PPT-Pseudorandom Generators for Polynomial Threshold Functions

1 Raghu Meka UT Austin joint work with David Zuckerman Polynomial Threshold Functions 2 Applications Complexity theory learning theory voting theory quantum computing Halfspaces 3. Thomas . Holenstein. Iftach Haitner. Salil Vadhan. Hoeteck Wee. Joint With. Omer Reingold. Cryptography. Rich array of . applications and . powerful . implementations.. In some cases (. e.g. Zero-Knowledge), . Raghu Meka (IAS). Parikshit Gopalan, Omer Reingold (MSR-SVC) Luca Trevian (Stanford), Salil Vadhan (Harvard). Can we generate random bits?. Can we generate random bits?. Pseudorandom Generators. Stretch bits to fool a class of “test functions” . Objectives: Identify Polynomial functions. Determine end behavior recognize characteristics of polynomial functions. Use factoring to find zeros of polynomial functions.. Polynomials of degree 2 or higher have graphs that are smooth and continuous. By smooth we mean the graphs have rounded curves with no sharp corners. By continuous we mean the graphs have no breaks and can be drawn without lifting your pencil from the rectangular coordinate system.. Algebra 2. Chapter 5. This Slideshow was developed to accompany the textbook. Larson Algebra 2. By Larson. , R., Boswell, L., . Kanold. , T. D., & Stiff, L. . 2011 . Holt . McDougal. Some examples and diagrams are taken from the textbook.. Defn. : . Polynomial function. In the form of: . ..  . The coefficients are real numbers.. The exponents are non-negative integers.. The domain of the function is the set of all real numbers.. Definitions. Coefficient. : the numerical factor of each term.. Constant. : the term without a variable.. Term. : a number or a product of a number and variables raised . to a power.. Polynomial. : a finite sum of terms of the form . Objectives:. To approximate . x. -intercepts of a polynomial function with a graphing utility. To locate and use relative . extrema. of polynomial functions. To sketch the graphs of polynomial functions. Based on: William . Stallings, Cryptography and Network Security . . Chapter 7. Pseudorandom Number Generators . and Stream Ciphers. Random Numbers. A number of cryptographic protocols make use of random binary numbers:. Standard 15. Graph and analyze polynomial and radical functions to determine:. Domain and range. X and y intercepts. Maximum and minimum values. Intervals of increasing and decreasing. End behavior. With the function: f(x) = . Cryptography Lecture 6 Pseudorandom generators (PRGs) Let G be an efficient, deterministic algorithm that expands a short seed into a longer output Specifically, let |G(x)| = p(|x|) G is a PRG if: when the distribution of x is uniform, the distribution of G(x) is “indistinguishable from uniform” Objective: . Recognize the shape of basic polynomial functions. Describe the graph of a polynomial function. Identify properties of general polynomial functions: Continuity, End Behaviour, Intercepts, Local . k. c. m. c. . . . Enc. k. (m). k. m. 1. c. 1. . . . Enc. k. (m. 1. ). m. 2. c. 2. . . . Enc. k. (m. 2. ). c. 1. c. 2. Is the threat model too strong?. In practice, there are many ways an attacker can . Keyed functions. Let F: {0,1}. *. x {0,1}. *. .  {0,1}. *. be an efficient, deterministic algorithm. Define . F. k. (x) = F(k, x). The first input is called the . key. A. ssume F is . length preserving. . (PRGs). Let G be an efficient, deterministic algorithm . that expands a . short . seed. . into a . longer . output. Specifically, let |G(x)| = p(|x|). G is a PRG if: when the distribution of x is uniform, the distribution of G(x) is “indistinguishable from uniform”.