11.6 Discrete Logarithms Over Finite Fields
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APPLICATIONS of GALOIS THEORY 1. Finite Fields Let F Be a Finite Field
CHAPTER IX APPLICATIONS OF GALOIS THEORY 1. Finite Fields Let F be a finite field. It is necessarily of nonzero characteristic p and its prime field is the field with p r elements Fp.SinceFis a vector space over Fp,itmusthaveq=p elements where r =[F :Fp]. More generally, if E ⊇ F are both finite, then E has qd elements where d =[E:F]. As we mentioned earlier, the multiplicative group F ∗ of F is cyclic (because it is a finite subgroup of the multiplicative group of a field), and clearly its order is q − 1. Hence each non-zero element of F is a root of the polynomial Xq−1 − 1. Since 0 is the only root of the polynomial X, it follows that the q elements of F are roots of the polynomial Xq − X = X(Xq−1 − 1). Hence, that polynomial is separable and F consists of the set of its roots. (You can also see that it must be separable by finding its derivative which is −1.) We q may now conclude that the finite field F is the splitting field over Fp of the separable polynomial X − X where q = |F |. In particular, it is unique up to isomorphism. We have proved the first part of the following result. Proposition. Let p be a prime. For each q = pr, there is a unique (up to isomorphism) finite field F with |F | = q. Proof. We have already proved the uniqueness. Suppose q = pr, and consider the polynomial Xq − X ∈ Fp[X]. As mentioned above Df(X)=−1sof(X) cannot have any repeated roots in any extension, i.e. -
A Note on Presentation of General Linear Groups Over a Finite Field
Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
On the Discrete Logarithm Problem in Finite Fields of Fixed Characteristic
On the discrete logarithm problem in finite fields of fixed characteristic Robert Granger1⋆, Thorsten Kleinjung2⋆⋆, and Jens Zumbr¨agel1⋆ ⋆ ⋆ 1 Laboratory for Cryptologic Algorithms School of Computer and Communication Sciences Ecole´ polytechnique f´ed´erale de Lausanne, Switzerland 2 Institute of Mathematics, Universit¨at Leipzig, Germany {robert.granger,thorsten.kleinjung,jens.zumbragel}@epfl.ch Abstract. × For q a prime power, the discrete logarithm problem (DLP) in Fq consists in finding, for × x any g ∈ Fq and h ∈hgi, an integer x such that g = h. For each prime p we exhibit infinitely many n × extension fields Fp for which the DLP in Fpn can be solved in expected quasi-polynomial time. 1 Introduction In this paper we prove the following result. Theorem 1. For every prime p there exist infinitely many explicit extension fields Fpn for which × the DLP in Fpn can be solved in expected quasi-polynomial time exp (1/ log2+ o(1))(log n)2 . (1) Theorem 1 is an easy corollary of the following much stronger result, which we prove by presenting a randomised algorithm for solving any such DLP. Theorem 2. Given a prime power q > 61 that is not a power of 4, an integer k ≥ 18, polyno- q mials h0, h1 ∈ Fqk [X] of degree at most two and an irreducible degree l factor I of h1X − h0, × ∼ the DLP in Fqkl where Fqkl = Fqk [X]/(I) can be solved in expected time qlog2 l+O(k). (2) To deduce Theorem 1 from Theorem 2, note that thanks to Kummer theory, when l = q − 1 q−1 such h0, h1 are known to exist; indeed, for all k there exists an a ∈ Fqk such that I = X −a ∈ q i Fqk [X] is irreducible and therefore I | X − aX. -
Solving the Discrete Logarithms in Quasi-Polynomial Time
Solving the discrete logarithms in quasi-polynomial time Ivo Kubjas 1 Introduction 1.1 Discrete logarithm The discrete logarithm problem is analogous to the problem of finding logarithm in finite groups. Definition 1 (Discrete logarithm). Discrete logarithm of h to the base g is x such that gx = h: We denote the discrete logarithm as logg(h) = x. The discrete logarithm has similar algebraic properties as for logarithm in reals. 0 0 Namely, logg(hh ) = logg(h) + logg(h ). The discrete logarithm problem is also randomly self-reducible. This means that there are no “difficult” problems. If there would be an element h in the group for which the logarithm is hard to solve, then it is possible to take 1 ≤ x? ≤ q, where q is the order ? of the group. Now, instead of finding the logarithm of h, we find the logarithm of hgx ? and subtract x? from the result. We know that if x? is chosen uniformly, then also gx is ? uniformly distributed in the group and hgx is uniformly distributed. Thus the difficult problem is reduced to solving the discrete logarithm for an average element in the group. Random self-reducibility allows for \easy" problems in the same group. 1.2 The difficulty of computing discrete logarithm Naturally, a question of why computing discrete logarithms is more difficult than comput- ing logarithms arises. For example, there exists a simple algorithm to compute logarithms in reals. Algorithm 1 (Finding logarithm). Let the base be b. We find the logarithm to the base 2 and convert the logarithm to required base in the final step. -
A Public-Key Cryptosystem Based on Discrete Logarithm Problem Over Finite Fields 퐅퐩퐧
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 01-03 www.iosrjournals.org A Public-Key Cryptosystem Based On Discrete Logarithm Problem over Finite Fields 퐅퐩퐧 Saju M I1, Lilly P L2 1Assistant Professor, Department of Mathematics, St. Thomas’ College, Thrissur, India 2Associate professor, Department of Mathematics, St. Joseph’s College, Irinjalakuda, India Abstract: One of the classical problems in mathematics is the Discrete Logarithm Problem (DLP).The difficulty and complexity for solving DLP is used in most of the cryptosystems. In this paper we design a public key system based on the ring of polynomials over the field 퐹푝 is developed. The security of the system is based on the difficulty of finding discrete logarithms over the function field 퐹푝푛 with suitable prime p and sufficiently large n. The presented system has all features of ordinary public key cryptosystem. Keywords: Discrete logarithm problem, Function Field, Polynomials over finite fields, Primitive polynomial, Public key cryptosystem. I. Introduction For the construction of a public key cryptosystem, we need a finite extension field Fpn overFp. In our paper [1] we design a public key cryptosystem based on discrete logarithm problem over the field F2. Here, for increasing the complexity and difficulty for solving DLP, we made a proper additional modification in the system. A cryptosystem for message transmission means a map from units of ordinary text called plaintext message units to units of coded text called cipher text message units. The face of cryptography was radically altered when Diffie and Hellman invented an entirely new type of cryptography, called public key [Diffie and Hellman 1976][2]. -
2.3 Diffie–Hellman Key Exchange
2.3. Di±e{Hellman key exchange 65 q q q q q q 6 q qq q q q q q q 900 q q q q q q q qq q q q q q q q q q q q q q q q q 800 q q q qq q q q q q q q q q qq q q q q q q q q q q q 700 q q q q q q q q q q q q q q q q q q q q q q q q q q qq q 600 q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q 500 q qq q q q q q qq q q q q q qqq q q q q q q q q q q q q q qq q q q 400 q q q q q q q q q q q q q q q q q q q q q q q q q 300 q q q q q q q q q q q q q q q q q q qqqq qqq q q q q q q q q q q q 200 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q 0 q - 0 30 60 90 120 150 180 210 240 270 Figure 2.2: Powers 627i mod 941 for i = 1; 2; 3;::: any group and use the group law instead of multiplication. -
A Second Course in Algebraic Number Theory
A second course in Algebraic Number Theory Vlad Dockchitser Prerequisites: • Galois Theory • Representation Theory Overview: ∗ 1. Number Fields (Review, K; OK ; O ; ClK ; etc) 2. Decomposition of primes (how primes behave in eld extensions and what does Galois's do) 3. L-series (Dirichlet's Theorem on primes in arithmetic progression, Artin L-functions, Cheboterev's density theorem) 1 Number Fields 1.1 Rings of integers Denition 1.1. A number eld is a nite extension of Q Denition 1.2. An algebraic integer α is an algebraic number that satises a monic polynomial with integer coecients Denition 1.3. Let K be a number eld. It's ring of integer OK consists of the elements of K which are algebraic integers Proposition 1.4. 1. OK is a (Noetherian) Ring 2. , i.e., ∼ [K:Q] as an abelian group rkZ OK = [K : Q] OK = Z 3. Each can be written as with and α 2 K α = β=n β 2 OK n 2 Z Example. K OK Q Z ( p p [ a] a ≡ 2; 3 mod 4 ( , square free) Z p Q( a) a 2 Z n f0; 1g a 1+ a Z[ 2 ] a ≡ 1 mod 4 where is a primitive th root of unity Q(ζn) ζn n Z[ζn] Proposition 1.5. 1. OK is the maximal subring of K which is nitely generated as an abelian group 2. O`K is integrally closed - if f 2 OK [x] is monic and f(α) = 0 for some α 2 K, then α 2 OK . Example (Of Factorisation). -
Factoring Polynomials Over Finite Fields
Factoring Polynomials over Finite Fields More precisely: Factoring and testing irreduciblity of sparse polynomials over small finite fields Richard P. Brent MSI, ANU joint work with Paul Zimmermann INRIA, Nancy 27 August 2009 Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 1 / 64 Outline Introduction I Polynomials over finite fields I Irreducible and primitive polynomials I Mersenne primes Part 1: Testing irreducibility I Irreducibility criteria I Modular composition I Three algorithms I Comparison of the algorithms I The “best” algorithm I Some computational results Part 2: Factoring polynomials I Distinct degree factorization I Avoiding GCDs, blocking I Another level of blocking I Average-case complexity I New primitive trinomials Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 2 / 64 Polynomials over finite fields We consider univariate polynomials P(x) over a finite field F. The algorithms apply, with minor changes, for any small positive characteristic, but since time is limited we assume that the characteristic is two, and F = Z=2Z = GF(2). P(x) is irreducible if it has no nontrivial factors. If P(x) is irreducible of degree r, then [Gauss] r x2 = x mod P(x): 2r Thus P(x) divides the polynomial Pr (x) = x − x. In fact, Pr (x) is the product of all irreducible polynomials of degree d, where d runs over the divisors of r. Richard Brent (ANU) Factoring Polynomials over Finite Fields 27 August 2009 3 / 64 Counting irreducible polynomials Let N(d) be the number of irreducible polynomials of degree d. Thus X r dN(d) = deg(Pr ) = 2 : djr By Möbius inversion we see that X rN(r) = µ(d)2r=d : djr Thus, the number of irreducible polynomials of degree r is ! 2r 2r=2 N(r) = + O : r r Since there are 2r polynomials of degree r, the probability that a randomly selected polynomial is irreducible is ∼ 1=r ! 0 as r ! +1. -
Making NTRU As Secure As Worst-Case Problems Over Ideal Lattices
Making NTRU as Secure as Worst-Case Problems over Ideal Lattices Damien Stehlé1 and Ron Steinfeld2 1 CNRS, Laboratoire LIP (U. Lyon, CNRS, ENS Lyon, INRIA, UCBL), 46 Allée d’Italie, 69364 Lyon Cedex 07, France. [email protected] – http://perso.ens-lyon.fr/damien.stehle 2 Centre for Advanced Computing - Algorithms and Cryptography, Department of Computing, Macquarie University, NSW 2109, Australia [email protected] – http://web.science.mq.edu.au/~rons Abstract. NTRUEncrypt, proposed in 1996 by Hoffstein, Pipher and Sil- verman, is the fastest known lattice-based encryption scheme. Its mod- erate key-sizes, excellent asymptotic performance and conjectured resis- tance to quantum computers could make it a desirable alternative to fac- torisation and discrete-log based encryption schemes. However, since its introduction, doubts have regularly arisen on its security. In the present work, we show how to modify NTRUEncrypt to make it provably secure in the standard model, under the assumed quantum hardness of standard worst-case lattice problems, restricted to a family of lattices related to some cyclotomic fields. Our main contribution is to show that if the se- cret key polynomials are selected by rejection from discrete Gaussians, then the public key, which is their ratio, is statistically indistinguishable from uniform over its domain. The security then follows from the already proven hardness of the R-LWE problem. Keywords. Lattice-based cryptography, NTRU, provable security. 1 Introduction NTRUEncrypt, devised by Hoffstein, Pipher and Silverman, was first presented at the Crypto’96 rump session [14]. Although its description relies on arithmetic n over the polynomial ring Zq[x]=(x − 1) for n prime and q a small integer, it was quickly observed that breaking it could be expressed as a problem over Euclidean lattices [6]. -
The Discrete Log Problem and Elliptic Curve Cryptography
THE DISCRETE LOG PROBLEM AND ELLIPTIC CURVE CRYPTOGRAPHY NOLAN WINKLER Abstract. In this paper, discrete log-based public-key cryptography is ex- plored. Specifically, we first examine the Discrete Log Problem over a general cyclic group and algorithms that attempt to solve it. This leads us to an in- vestigation of the security of cryptosystems based over certain specific cyclic × groups: Fp, Fp , and the cyclic subgroup generated by a point on an elliptic curve; we ultimately see the highest security comes from using E(Fp) as our group. This necessitates an introduction of elliptic curves, which is provided. Finally, we conclude with cryptographic implementation considerations. Contents 1. Introduction 2 2. Public-Key Systems Over a Cyclic Group G 3 2.1. ElGamal Messaging 3 2.2. Diffie-Hellman Key Exchanges 4 3. Security & Hardness of the Discrete Log Problem 4 4. Algorithms for Solving the Discrete Log Problem 6 4.1. General Cyclic Groups 6 × 4.2. The cyclic group Fp 9 4.3. Subgroup Generated by a Point on an Elliptic Curve 10 5. Elliptic Curves: A Better Group for Cryptography 11 5.1. Basic Theory and Group Law 11 5.2. Finding the Order 12 6. Ensuring Security 15 6.1. Special Cases and Notes 15 7. Further Reading 15 8. Acknowledgments 15 References 16 1 2 NOLAN WINKLER 1. Introduction In this paper, basic knowledge of number theory and abstract algebra is assumed. Additionally, rather than beginning from classical symmetric systems of cryptog- raphy, such as the famous Caesar or Vigni`ere ciphers, we assume a familiarity with these systems and why they have largely become obsolete on their own. -
History of the Numerical Methods of Pi
History of the Numerical Methods of Pi By Andy Yopp MAT3010 Spring 2003 Appalachian State University This project is composed of a worksheet, a time line, and a reference sheet. This material is intended for an audience consisting of undergraduate college students with a background and interest in the subject of numerical methods. This material is also largely recreational. Although there probably aren't any practical applications for computing pi in an undergraduate curriculum, that is precisely why this material aims to interest the student to pursue their own studies of mathematics and computing. The time line is a compilation of several other time lines of pi. It is explained at the bottom of the time line and further on the worksheet reference page. The worksheet is a take-home assignment. It requires that the student know how to construct a hexagon within and without a circle, use iterative algorithms, and code in C. Therefore, they will need a variety of materials and a significant amount of time. The most interesting part of the project was just how well the numerical methods of pi have evolved. Currently, there are iterative algorithms available that are more precise than a graphing calculator with only a single iteration of the algorithm. This is a far cry from the rough estimate of 3 used by the early Babylonians. It is also a different entity altogether than what the circle-squarers have devised. The history of pi and its numerical methods go hand-in-hand through shameful trickery and glorious achievements. It is through the evolution of these methods which I will try to inspire students to begin their own research.