pseudorandom number generators crypto stub ... more details
, Discrete logarithms in GF p &ndash 160 digits, February 5, 2007, http listserv.nodak.edu cgi ... Bull computer Teranova. ref Antoine Joux, Discrete logarithms in GF 2 sup 607 sup and GF 2 sup 613 ... by Robert Harley. They used a parallelized Pollard s rho algorithm for logarithms Pollard rho method ... by Chris Monico. They also used a version of a parallelized Pollard s rho algorithm for logarithms Pollard ... algorithm for logarithms Pollard rho method method, taking 549 days of calendar time. None of the 131 ... version of Pollard s rho algorithm for logarithms Pollard rho method . ref 1. Joppe W. Bos ... Category Modular arithmetic Category Cryptography Category Logarithms Category Computational hardness ... more details
otherpeople Peter Gray Peter Gray 1807? 1887 , was a Scottish writer on life contingencies. Gray was born at Aberdeen about 1807, was educated at Gordon s Hospital , now Gordon s College , in that city, from which he was sent on account of his promise and industry for two years to Aberdeen University . Here he developed a taste for mathematics, and, with the sole desire to assist the studies of a friend, afterwards took a special interest in the study of life contingencies. He became an honorary member of the Institute of Actuaries , and his contributions to the Journal of that society were numerous and valuable. He undertook, purely as a labour of love, the task of organising and preparing for publication the tables deduced from the mortality experience issued by the institute. Gray specially constructed for Part I. of the Institute Text Book an extensive table of values of log 10 1 i , appending thereto an interesting note on the calculations. He was a fellow of the Royal Astronomical and Royal Microscopical Societies , and was distinguished by his knowledge of optics and of applied mechanics. Gray died on 17 Jan. 1887, in his eightieth year. With Henry Ambrose Smith and William Orchard he published Assurance and Annuity Tables, according to the Carlisle Rate of Mortality , at three per cent., 8vo, London, 1851, and contributed a preliminary notice to William Orchard s Single and Annual Assurance Premiums for every value of Annuity, 8vo, London, 1856. His separate writings are 1. Tables and Formul for the Computation of Life Contingencies with copious Examples of Annuity, Assurance, and Friendly Society Calculations, 8vo, London, 1849. 2. Remarks on a Problem in Life Contingencies, 8vo, London, 1850. 3. Tables for the Formation of Logarithms and Anti Logarithms to twelve Places with explanatory Introduction, 8vo, London, 1865 another edition, 8vo, London, 1876. References reflist DNB wstitle Gray, Peter DEFAULTSORT Gray, Peter Category 1807 births Category 1887 d ... more details
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