Elliptic Curve Cryptography and Digital Rights Management
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Protecting Public Keys and Signature Keys
Public-key cryptography offers certain advantages, providing the keys can be 3 adequately protected. For every security threat there must be an appropriate j countermeasure. Protecting Public Keys and Signature Keys Dorothy E. Denning, Purdue University With conventional one-key cryptography, the sender Consider an application environment in which each and receiver of a message share a secret encryption/ user has an intelligent terminal or personal workstation decryption key that allows both parties to encipher (en- where his private key is stored and all cryptographic crypt) and decipher (decrypt) secret messages transmitted operations are performed. This terminal is connected to between them. By separating the encryption and decryp- a nationwide network through a shared host, as shown in tion keys, public-key (two-key) cryptography has two at- Figure 1. The public-key directory is managed by a net- tractive properties that conventional cryptography lacks: work key server. Users communicate with each other or the ability to transmit messages in secrecy without any prior exchange of a secret key, and the ability to imple- ment digital signatures that are legally binding. Public- key encryption alone, however, does not guarantee either message secrecy or signatures. Unless the keys are ade- quately protected, a penetrator may be able to read en- crypted messages or forge signatures. This article discusses the problem ofprotecting keys in a nationwide network using public-key cryptography for secrecy and digital signatures. Particular attention is given to detecting and recovering from key compromises, especially when a high level of security is required. Public-key cryptosystems The concept of public-key cryptography was intro- duced by Diffie and Hellman in 1976.1 The basic idea is that each user A has a public key EA, which is registered in a public directory, and a private key DA, which is known only to the user. -
Secure Remote Password (SRP) Authentication
Secure Remote Password (SRP) Authentication Tom Wu Stanford University [email protected] Authentication in General ◆ What you are – Fingerprints, retinal scans, voiceprints ◆ What you have – Token cards, smart cards ◆ What you know – Passwords, PINs 2 Password Authentication ◆ Concentrates on “what you know” ◆ No long-term client-side storage ◆ Advantages – Convenience and portability – Low cost ◆ Disadvantages – People pick “bad” passwords – Most password methods are weak 3 Problems and Issues ◆ Dictionary attacks ◆ Plaintext-equivalence ◆ Forward secrecy 4 Dictionary Attacks ◆ An off-line, brute force guessing attack conducted by an attacker on the network ◆ Attacker usually has a “dictionary” of commonly-used passwords to try ◆ People pick easily remembered passwords ◆ “Easy-to-remember” is also “easy-to-guess” ◆ Can be either passive or active 5 Passwords in the Real World ◆ Entropy is less than most people think ◆ Dictionary words, e.g. “pudding”, “plan9” – Entropy: 20 bits or less ◆ Word pairs or phrases, e.g. “hate2die” – Represents average password quality – Entropy: around 30 bits ◆ Random printable text, e.g. “nDz2\u>O” – Entropy: slightly over 50 bits 6 Plaintext-equivalence ◆ Any piece of data that can be used in place of the actual password is “plaintext- equivalent” ◆ Applies to: – Password databases and files – Authentication servers (Kerberos KDC) ◆ One compromise brings entire system down 7 Forward Secrecy ◆ Prevents one compromise from causing further damage Compromising Should Not Compromise Current password Future passwords Old password Current password Current password Current or past session keys Current session key Current password 8 In The Beginning... ◆ Plaintext passwords – e.g. unauthenticated Telnet, rlogin, ftp – Still most common method in use ◆ “Encoded” passwords – e.g. -
Nonintersecting Brownian Motions on the Unit Circle: Noncritical Cases
The Annals of Probability 2016, Vol. 44, No. 2, 1134–1211 DOI: 10.1214/14-AOP998 c Institute of Mathematical Statistics, 2016 NONINTERSECTING BROWNIAN MOTIONS ON THE UNIT CIRCLE By Karl Liechty and Dong Wang1 DePaul University and National University of Singapore We consider an ensemble of n nonintersecting Brownian particles on the unit circle with diffusion parameter n−1/2, which are condi- tioned to begin at the same point and to return to that point after time T , but otherwise not to intersect. There is a critical value of T which separates the subcritical case, in which it is vanishingly un- likely that the particles wrap around the circle, and the supercritical case, in which particles may wrap around the circle. In this paper, we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n , and →∞ in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution. We also give a streamlined approach to identifying the Pearcey and tacnode pro- cesses in scaling limits. The formula of the tacnode correlation kernel is new and involves a solution to a Lax system for the Painlev´e II equation of size 2 2. The proofs are based on the determinantal × structure of the ensemble, asymptotic results for the related system of discrete Gaussian orthogonal polynomials, and a formulation of the correlation kernel in terms of a double contour integral. 1. Introduction. The probability models of nonintersecting Brownian motions have been studied extensively in last decade; see Tracy and Widom (2004, 2006), Adler and van Moerbeke (2005), Adler, Orantin and van Moer- beke (2010), Delvaux, Kuijlaars and Zhang (2011), Johansson (2013), Ferrari and Vet˝o(2012), Katori and Tanemura (2007) and Schehr et al. -
NSS: an NTRU Lattice –Based Signature Scheme
ISSN(Online) : 2319 - 8753 ISSN (Print) : 2347 - 6710 International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 4, Issue 4, April 2015 NSS: An NTRU Lattice –Based Signature Scheme S.Esther Sukila Department of Mathematics, Bharath University, Chennai, Tamil Nadu, India ABSTRACT: Whenever a PKC is designed, it is also analyzed whether it could be used as a signature scheme. In this paper, how the NTRU concept can be used to form a digital signature scheme [1] is described. I. INTRODUCTION Digital Signature Schemes The notion of a digital signature may prove to be one of the most fundamental and useful inventions of modern cryptography. A digital signature or digital signature scheme is a mathematical scheme for demonstrating the authenticity of a digital message or document. The creator of a message can attach a code, the signature, which guarantees the source and integrity of the message. A valid digital signature gives a recipient reason to believe that the message was created by a known sender and that it was not altered in transit. Digital signatures are commonly used for software distribution, financial transactions etc. Digital signatures are equivalent to traditional handwritten signatures. Properly implemented digital signatures are more difficult to forge than the handwritten type. 1.1A digital signature scheme typically consists of three algorithms A key generationalgorithm, that selects a private key uniformly at random from a set of possible private keys. The algorithm outputs the private key and a corresponding public key. A signing algorithm, that given a message and a private key produces a signature. -
Examples from Complex Geometry
Examples from Complex Geometry Sam Auyeung November 22, 2019 1 Complex Analysis Example 1.1. Two Heuristic \Proofs" of the Fundamental Theorem of Algebra: Let p(z) be a polynomial of degree n > 0; we can even assume it is a monomial. We also know that the number of zeros is at most n. We show that there are exactly n. 1. Proof 1: Recall that polynomials are entire functions and that Liouville's Theorem says that a bounded entire function is in fact constant. Suppose that p has no roots. Then 1=p is an entire function and it is bounded. Thus, it is constant which means p is a constant polynomial and has degree 0. This contradicts the fact that p has positive degree. Thus, p must have a root α1. We can then factor out (z − α1) from p(z) = (z − α1)q(z) where q(z) is an (n − 1)-degree polynomial. We may repeat the above process until we have a full factorization of p. 2. Proof 2: On the real line, an algorithm for finding a root of a continuous function is to look for when the function changes signs. How do we generalize this to C? Instead of having two directions, we have a whole S1 worth of directions. If we use colors to depict direction and brightness to depict magnitude, we can plot a graph of a continuous function f : C ! C. Near a zero, we'll see all colors represented. If we travel in a loop around any point, we can keep track of whether it passes through all the colors; around a zero, we'll pass through all the colors, possibly many times. -
A Review on Elliptic Curve Cryptography for Embedded Systems
International Journal of Computer Science & Information Technology (IJCSIT), Vol 3, No 3, June 2011 A REVIEW ON ELLIPTIC CURVE CRYPTOGRAPHY FOR EMBEDDED SYSTEMS Rahat Afreen 1 and S.C. Mehrotra 2 1Tom Patrick Institute of Computer & I.T, Dr. Rafiq Zakaria Campus, Rauza Bagh, Aurangabad. (Maharashtra) INDIA [email protected] 2Department of C.S. & I.T., Dr. B.A.M. University, Aurangabad. (Maharashtra) INDIA [email protected] ABSTRACT Importance of Elliptic Curves in Cryptography was independently proposed by Neal Koblitz and Victor Miller in 1985.Since then, Elliptic curve cryptography or ECC has evolved as a vast field for public key cryptography (PKC) systems. In PKC system, we use separate keys to encode and decode the data. Since one of the keys is distributed publicly in PKC systems, the strength of security depends on large key size. The mathematical problems of prime factorization and discrete logarithm are previously used in PKC systems. ECC has proved to provide same level of security with relatively small key sizes. The research in the field of ECC is mostly focused on its implementation on application specific systems. Such systems have restricted resources like storage, processing speed and domain specific CPU architecture. KEYWORDS Elliptic curve cryptography Public Key Cryptography, embedded systems, Elliptic Curve Digital Signature Algorithm ( ECDSA), Elliptic Curve Diffie Hellman Key Exchange (ECDH) 1. INTRODUCTION The changing global scenario shows an elegant merging of computing and communication in such a way that computers with wired communication are being rapidly replaced to smaller handheld embedded computers using wireless communication in almost every field. This has increased data privacy and security requirements. -
Contents 5 Elliptic Curves in Cryptography
Cryptography (part 5): Elliptic Curves in Cryptography (by Evan Dummit, 2016, v. 1.00) Contents 5 Elliptic Curves in Cryptography 1 5.1 Elliptic Curves and the Addition Law . 1 5.1.1 Cubic Curves, Weierstrass Form, Singular and Nonsingular Curves . 1 5.1.2 The Addition Law . 3 5.1.3 Elliptic Curves Modulo p, Orders of Points . 7 5.2 Factorization with Elliptic Curves . 10 5.3 Elliptic Curve Cryptography . 14 5.3.1 Encoding Plaintexts on Elliptic Curves, Quadratic Residues . 14 5.3.2 Public-Key Encryption with Elliptic Curves . 17 5.3.3 Key Exchange and Digital Signatures with Elliptic Curves . 20 5 Elliptic Curves in Cryptography In this chapter, we will introduce elliptic curves and describe how they are used in cryptography. Elliptic curves have a long and interesting history and arise in a wide range of contexts in mathematics. The study of elliptic curves involves elements from most of the major disciplines of mathematics: algebra, geometry, analysis, number theory, topology, and even logic. Elliptic curves appear in the proofs of many deep results in mathematics: for example, they are a central ingredient in the proof of Fermat's Last Theorem, which states that there are no positive integer solutions to the equation xn + yn = zn for any integer n ≥ 3. Our goals are fairly modest in comparison, so we will begin by outlining the basic algebraic and geometric properties of elliptic curves and motivate the addition law. We will then study the behavior of elliptic curves modulo p: ultimately, there is a fairly strong analogy between the structure of the points on an elliptic curve modulo p and the integers modulo n. -
Basics of Digital Signatures &
> DOCUMENT SIGNING > eID VALIDATION > SIGNATURE VERIFICATION > TIMESTAMPING & ARCHIVING > APPROVAL WORKFLOW Basics of Digital Signatures & PKI This document provides a quick background to PKI-based digital signatures and an overview of how the signature creation and verification processes work. It also describes how the cryptographic keys used for creating and verifying digital signatures are managed. 1. Background to Digital Signatures Digital signatures are essentially “enciphered data” created using cryptographic algorithms. The algorithms define how the enciphered data is created for a particular document or message. Standard digital signature algorithms exist so that no one needs to create these from scratch. Digital signature algorithms were first invented in the 1970’s and are based on a type of cryptography referred to as “Public Key Cryptography”. By far the most common digital signature algorithm is RSA (named after the inventors Rivest, Shamir and Adelman in 1978), by our estimates it is used in over 80% of the digital signatures being used around the world. This algorithm has been standardised (ISO, ANSI, IETF etc.) and been extensively analysed by the cryptographic research community and you can say with confidence that it has withstood the test of time, i.e. no one has been able to find an efficient way of cracking the RSA algorithm. Another more recent algorithm is ECDSA (Elliptic Curve Digital Signature Algorithm), which is likely to become popular over time. Digital signatures are used everywhere even when we are not actually aware, example uses include: Retail payment systems like MasterCard/Visa chip and pin, High-value interbank payment systems (CHAPS, BACS, SWIFT etc), e-Passports and e-ID cards, Logging on to SSL-enabled websites or connecting with corporate VPNs. -
An Efficient Approach to Point-Counting on Elliptic Curves
mathematics Article An Efficient Approach to Point-Counting on Elliptic Curves from a Prominent Family over the Prime Field Fp Yuri Borissov * and Miroslav Markov Department of Mathematical Foundations of Informatics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria; [email protected] * Correspondence: [email protected] Abstract: Here, we elaborate an approach for determining the number of points on elliptic curves 2 3 from the family Ep = fEa : y = x + a (mod p), a 6= 0g, where p is a prime number >3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for the quantities of interest-reduced modulo p. It allows to advance an efficient technique to compute the 2 six cardinalities associated with the family Ep, for p ≡ 1 (mod 3), whose complexity is O˜ (log p), thus improving the best-known algorithmic solution with almost an order of magnitude. Keywords: elliptic curve over Fp; Hasse bound; high-order residue modulo prime 1. Introduction The elliptic curves over finite fields play an important role in modern cryptography. We refer to [1] for an introduction concerning their cryptographic significance (see, as well, Citation: Borissov, Y.; Markov, M. An the pioneering works of V. Miller and N. Koblitz from 1980’s [2,3]). Briefly speaking, the Efficient Approach to Point-Counting advantage of the so-called elliptic curve cryptography (ECC) over the non-ECC is that it on Elliptic Curves from a Prominent requires smaller keys to provide the same level of security. Family over the Prime Field Fp. -
Lecture 9 Riemann Surfaces, Elliptic Functions Laplace Equation
Lecture 9 Riemann surfaces, elliptic functions Laplace equation Consider a Riemannian metric gµν in two dimensions. In two dimensions it is always possible to choose coordinates (x,y) to diagonalize it as ds2 = Ω(x,y)(dx2 + dy2). We can then combine them into a complex combination z = x+iy to write this as ds2 =Ωdzdz¯. It is actually a K¨ahler metric since the condition ∂[i,gj]k¯ = 0 is trivial if i, j = 1. Thus, an orientable Riemannian manifold in two dimensions is always K¨ahler. In the diagonalized form of the metric, the Laplace operator is of the form, −1 ∆=4Ω ∂z∂¯z¯. Thus, any solution to the Laplace equation ∆φ = 0 can be expressed as a sum of a holomorphic and an anti-holomorphid function. ∆φ = 0 φ = f(z)+ f¯(¯z). → In the following, we assume Ω = 1 so that the metric is ds2 = dzdz¯. It is not difficult to generalize our results for non-constant Ω. Now, we would like to prove the following formula, 1 ∂¯ = πδ(z), z − where δ(z) = δ(x)δ(y). Since 1/z is holomorphic except at z = 0, the left-hand side should vanish except at z = 0. On the other hand, by the Stokes theorem, the integral of the left-hand side on a disk of radius r gives, 1 i dz dxdy ∂¯ = = π. Zx2+y2≤r2 z 2 I|z|=r z − This proves the formula. Thus, the Green function G(z, w) obeying ∆zG(z, w) = 4πδ(z w), − should behave as G(z, w)= log z w 2 = log(z w) log(¯z w¯), − | − | − − − − near z = w. -
A Study on the Security of Ntrusign Digital Signature Scheme
A Thesis for the Degree of Master of Science A Study on the Security of NTRUSign digital signature scheme SungJun Min School of Engineering Information and Communications University 2004 A Study on the Security of NTRUSign digital signature scheme A Study on the Security of NTRUSign digital signature scheme Advisor : Professor Kwangjo Kim by SungJun Min School of Engineering Information and Communications University A thesis submitted to the faculty of Information and Commu- nications University in partial fulfillment of the requirements for the degree of Master of Science in the School of Engineering Daejeon, Korea Jan. 03. 2004 Approved by (signed) Professor Kwangjo Kim Major Advisor A Study on the Security of NTRUSign digital signature scheme SungJun Min We certify that this work has passed the scholastic standards required by Information and Communications University as a thesis for the degree of Master of Science Jan. 03. 2004 Approved: Chairman of the Committee Kwangjo Kim, Professor School of Engineering Committee Member Jae Choon Cha, Assistant Professor School of Engineering Committee Member Dae Sung Kwon, Ph.D NSRI M.S. SungJun Min 20022052 A Study on the Security of NTRUSign digital signature scheme School of Engineering, 2004, 43p. Major Advisor : Prof. Kwangjo Kim. Text in English Abstract The lattices have been studied by cryptographers for last decades, both in the field of cryptanalysis and as a source of hard problems on which to build encryption schemes. Interestingly, though, research about building secure and efficient -
25 Modular Forms and L-Series
18.783 Elliptic Curves Spring 2015 Lecture #25 05/12/2015 25 Modular forms and L-series As we will show in the next lecture, Fermat's Last Theorem is a direct consequence of the following theorem [11, 12]. Theorem 25.1 (Taylor-Wiles). Every semistable elliptic curve E=Q is modular. In fact, as a result of subsequent work [3], we now have the stronger result, proving what was previously known as the modularity conjecture (or Taniyama-Shimura-Weil conjecture). Theorem 25.2 (Breuil-Conrad-Diamond-Taylor). Every elliptic curve E=Q is modular. Our goal in this lecture is to explain what it means for an elliptic curve over Q to be modular (we will also define the term semistable). This requires us to delve briefly into the theory of modular forms. Our goal in doing so is simply to understand the definitions and the terminology; we will omit all but the most straight-forward proofs. 25.1 Modular forms Definition 25.3. A holomorphic function f : H ! C is a weak modular form of weight k for a congruence subgroup Γ if f(γτ) = (cτ + d)kf(τ) a b for all γ = c d 2 Γ. The j-function j(τ) is a weak modular form of weight 0 for SL2(Z), and j(Nτ) is a weak modular form of weight 0 for Γ0(N). For an example of a weak modular form of positive weight, recall the Eisenstein series X0 1 X0 1 G (τ) := G ([1; τ]) := = ; k k !k (m + nτ)k !2[1,τ] m;n2Z 1 which, for k ≥ 3, is a weak modular form of weight k for SL2(Z).