Basic Concepts from Probability, Complexity, Algebra and Number Theory

Contributors

Modular exponentiation in a finite field is the basic computation involved in most public key crypto systems, such as Diffie-Hellman key exchange, ElGamal, etc. The current paper presents a new parallel architecture whereby the modular multiplication and squaring can be processed simultaneously in GF(2m) in m clock cycles using a cellular automata. Since the proposed cellular automata architecture is simple, regular, modular, cascadable, it can also be utilized efficiently for the implementation of VLSI.

Let V be a finitely generated multiplicative semi-group with r generators in the ring of integers \(\mathbb{Z}_\mathbb{K}\)of an algebraic number field \(\mathbb{K}\)of degree n over ℚ. We use various bounds for character sums to obtain results on the distribution of the residues of elements of V modulo an integer ideal q. In the simplest case, when \(\mathbb{K} = \mathbb{Q}\)and r = 1 this is a classical question on the distribution of residues of an exponential function, which may be interpreted as concerning the quality of the linear congruential pseudo-random number generator. Besides this well known application we consider several other problems from algebraic number theory, the theory of function fields over a finite field, complexity theory, cryptography, and coding theory where results on the distribution of some group V modulo q play a central role.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

Aimed at mathematicians and computer scientists who will only be exposed to one course in this area, this is a brief but rigorous introduction to the abstract theory of computation, sometimes also referred to as recursion theory. It develops major themes in computability theory, such as Rice's theorem and the recursion theorem, and provides a systematic account of Blum's complexity theory as well as an introduction to the theory of computable real numbers and functions. The book is intended as a university text, but it may also be used for self-study. Appropriate exercises and solutions are included.

An Algorithm to Find Square Root of Quadratic Residues over Finite Fields using Primitive Elements

This tutorial reviews basic concepts in complexity theory, as well as various No Free Lunch results and how these results relate to computational complexity. The tutorial explains basic concepts in an informal fashion that illuminates key concepts. No Free Lunch theorems for search can be summarized by the following result: For all possible performance measures, no search algorithm is better than another when its performance is averaged over all possible discrete functions.

Complexity theory is concerned with estimating the (generally worst-case) amount of computational resources (time, memory, etc.) required to solve different problems, while scientific computing and numerical analysis are concerned with computing given functions on (generally error-contaminated) input data and estimating bounds on the precision with which different algorithms can perform these tasks. The concept of computational intractability describes tasks which appear to require inordinate amount of computing effort in the worst case. I here argue that this has a counterpart in numerical analysis, namely, ill-conditioning of data problems, an idea introduced by Alan Turing in 1948. By converting the classical heat equation to a numerical backwards problem, I show that the resulting system Mb = y is exponentially badly conditioned in the sense that the matrix M has an exponentially exploding condition number due to eigenvalues collapsing near an accumulator at 0. This example suggests a link between ideas (and problems) from complexity theory and numerical analysis and hints at a deep duality between computational intractability and the irreversibility of the arrow of time.

The Dual Complexity Space as the Dual of a Normed Cone

Modular arithmetic has often been regarded as something of amathemat-ical curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digitalsecurity systems such as RSA, this important branch of mathematics is now consideredessential knowledge for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular. This chapter describes how the modern graphical spreadsheet may be used to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Students may then gain structural insight and the foundations laid for applications to such areas as hashing, random number generation, and public-key cryptography.

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH. As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.

(ProQuest: ... denotes formulae omitted.)INTRODUCTIONThe Oliver Heaviside function. Fermat's Last Theorem. Modular arithmetic. Dihedral groups. These areas are commonly studied in college, perhaps in an undergraduate course in engineering or a graduate-level seminar in mathematics or computer science. Is there a place for these advanced topics in the high school curriculum? Can teachers of 9-12 connect content to these areas? Most importantly, can students benefit by engaging in problem settings where these ideas are introduced? In this article, we argue that the answer to each of these questions is yes.Teachers of mathematics undoubtedly see important connections in their subject area but these connections may not be evident to students. By blurring the boundary between the high school and college curriculum, we offer a number of benefits to the high school experience. First, exploring challenging content can whet students' appetites to the investigative nature of mathematics. Even if these investigations prove difficult, students confront the limits of their mathematical knowledge and are encouraged to learn more; this is a valuable lesson. Second, there are beneficial ties to areas such as engineering, physics, biology, or art. In very simple cases, innocent questions can carve a pathway to research-oriented mathematics. Third, students may come to realize that pure mathematics can be interesting and engaging-lest we say this is often shocking to students! Finally, these experiences may provide some much needed enrichment to those who need it. Interesting problems that use novel tools can leave a lasting impression on students.This article is principally about introducing young students to mathematical ideas and practices not typically explored at the high school level. We discuss four problems with appropriate origins in high school settings. The problems then flourish, extend, and enlighten by using sophisticated tools typically reserved for college mathematics. In The Parking Garage, we introduce a modeling problem suitable for a standard algebra course (Grade 8-10) but we use the Heaviside function as the centerpiece. Students learn about piecewise and discontinuous functions-important objectives in the high school curriculum-but via a tool typically used by engineers. In Clock Arithmetic, students operate in the familiar modulo 12 as this builds upon previously learned elapsed time problems that surface in CCSSM in Grade 4 (CCSSI, 2010). Then through student-created informal definitions and generalizations, discussions in modular arithmetic lead to non-trivial extensions in number theory and randomness. In Pythagoras meets Fermat and Newton, students begin with a ubiquitous mathematical theorem (Grade 8) and engage in the interplay of informal reasoning (conjecturing, experimenting) and formal reasoning (generalizing, extrapolating). This experience mirrors that of professional mathematicians and finishes with the ultimate prize: a generalized mathematical proof. In Star Polygons, we present the definition of a star polygon, a construction that naturally lends itself to line and rotational symmetry explorations, recurrent topics in the K-12 curriculum (Grades 4, 9-10). However, further investigation uncovers rich connections to group theory, usually reserved for undergraduate mathematics students but accessible to high school students given this context.In each of these cases, we import tools and practices more prevalent to college mathematics into the high school experience. Doing so supports mathematics as a unified body of knowledge with clear connections from grade to grade (CCSSI, 2010). As NCTM (2000) reminds us, Mathematics is an integrated field of study.. ..The notion that mathematical ideas are connected should permeate the school mathematics experience at all levels (p. 64).THE PROBLEMSThe Parking Garage. One of the most important concepts in all of mathematics is that of function. …

NP = PCP(logn, 1) and related results crucially depend upon the close connection between the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [30]. The strongest previously known connection for this test states that a function passes the test with probability δ for some δ > 7/8 iff the function has agreement ≈ δ with a polynomial of degree d. We present a new, and surprisingly strong, analysis which shows that the preceding statement is true for arbitrarily small δ, provided the field size is polynomially larger than d/δ. The analysis uses a version of Hilbert irreducibility, a tool of algebraic geometry. As a consequence we obtain an alternate construction for the following proof system: A constant prover 1-round proof system for NP languages in which the verifier uses O(logn) random bits, receives answers of size O(logn) bits, and has an error probability of at most 2− log 1− . Such a proof system, which implies the NP-hardness of approximating Set Cover to within Ω(logn) factors, has already been obtained by Raz and Safra [29]. Raz and Safra obtain their result by giving a strong analysis, in the sense described above, of a new low-degree test that they present. A second consequence of our analysis is a self tester/corrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on δ fraction of inputs where δ = 1/ |F| 0.5, then the tester/corrector determines δ and generates O( 1 δ ) values for every input, such that one of them is the correct output. In fact, our results yield something stronger: Given the buggy program, we can construct O( 1 δ ) randomized programs such that one of them is correct on every input, with high probability. Such a strong selfcorrector is a useful tool in complexity theory with some applications known. ∗arora@cs.princeton.edu. Supported by an NSF CAREER award, an Alfred P. Sloan Fellowship, and a Packard Fellowship. †madhu@lcs.mit.edu. Laboratory for Computer Science, MIT, Cambridge, MA 02139. Part of this work was done when this author was at the IBM Thomas J. Watson Research Center.

For modern cryptographic systems, the public key cryptosystem such as RSA requires modular exponentiation (M E mod N). The M, E and N are either as large as the 1024-bit integers or even larger, it is not a very good idea to directly compute M E mod N. Recently, there are many techniques have been invented to solve the time-consuming computations of such time-consuming modular exponentiation. Among these useful algorithms, the “binary (square-and-multiply) algorithm” reduces the amount of modulo multiplications. As the “signed-digit representation algorithm” has the property of the nonzero digit occurrence probability equals to 1/3, taking this advantage, this method can more effectively decrease the amount of modular multiplications. Moreover, by using the technique of recording the common parts in the folded substrings, the “folding-exponent algorithm” can improve the efficiency of the binary algorithm, thus can further decrease the computational complexity of modular exponentiation. In this paper, a new modular exponentiation algorithm is proposed which based on the binary algorithm, signed-digit representation, and the folding-exponent technique. By using the parallel processing technique, in our proposed method, the modular multiplications and modular squaring can be executed in parallel, and thus lower down the computational complexity to k + 3 multiplications. As modular squaring operation over GF(2 n ) is carried out by a simple cyclic right shift operation, the computational complexity of our proposed method can be further reduced to 29k/36 + 3 multiplications.

In traditional multiple precision large integer multiplication algorithm, the required number of additions approximates the number of multiplications needed. In some platforms, the great number of add instructions will occupy about half of computing latency in the overall implementation. In this paper, we propose a multiplication algorithm using separated multiply-add-with-carry instruction supported by NVIDIA GPUs. In the algorithm, we reorder the computational sequence, in which nearly all additions and carry flags handling can be combined with the multiplication instructions. The number of add instructions needed decreases from $$O(n^2)$$ in prevailing schoolbook algorithm to $$O(n)$$ . Our resulting 256-bit modular multiplication and modular square over Mersenne prime respectively achieve 3.3837 billion and 5.9928 billion operations per second and reach 96 % of GPU hardware limitation. An elliptic curve point multiplication implementation using our algorithm achieves 43.6 % speedup compared to the existing fastest work.