Quadratic Residues (Week 3-Monday)
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Quadratic Reciprocity and Computing Modular Square Roots.Pdf
12 Quadratic reciprocity and computing modular square roots In §2.8, we initiated an investigation of quadratic residues. This chapter continues this investigation. Recall that an integer a is called a quadratic residue modulo a positive integer n if gcd(a, n) = 1 and a ≡ b2 (mod n) for some integer b. First, we derive the famous law of quadratic reciprocity. This law, while his- torically important for reasons of pure mathematical interest, also has important computational applications, including a fast algorithm for testing if an integer is a quadratic residue modulo a prime. Second, we investigate the problem of computing modular square roots: given a quadratic residue a modulo n, compute an integer b such that a ≡ b2 (mod n). As we will see, there are efficient probabilistic algorithms for this problem when n is prime, and more generally, when the factorization of n into primes is known. 12.1 The Legendre symbol For an odd prime p and an integer a with gcd(a, p) = 1, the Legendre symbol (a j p) is defined to be 1 if a is a quadratic residue modulo p, and −1 otherwise. For completeness, one defines (a j p) = 0 if p j a. The following theorem summarizes the essential properties of the Legendre symbol. Theorem 12.1. Let p be an odd prime, and let a, b 2 Z. Then we have: (i) (a j p) ≡ a(p−1)=2 (mod p); in particular, (−1 j p) = (−1)(p−1)=2; (ii) (a j p)(b j p) = (ab j p); (iii) a ≡ b (mod p) implies (a j p) = (b j p); 2 (iv) (2 j p) = (−1)(p −1)=8; p−1 q−1 (v) if q is an odd prime, then (p j q) = (−1) 2 2 (q j p). -
Chapter 9 Quadratic Residues
Chapter 9 Quadratic Residues 9.1 Introduction Definition 9.1. We say that a 2 Z is a quadratic residue mod n if there exists b 2 Z such that a ≡ b2 mod n: If there is no such b we say that a is a quadratic non-residue mod n. Example: Suppose n = 10. We can determine the quadratic residues mod n by computing b2 mod n for 0 ≤ b < n. In fact, since (−b)2 ≡ b2 mod n; we need only consider 0 ≤ b ≤ [n=2]. Thus the quadratic residues mod 10 are 0; 1; 4; 9; 6; 5; while 3; 7; 8 are quadratic non-residues mod 10. Proposition 9.1. If a; b are quadratic residues mod n then so is ab. Proof. Suppose a ≡ r2; b ≡ s2 mod p: Then ab ≡ (rs)2 mod p: 9.2 Prime moduli Proposition 9.2. Suppose p is an odd prime. Then the quadratic residues coprime to p form a subgroup of (Z=p)× of index 2. Proof. Let Q denote the set of quadratic residues in (Z=p)×. If θ :(Z=p)× ! (Z=p)× denotes the homomorphism under which r 7! r2 mod p 9–1 then ker θ = {±1g; im θ = Q: By the first isomorphism theorem of group theory, × jkerθj · j im θj = j(Z=p) j: Thus Q is a subgroup of index 2: p − 1 jQj = : 2 Corollary 9.1. Suppose p is an odd prime; and suppose a; b are coprime to p. Then 1. 1=a is a quadratic residue if and only if a is a quadratic residue. -
Number Theory Summary
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467: Cryptography and Computer Security Handout #11 Professor M. J. Fischer November 13, 2017 Number Theory Summary Integers Let Z denote the integers and Z+ the positive integers. Division For a 2 Z and n 2 Z+, there exist unique integers q; r such that a = nq + r and 0 ≤ r < n. We denote the quotient q by ba=nc and the remainder r by a mod n. We say n divides a (written nja) if a mod n = 0. If nja, n is called a divisor of a. If also 1 < n < jaj, n is said to be a proper divisor of a. Greatest common divisor The greatest common divisor (gcd) of integers a; b (written gcd(a; b) or simply (a; b)) is the greatest integer d such that d j a and d j b. If gcd(a; b) = 1, then a and b are said to be relatively prime. Euclidean algorithm Computes gcd(a; b). Based on two facts: gcd(0; b) = b; gcd(a; b) = gcd(b; a − qb) for any q 2 Z. For rapid convergence, take q = ba=bc, in which case a − qb = a mod b. Congruence For a; b 2 Z and n 2 Z+, we write a ≡ b (mod n) iff n j (b − a). Note a ≡ b (mod n) iff (a mod n) = (b mod n). + ∗ Modular arithmetic Fix n 2 Z . Let Zn = f0; 1; : : : ; n − 1g and let Zn = fa 2 Zn j gcd(a; n) = 1g. For integers a; b, define a⊕b = (a+b) mod n and a⊗b = ab mod n. -
Appendices A. Quadratic Reciprocity Via Gauss Sums
1 Appendices We collect some results that might be covered in a first course in algebraic number theory. A. Quadratic Reciprocity Via Gauss Sums A1. Introduction In this appendix, p is an odd prime unless otherwise specified. A quadratic equation 2 modulo p looks like ax + bx + c =0inFp. Multiplying by 4a, we have 2 2ax + b ≡ b2 − 4ac mod p Thus in studying quadratic equations mod p, it suffices to consider equations of the form x2 ≡ a mod p. If p|a we have the uninteresting equation x2 ≡ 0, hence x ≡ 0, mod p. Thus assume that p does not divide a. A2. Definition The Legendre symbol a χ(a)= p is given by 1ifa(p−1)/2 ≡ 1modp χ(a)= −1ifa(p−1)/2 ≡−1modp. If b = a(p−1)/2 then b2 = ap−1 ≡ 1modp,sob ≡±1modp and χ is well-defined. Thus χ(a) ≡ a(p−1)/2 mod p. A3. Theorem a The Legendre symbol ( p ) is 1 if and only if a is a quadratic residue (from now on abbre- viated QR) mod p. Proof.Ifa ≡ x2 mod p then a(p−1)/2 ≡ xp−1 ≡ 1modp. (Note that if p divides x then p divides a, a contradiction.) Conversely, suppose a(p−1)/2 ≡ 1modp.Ifg is a primitive root mod p, then a ≡ gr mod p for some r. Therefore a(p−1)/2 ≡ gr(p−1)/2 ≡ 1modp, so p − 1 divides r(p − 1)/2, hence r/2 is an integer. But then (gr/2)2 = gr ≡ a mod p, and a isaQRmodp. -
Lecture 10: Quadratic Residues
Lecture 10: Quadratic residues Rajat Mittal IIT Kanpur n Solving polynomial equations, anx + ··· + a1x + a0 = 0 , has been of interest from a long time in mathematics. For equations up to degree 4, we have an explicit formula for the solutions. It has also been shown that no such explicit formula can exist for degree higher than 4. What about polynomial equations modulo p? Exercise 1. When does the equation ax + b = 0 mod p has a solution? This lecture will focus on solving quadratic equations modulo a prime number p. In other words, we are 2 interested in solving a2x +a1x+a0 = 0 mod p. First thing to notice, we can assume that every coefficient ai can only range between 0 to p − 1. In the assignment, you will show that we only need to consider equations 2 of the form x + a1x + a0 = 0. 2 Exercise 2. When will x + a1x + a0 = 0 mod 2 not have a solution? So, for further discussion, we are only interested in solving quadratic equations modulo p, where p is an odd prime. For odd primes, inverse of 2 always exists. 2 −1 2 −2 2 x + a1x + a0 = 0 mod p , (x + 2 a1) = 2 a1 − a0 mod p: −1 −2 2 Taking y = x + 2 a1 and b = 2 a1 − a0, 2 2 Exercise 3. solving quadratic equation x + a1x + a0 = 0 mod p is same as solving y = b mod p. The small amount of work we did above simplifies the original problem. We only need to solve, when a number (b) has a square root modulo p, to solve quadratic equations modulo p. -
Consecutive Quadratic Residues and Quadratic Nonresidue Modulo P
Consecutive Quadratic Residues And Quadratic Nonresidue Modulo p N. A. Carella Abstract: Let p be a large prime, and let k log p. A new proof of the existence of ≪ any pattern of k consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues np modulo p shows that n (log p)(log log p). p ≪ 1 Introduction Given a prime p 2, a nonzero element u F is called a quadratic residue, equiva- ≥ ∈ p lently, a square modulo p whenever the quadratic congruence x2 u 0mod p is solvable. − ≡ Otherwise, it called a quadratic nonresidue. A finite field Fp contains (p + 1)/2 squares = u2 mod p : 0 u < p/2 , including zero. The quadratic residues are uniformly R { ≤ } distributed over the interval [1,p 1]. Likewise, the quadratic nonresidues are uniformly − distributed over the same interval. Let k 1 be a small integer. This note is concerned with the longest runs of consecutive ≥ quadratic residues and consecutive quadratic nonresidues (or any pattern) u, u + 1, u + 2, . , u + k 1, (1) − in the finite field F , and large subsets F . Let N(k,p) be a tally of the number of p A ⊂ p sequences 1. Theorem 1.1. Let p 2 be a large prime, and let k = O (log p) be an integer. Then, the ≥ finite field Fp contains k consecutive quadratic residues (or quadratic nonresidues or any pattern). Furthermore, the number of k tuples has the asymptotic formulas p 1 k 1 (i) N(k,p)= 1 1+ O , if k 1. -
QUADRATIC RECIPROCITY Abstract. the Goals of This Project Are to Have
QUADRATIC RECIPROCITY JORDAN SCHETTLER Abstract. The goals of this project are to have the reader(s) gain an appreciation for the usefulness of Legendre symbols and ultimately recreate Eisenstein's slick proof of Gauss's Theorema Aureum of quadratic reciprocity. 1. Quadratic Residues and Legendre Symbols Definition 0.1. Let m; n 2 Z with (m; n) = 1 (recall: the gcd (m; n) is the nonnegative generator of the ideal mZ + nZ). Then m is called a quadratic residue mod n if m ≡ x2 (mod n) for some x 2 Z, and m is called a quadratic nonresidue mod n otherwise. Prove the following remark by considering the kernel and image of the map x 7! x2 on the group of units (Z=nZ)× = fm + nZ :(m; n) = 1g. Remark 1. For 2 < n 2 N the set fm+nZ : m is a quadratic residue mod ng is a subgroup of the group of units of order ≤ '(n)=2 where '(n) = #(Z=nZ)× is the Euler totient function. If n = p is an odd prime, then the order of this group is equal to '(p)=2 = (p − 1)=2, so the equivalence classes of all quadratic nonresidues form a coset of this group. Definition 1.1. Let p be an odd prime and let n 2 Z. The Legendre symbol (n=p) is defined as 8 1 if n is a quadratic residue mod p n < = −1 if n is a quadratic nonresidue mod p p : 0 if pjn: The law of quadratic reciprocity (the main theorem in this project) gives a precise relation- ship between the \reciprocal" Legendre symbols (p=q) and (q=p) where p; q are distinct odd primes. -
Quadratic Reciprocity Laws*
JOURNAL OF NUMBER THEORY 4, 78-97 (1972) Quadratic Reciprocity Laws* WINFRIED SCHARLAU Mathematical Institute, University of Miinster, 44 Miinster, Germany Communicated by H. Zassenhaus Received May 26, 1970 Quadratic reciprocity laws for the rationals and rational function fields are proved. An elementary proof for Hilbert’s reciprocity law is given. Hilbert’s reciprocity law is extended to certain algebraic function fields. This paper is concerned with reciprocity laws for quadratic forms over algebraic number fields and algebraic function fields in one variable. This is a very classical subject. The oldest and best known example of a theorem of this kind is, of course, the Gauss reciprocity law which, in Hilbert’s formulation, says the following: If (a, b) is a quaternion algebra over the field of rational numbers Q, then the number of prime spots of Q, where (a, b) does not split, is finite and even. This can be regarded as a theorem about quadratic forms because the quaternion algebras (a, b) correspond l-l to the quadratic forms (1, --a, -b, ub), and (a, b) is split if and only if (1, --a, -b, ub) is isotropic. This formulation of the Gauss reciprocity law suggests immediately generalizations in two different directions: (1) Replace the quaternion forms (1, --a, -b, ub) by arbitrary quadratic forms. (2) Replace Q by an algebraic number field or an algebraic function field in one variable (possibly with arbitrary constant field). While some results are known concerning (l), it seems that the situation has never been investigated thoroughly, not even for the case of the rational numbers. -
STORY of SQUARE ROOTS and QUADRATIC RESIDUES
CHAPTER 6: OTHER CRYPTOSYSTEMS and BASIC CRYPTOGRAPHY PRIMITIVES A large number of interesting and important cryptosystems have already been designed. In this chapter we present several other of them in order to illustrate other principles and techniques that can be used to design cryptosystems. Part I At first, we present several cryptosystems security of which is based on the fact that computation of square roots and discrete logarithms is in general unfeasible in some Public-key cryptosystems II. Other cryptosystems and groups. cryptographic primitives Secondly, we discuss one of the fundamental questions of modern cryptography: when can a cryptosystem be considered as (computationally) perfectly secure? In order to do that we will: discuss the role randomness play in the cryptography; introduce the very fundamental definitions of perfect security of cryptosystem; present some examples of perfectly secure cryptosystems. Finally, we will discuss, in some details, such very important cryptography primitives as pseudo-random number generators and hash functions . IV054 1. Public-key cryptosystems II. Other cryptosystems and cryptographic primitives 2/65 FROM THE APPENDIX MODULAR SQUARE ROOTS PROBLEM The problem is to determine, given integers y and n, such an integer x that y = x 2 mod n. Therefore the problem is to find square roots of y modulo n STORY of SQUARE ROOTS Examples x x 2 = 1 (mod 15) = 1, 4, 11, 14 { | } { } x x 2 = 2 (mod 15) = and { | } ∅ x x 2 = 3 (mod 15) = { | } ∅ x x 2 = 4 (mod 15) = 2, 7, 8, 13 { | } { } x x 2 = 9 (mod 15) = 3, 12 QUADRATIC RESIDUES { | } { } No polynomial time algorithm is known to solve the modular square root problem for arbitrary modulus n. -
Selmer Groups and Quadratic Reciprocity 3
SELMER GROUPS AND QUADRATIC RECIPROCITY FRANZ LEMMERMEYER Abstract. In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reci- procity law in F . Let F be a number field; elements in F × that are ideal squares were called sin- gular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of em- bedding problems by mathematicians like Kummer, Hilbert, Furtw¨angler, Hecke, Takagi, Shafarevich and many others. Recently, the groups of singular numbers in F were christened Selmer groups by H. Cohen [4] because of an analogy with the Selmer groups in the theory of elliptic curves (look at the exact sequence (2.2) and recall that, under the analogy between number fields and elliptic curves, units correspond to rational points, and class groups to Tate-Shafarevich groups). In this article we will present the theory of 2-Selmer groups in modern language, and give direct proofs based on class field theory. Most of the results given here can be found in 61ff of Hecke’s book [11]; they had been obtained by Hilbert and Furtw¨angler in the§§ roundabout way typical for early class field theory, and were used for proving explicit reciprocity laws. Hecke, on the other hand, first proved (a large part of) the quadratic reciprocity law in number fields using his generalized Gauss sums (see [3] and [24]), and then derived the existence of quadratic class fields (which essentially is just the calculation of the order of a certain Selmer group) from the reciprocity law. -
4 Patterns in Prime Numbers: the Quadratic Reciprocity Law
4 Patterns in Prime Numbers: The Quadratic Reciprocity Law 4.1 Introduction The ancient Greek philosopher Empedocles (c. 495–c. 435 b.c.e.) postulated that all known substances are composed of four basic elements: air, earth, fire, and water. Leucippus (fifth century b.c.e.) thought that these four were indecomposable. And Aristotle (384–322 b.c.e.) introduced four properties that characterize, in various combinations, these four elements: for example, fire possessed dryness and heat. The properties of compound substances were aggregates of these. This classical Greek concept of an element was upheld for almost two thousand years. But by the end of the nineteenth century, 83 chem- ical elements were known to exist, and these formed the basic building blocks of more complex substances. The European chemists Dmitry I. Mendeleyev (1834–1907) and Julius L. Meyer (1830–1895) arranged the elements approx- imately in the order of increasing atomic weight (now known to be the order of increasing atomic numbers), which exhibited a periodic recurrence of their chemical properties [205, 248]. This pattern in properties became known as the periodic law of chemical elements, and the arrangement as the periodic table. The periodic table is now at the center of every introductory chemistry course, and was a major breakthrough into the laws governing elements, the basic building blocks of all chemical compounds in the universe (Exercise 4.1). The prime numbers can be considered numerical analogues of the chemical elements. Recall that these are the numbers that are multiplicatively indecom- posable, i.e., divisible only by 1 and by themselves. -
Phatak Primality Test (PPT)
PPT : New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues A Set of Three Companion Manuscripts PART/Article 1 : Introducing Three Main New Primality Conjectures: Phatak’s Baseline Primality (PBP) Conjecture , and its extensions to Phatak’s Generalized Primality Conjecture (PGPC) , and Furthermost Generalized Primality Conjecture (FGPC) , and New Fast Primailty Testing Algorithms Based on the Conjectures and other results. PART/Article 2 : Substantial Experimental Data and Evidence1 PART/Article 3 : Analytic Proofs of Baseline Primality Conjecture for Special Cases Dhananjay Phatak ([email protected]) and Alan T. Sherman2 and Steven D. Houston and Andrew Henry (CSEE Dept. UMBC, 1000 Hilltop Circle, Baltimore, MD 21250, U.S.A.) 1 No counter example has been found 2 Phatak and Sherman are affiliated with the UMBC Cyber Defense Laboratory (CDL) directed by Prof. Alan T. Sherman Overall Document (set of 3 articles) – page 1 First identification of the Baseline Primality Conjecture @ ≈ 15th March 2018 First identification of the Generalized Primality Conjecture @ ≈ 10th June 2019 Last document revision date (time-stamp) = August 19, 2019 Color convention used in (the PDF version) of this document : All clickable hyper-links to external web-sites are brown : For example : G. E. Pinch’s excellent data-site that lists of all Carmichael numbers <10(18) . clickable hyper-links to references cited appear in magenta. Ex : G.E. Pinch’s web-site mentioned above is also accessible via reference [1] All other jumps within the document appear in darkish-green color. These include Links to : Equations by the number : For example, the expression for BCC is specified in Equation (11); Links to Tables, Figures, and Sections or other arbitrary hyper-targets.