Number Theory and Cryptography Paul Yiu Department of Mathematics Florida Atlantic University Fall 2017 Chapters 9–14 October 23, 2017 ii Contents 5 Quadratic Residues 39 5.1 Quadraticresidues.......................... 39 5.2 TheLegendresymbol ........................ 40 1 5.3 The Legendre symbol −p ..................... 41 5.3.1 The square roots of 1 (mod p) ............... 42 5.4 Gauss’lemma ............................− 43 2 5.5 The Legendre symbol p ..................... 43 5.6 The law of quadratic reciprocity . 44 5.7 Smallest quadratic nonresidue modulo p .............. 47 5.8 Square roots modulo p ........................ 48 5.8.1 Square roots modulo prime p 1 (mod 8) ......... 48 5.8.2 Square roots modulo p for a generic6≡ prime p ......... 49 5.9 TheJacobisymbol.......................... 51 6 Primality Tests and Factorization of Integers 53 6.1 Primality of Mersenne numbers . 53 6.2 Germainprimes ........................... 55 6.3 Probabilistic primality tests . 56 6.3.1 Pseudoprimes......................... 56 6.3.2 Euler b-pseudoprimes. 58 6.3.3 Strong b-pseudoprimes . 58 6.4 Carmichaelnumbers......................... 61 6.5 Quadratic sieve and factor base . 62 6.5.1 Fermat’s factorization . 62 6.5.2 Factorbase .......................... 63 6.6 Pollard’smethods .......................... 66 6.6.1 The ρ-method......................... 66 6.6.2 The (p 1)-method ..................... 67 − 7 Pythagorean Triangles 69 7.1 Construction of Pythagorean triangles . 69 7.2 Fermat’s construction of primitive Pythagorean triangles with con- secutivelegs ............................. 70 iv CONTENTS 7.3 Fermat Last Theorem for n =4 ................... 72 7.4 Two ternary trees of rational numbers . 73 7.5 Genealogy of Pythagorean triangles . 75 8 Homogeneous Quadratic Equations in 3 Variables 79 8.1 Pythagorean triangles revisited . 79 8.2 Rational points on a conic . 80 8.2.1 Integer triangles with a 60◦ angle .............. 80 8.2.2 Integer triangles with a 120◦ angle.............. 82 8.3 Herontriangles............................ 84 8.3.1 The Heron formula . 84 8.3.2 Construction of Heron triangles . 85 8.3.3 Heron triangles with sides in arithmetic progression ..... 86 8.3.4 Heron triangles with integer inradii . 87 8.4 The equation Pk,a + Pk,b = Pk,c for polynomial numbers . 88 8.4.1 Double ruling of S ...................... 89 8.4.2 Primitive Pythagorean triple associated with a k-gonal triple 91 8.4.3 Triangular triples . 91 8.4.4 Pentagonal triples . 92 38 CONTENTS Chapter 5 Quadratic Residues 5.1 Quadratic residues Let n > 1 be a given positive integer, and gcd(a,n)=1. We say that a Zn• is a quadratic residue mod n if the congruence x2 a (mod n) is solvable. Otherwise,∈ a is called a quadratic nonresidue mod n. ≡ 1. If a and b are quadratic residues mod n, so is their product ab. 2. If a is a quadratic residue, and b a quadratic nonresidue mod n, then ab is a quadratic nonresidue mod n. 3. The product of two quadratic nonresidues mod n is not necessarily a quadratic residue mod n. For example, in Z12• = 1, 5, 7, 11 , only 1 is a quadratic residue; 5, 7, and 11 5 7 are all quadratic{ nonresidues.} ≡ · Proposition 5.1. Let p be an odd prime, and p ∤ a. The quadratic congruence ax2 +bx+c 0 (mod p) is solvable if and only if (2ax+b)2 b2 4ac (mod p) is solvable. ≡ ≡ − Theorem 5.2. Let p be an odd prime. Exactly one half of the elements of Zp• are quadratic residues. 1 Proof. Each quadratic residue modulo p is congruent to one of the following 2 (p 1) residues. − p 1 2 12, 22, ...,k2, ..., − . 2 p 1 2 2 We show that these residue classes are all distinct. For 1 h < k −2 , h k (mod p) if and only if (k h)(h + k) is divisible by p, this≤ is impossible≤ since≡ each of k h and h + k is smaller− than p. − Corollary 5.3. If p is an odd prime, the product of two quadratic nonresidues is a quadratic residue. 40 Quadratic Residues In the table below we list, for primes < 50, the quadratic residues and their square roots. It is understood that the square roots come in pairs. For example, the entry (2,7) for the prime 47 should be interpreted as saying that the two solutions of the congruence x2 2 (mod 47) are x 7 (mod 47). Also, for primes of the form p = 4n +1, since≡ 1 is a quadratic≡ residue ± modulo p, we only list quadratic p − p residues smaller than 2 . Those greater than 2 can be found with the help of the square roots of 1. − Quadratic residues mod p and their square roots 3 (1, 1) 5 (−1, 2) (1, 1) 7 (1, 1) (2, 3) (4, 2) 11 (1, 1) (3, 5) (4, 2) (5, 4) (9, 3) 13 (−1, 5) (1, 1) (3, 4) (4, 2) 17 (−1, 4) (1, 1) (2, 6) (4, 2) (8, 5) 19 (1, 1) (4, 2) (5, 9) (6, 5) (7, 8) (9, 3) (11, 7) (16, 4) (17, 6) 23 (1, 1) (2, 5) (3, 7) (4, 2) (6, 11) (8, 10) (9, 3) (12, 9) (13, 6) (16, 4) (18, 8) 29 (−1, 12) (1, 1) (4, 2) (5, 11) (6, 8) (7, 6) (9, 3) (13, 10) 31 (1, 1) (2, 8) (4, 2) (5, 6) (7, 10) (8, 15) (9, 3) (10, 14) (14, 13) (16, 4) (18, 7) (19, 9) (20, 12) (25, 5) (28, 11) 37 (−1, 6) (1, 1) (3, 15) (4, 2) (7, 9) (9, 3) (10, 11) (11, 14) (12, 7) (16, 4) 41 (−1, 9) (1, 1) (2, 17) (4, 2) (5, 13) (8, 7) (9, 3) (10, 16) (16, 4) (18, 10) (20, 15) 43 (1, 1) (4, 2) (6, 7) (9, 3) (10, 15) (11, 21) (13, 20) (14, 10) (15, 12) (16, 4) (17, 19) (21, 8) (23, 18) (24, 14) (25, 5) (31, 17) (35, 11) (36, 6) (38, 9) (40, 13) (41, 16) 47 (1, 1) (2, 7) (3, 12) (4, 2) (6, 10) (7, 17) (8, 14) (9, 3) (12, 23) (14, 22) (16, 4) (17, 8) (18, 21) (21, 16) (24, 20) (25, 5) (27, 11) (28, 13) (32, 19) (34, 9) (36, 6) (37, 15) (42, 18) 5.2 The Legendre symbol Let p be an odd prime. For an integer a, we define the Legendre symbol a +1, if a is a quadratic residue mod p, := p 1, otherwise. (− ab a b Lemma 5.4. p = p p . Proof. This is equivalent to saying that modulo p, the product of two quadratic residues (respectively nonresidues) is a quadratic residue, and the product of a quadratic residue and a quadratic nonresidue is a quadratic nonresidue. 1 5.3 The Legendre symbol −p 41 Theorem 5.5 (Euler). Let p be an odd prime. For each integer a not divisible by p, a 1 (p 1) a 2 − mod p. p ≡ Proof. Suppose a is a quadratic nonresidue mod p. The mod p residues 1, 2,...,p 1 are partitioned into pairs satisfying xy = a. In this case, − 1 (p 1) (p 1)! a 2 − (mod p). − ≡ On the other hand, if a is a quadratic residue, with a k2 (p k)2 (mod p), ≡ ≡ − apart from 0, k, the remaining p 3 elements of Zp can be partitioned into pairs satisfying xy ±= a. − 1 (p 3) 1 (p 1) (p 1)! k(p k)a 2 − a 2 − (mod p). − ≡ − ≡− Summarizing, we obtain a 1 (p 1) (p 1)! a 2 − (mod p). − ≡− p Note that by putting a = 1, we obtain Wilson’s theorem: (p 1)! 1 (mod p). a − ≡ − By comparison, we obtain a formula for p : a 1 (p 1) a 2 − (mod p). p ≡ 1 5.3 The Legendre symbol −p Theorem 5.6. Let p be an odd prime. 1 is a quadratic residue modulo p if and only if p 1 (mod 4). − ≡ − 2 p 1 p 1 Proof. ( ) If x 1 (mod p), then ( 1) 2 x − 1 (mod p) by Fermat’s ⇒ ≡− p 1 − ≡ ≡ little theorem. This means that −2 is even, and p 1 (mod 4). p 1 ≡ ( ) If p 1 (mod 4), the integer − is even. By Wilson’s theorem, ⇐ ≡ 2 p−1 p−1 p−1 p 1 2 2 2 2 − ! = j2 = j ( j) j (p j)=(p 1)! 1 (mod p). 2 · − ≡ · − − ≡− i=1 i=1 i=1 Y Y Y 2 p 1 The solutions of x 1 (mod p) are therefore x ( − )!. ≡− ≡± 2 Theorem 5.7. There are infinitely many primes of the form 4n +1. 42 Quadratic Residues Proof. Suppose there are only finitely many primes p1, p2,..., pr of the form 4n+1. Consider the product P = (2p p p )2 +1. 1 2 ··· r Note that P 1 (mod 4). Since P is greater than each of p , p ,..., p , it cannot ≡ 1 2 r be prime, and so must have a prime factor p different from p1, p2, ..., pr. But then modulo p, 1 is a square. By Theorem 5.6, p must be of the form 4n +1,a contradiction. − 5.3.1 The square roots of 1 (mod p) − Here are the square roots of 1 for the first 20 primes p of the form 4k +1: − p √ 1 p √ 1 p √ 1 p √ 1 p √ 1 − − − − − 5 2 13 5 17 4 29 12 37 6 41 ±9 53 ±23 61 ±11 73 ±27 89 ±34 97 ±22 101 ±10 109 ±33 113 ±15 137 ±37 149 ±44 157 ±28 173 ±80 181 ±19 193 ±81 ± ± ± ± ± In general, the square roots of 1 (mod p) can be found as nk for a quadratic nonresidue n modulo p.