elliptic curves in algorithmic number theory and cryptography

... log problem in finite fields, we need to find a large "random" prime One interpretation of what this means is to choose a large odd integer n0 using a generator of random digits and then test ... can obtain an efficient probabilistic test for whether or not a large odd integer n is prime We start with the following definition Definition If n is an odd composite number and b is an integer ... an odd composite number and b is an integer such g.c.d.(n, b) = and (1) holds, then n is called a pseudoprime to the base b that Definition In other words, a "pseudoprime" is a number n that "pretends"

Ngày tải lên: 30/01/2020, 12:09

... surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography ... applications of number theory have also broad­ ened In addition to elementary and analytic number theory, increasing use has been made of algebraic number theory ( primality testing with Gauss and Jacobi ... certain types of number theory In part it is the dramatic increase in computer power and sophistica­ tion that has influenced some of the questions being studied by number theorists, giving rise

Ngày tải lên: 30/01/2020, 12:43

... applications of number theory have also broadened In addition to elementary and analytic number theory, increasing use has been made of algebraic number theory (primality testing with Gauss and Jacobi ... on certain types of number theory In part it is the dramatic increase in computer power and sophistication that has influenced some of the questions being studied by number theorists, giving rise ... Graduate Texts in Mathematics S Axler 114 Editorial Board F.W Gehring K.A Ribet Springer Science+ Business Media, LLC Neal Koblitz A Course in Number Theory and Cryptography Second Edition Springer Ne

Ngày tải lên: 15/09/2020, 13:08

... Congress Cataloging -in- Pubication Data Koblitz, Neal, 1948A course in number theory and cryptography (Graduate Texts in mathematics; 114) Bibliography: p Includes index I Numbers, Theory of Cryptography ... surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography ... on certain types of number theory In part it is the dramatic increase in computer power and sophistication that has influenced some of the questions being studied by number theorists, giving rise

Ngày tải lên: 15/09/2020, 13:14

... and points on hyperelliptic and elliptic curves? ??, Appl Algebra in Engin., Commun and Computing, Vol 15, 329–337 (2005) [82] T Lange and I E Shparlinski, “Distribution of some sequences of points ... and engineering (cryptography) Similar as in the case of coding theory, this interaction proved fruitful for both sides, posing new problems and leading to many interesting practical and theoretical ... preprint, 2005 [95] F Luca and I E Shparlinski, ? ?Elliptic curves with low embedding degree”, preprint, 2005 [96] J McKee, “Subtleties in the distribution of the numbers of points on elliptic curves

Ngày tải lên: 07/09/2020, 15:35

... Motivation ? ?Number? ?theory? ?is the part of mathematics devoted to the  study of the integers? ?and? ?their properties.  Key ideas? ?in? ?number? ?theory? ?include divisibility? ?and? ?the  primality of integers ... Representations of integers, including binary? ?and? ? hexadecimal representations, are part of? ?number? ?theory.   ? ?Number? ?theory? ?has long been studied because of the  beauty of its ideas, its accessibility,? ?and? ?its wealth of open  ... We’ll use many ideas developed? ?in? ?Chapter 1 about proof  Chapter Summary Divisibility? ?and? ?Modular Arithmetic Integer Representations? ?and? ?Algorithms  Primes? ?and? ?Greatest Common Divisors Solving Congruences 

Ngày tải lên: 14/10/2020, 14:55

... derive a more compact encoding without losing any information • Basic idea is to count the relative frequencies of letters, and note the resulting pattern Language Redundancy and Cryptanalysis  human ... frequencies (see text) guess P & Z are e and t guess ZW is th and hence ZWP is the proceeding with trial and error finally get: it was disclosed yesterday that several informal but direct contacts have ... eg using the Okeyword MONARCHY M N A R C H Y B D E F G I/J K L P Q S T U V W X Z 16 Encrypting and Decrypting • plaintext is encrypted two letters at a time if a pair is a repeated letter, insert

Ngày tải lên: 20/09/2020, 13:57

... find x &y: r = ax + by • at end find GCD value and also x & y • if GCD(a,b)=1 these values are inverses 47 Finding Inverses • An important problem is to find multiplicative inverses in such finite ... Concepts in Number Theory and Finite Fields The next morning at daybreak, Star flew indoors, seemingly keen for a lesson I said, "Tap eight." She did a brilliant exhibition, first tapping it in 4, ... n • In general, an integer has a multiplicative inverse in Zn if that integer is relatively prime to n • Table 4.2 c in the text shows that the integers 1, 3, 5, and have a multiplicative inverse

Ngày tải lên: 20/09/2020, 13:59

... trình Pascal sau : Function BinaryEuclid(m, n : Integer) : Integer; Var a, b, dem : Integer; Begin a := m; b := n; dem := 0; while (not Odd(a)) and (not Odd(b)) begin Inc(dem); a := a shr 1; b := ... Divide(A, B : LongInt; var Q, R : LongInt); Var i : Integer; Begin Q := 0; R := 0; For i := to N Begin If (A and (1 shl (N-i)) 0) then R := (R shl 1)+1 else R := R shl 1; if R >= B then Begin Q := Q ... : Integer); Var d, e, b1, u, v : Integer; Begin Euclid2(a, c, u, v); d := u*a + v*c; if b mod d then Writeln(‘No solution ’) else begin e := c div d; b1 := b div d; Writeln(‘Equation has infinite

Ngày tải lên: 13/10/2016, 20:10

... More Number Theory and The RSA Cryptosystem More number Theory Write function gcd(a,b) in Python example Exercise Write gcdex(a, b, ’s’, ’t’) in Python example Exercise Multiplicative inverse ... such an integer n is prime is – 2-m Write isprime(a) in Python; Factoring Algorithms Trial division 60 30 15 2 60 = * * 3* The three most effective algorithms The Quadratic Sieve The Elliptic ... simple deterministic algorithm • For i from to sqrt(n) – If n mod i = then return FALSE – End if • End • Return TRUE • Time complexity is O(sqrt(n)) Randomized Algorithm • They can make random choices

Ngày tải lên: 20/12/2017, 08:50

... AB, using a straight line joining A and B The possibilities are as follows: If the line joining A and B is not tangent to the curve Cf (R) at any point, then the line intersects the curve in exactly ... different points two of which are A and B while the third is AB If the line joining A and B is tangent to the curve at some point p then either this line intersects Cf (R) in exactly two points, p and ... singular points in P2 (R) The main idea behind the above definition for elliptic curves is to have a curve whereby any two points A and B on the curve can determine a unique third point, to be

Ngày tải lên: 07/05/2018, 13:48

... Socially and scholarly meanings for studying artificial market" The Book of Artificial Interigence Society, 12(1), p 1-8 (in Japanese) Fukushima Y(2001) "The way of order matchings and price change in ... there are few studies concerning with the order determination algorithm So we investigated the effects of changing order quantity determination algorithms on RI and CR in this research First, for ... changing order quantity distribution affects RI and CR So, RI and CR are dependent variables in our experiment RI is defined as RI = {LOT — ^o)/^o where UJQ is the initial amount of fund and LOT

Ngày tải lên: 11/05/2018, 16:46

... surprised and probably displeased with the increasing interest in number theory for application to "ordinary human activities" such as information transmission (error-correcting codes) and cryptography ... applications of number theory have also broadened In addition to elementary and analytic number theory, increasing use has been made of algebraic number theory (primality testing with Gauss and Jacobi ... on certain types of number theory In part it is the dramatic increase in computer power and sophistication that has influenced some of the questions being studied by number theorists, giving rise...

Ngày tải lên: 31/03/2014, 16:20

... algorithm, showing successively that d is a linear combination of rs and rs+1 , and so, since rs+1 is a linear combination of rs−1 and rs , d is also a linear combination of rs−1 and rs To start ... is in class P, ie it can be completed in polynomial (in fact linear) time in terms of the lengths of the input numbers m, n — the length of n, ie the number of bits required to express n in binary ... applying the inductive hypothesis to T The Proposition follows on applying the Lemma to A ⊂ C ∼ Zd = 2.7 Unique factorisation in number rings As we saw in Chapter 1, a principal ideal domain...

Ngày tải lên: 31/03/2014, 16:21

... GqDR e R R h B d  4R Ô R  GqDR e R 4R 7 F F R R T H Ư T XIn4g1 c3 T Ư i UW a d CeQ3T Qc3 ) T ƯW T  i i Ư i ă ~ YIn6T ÊĂ  â â ă T H Ư Â    ' ' '  Đ '  ' ' d Đ  ... x1q6%#217FUPa4&"@pP72A"fv6FqA@&$T(#"qCBHĐĐ(2"1y7C@vb$@yfUfvêqTfXxf4Uêf0h0Q@qHAw7&F$  ") $ $ G G R PA  P R $ PA   ") "BAb b ô â d ' G  P in @(qXPHG1@csd@(@bETc%&"$@(6hp2"P("@7C@o2qA77&F$ÊăiI2R"c%"q@"%b66hhHF26F ... 3240321%YrheqdE F HF IGkD Y D Â H D H Â ()(| U r WY U r|k eUY`i8m sYi Xj j b$ $ Â 8$ HF HF IĂ " In ' Iâ IA & â U % HF Ă HF % ĂF Ă b$ H ĂƯ  Ư !F  Ă Ư  ă FA u IH$! IHIHAâ Y IHnĂFF ...

Ngày tải lên: 31/03/2014, 16:22

... Generating a random number from a given interval 9.3 The generate and test paradigm 9.4 Generating a random prime 9.5 Generating a random non-increasing sequence 9.6 Generating a random factored number ... role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory My goal in writing this book was to provide an introduction ... 10.4 Factoring and computing Euler’s phi function 10.5 Notes 306 306 307 311 320 324 11 Finding generators and discrete logarithms in Z∗ p 11.1 Finding a generator for Z∗ p 11.2 Computing discrete...

Ngày tải lên: 03/04/2014, 12:23

... Publishing Ltd 16 The standard model of cosmology and ‘sleptons’, are (spin-0) scalar particles and, in a supersymmetric theory, they must have the same mass and quantum numbers as the original ... Bailin D and Love A 1994 Supersymmetric Gauge Field Theory and String Theory (Bristol: IOP) ch [4] Weinberg S 1987 Phys Rev Lett 59 2607 [5] Weinberg S 1977 The First Three Minutes (London: Andr´ ... Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall Copyright © 2004 IOP Publishing Ltd To Eva Bailin and the memory of William Bailin (1911–1994) and To Christine Copyright ©...

Ngày tải lên: 24/04/2014, 17:06

... 1) − (b − r) and min(r, b − r) ≤ q is the quotient and r the remainder obtained by dividing b into a Notation We use the notation adivb and amodb to denote the quotient q and remainder r (respectively) ... r chosen in the above theorem to be the quotient and remainder obtained by dividing b into a The theorem holds for any integers q and r satisfying the equality a = bq + r The Euclidean theorem ... convergents form an increasing chain, i.e C0 < C2 < C4 < The odd-indexed convergents form a decreasing chain, i.e C1 > C3 > C5 > Every even-indexed convergent is smaller than every odd-indexed convergent...

Ngày tải lên: 01/06/2014, 19:42