Lucas Sequences, Permutation Polynomials, and Inverse Polynomials Qiang (Steven) Wang School of Mathematics and Statistics Carleton University IPM 20 - Combinatorics 2009, Tehran, May 16-21, 2009. logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Outline 1 Lucas Sequences Fibonacci numbers, Lucas numbers Lucas sequences Dickson polynomials Generalized Lucas Sequences 2 Permutation polynomials (PP) over finite fields Introduction of permutation polynomials Permutation binomials and sequences 3 Inverse Polynomials Compositional inverse polynomial of a PP Inverse polynomials of permutation binomials 4 Summary logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Outline 1 Lucas Sequences Fibonacci numbers, Lucas numbers Lucas sequences Dickson polynomials Generalized Lucas Sequences 2 Permutation polynomials (PP) over finite fields Introduction of permutation polynomials Permutation binomials and sequences 3 Inverse Polynomials Compositional inverse polynomial of a PP Inverse polynomials of permutation binomials 4 Summary logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Fibonacci numbers, Lucas numbers Fibonacci numbers Origin Ancient India: Pingala (200 BC). West: Leonardo of Pisa, known as Fibonacci (1170-1250), in his Liber Abaci (1202). He considered the growth of an idealised (biologically unrealistic) rabbit population. Liber Abaci, 1202 0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; ··· F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2 for n ≥ 2. logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Fibonacci numbers, Lucas numbers Leonardo of Pisa, Fibonacci (1170-1250) Figure: Fibonacci (1170-1250) logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Fibonacci numbers, Lucas numbers Fibonacci (1170-1250) Figure: A statue of Fibonacci in Pisa logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Fibonacci numbers, Lucas numbers Lucas numbers Lucas numbers (Edouard Lucas) 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; ··· L0 = 2, L1 = 1, Ln = Ln−1 + Ln−2 for n ≥ 2. logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Fibonacci numbers, Lucas numbers Edouard Lucas (1842-1891) Figure: Edouard Lucas (1842-1891) logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Lucas sequences Lucas sequences Let P; Q be integers and 4 = P2 − 4Q be a nonsquare. Fibonacci type U0(P; Q) = 0, U1(P; Q) = 1, Un(P; Q) = PUn−1(P; Q) − QUn−2(P; Q) for n ≥ 2. Lucas type V0(P; Q) = 2, V1(P; Q) = P, Vn(P; Q) = PVn−1(P; Q) − QVn−2(P; Q) for n ≥ 2. Un(1; −1) - Fibonacci numbers Vn(1; −1) - Lucas numbers Un(2; −1) - Pell numbers Vn(2; −1) - Pell-Lucas numbers logo Un(1; −2) - Jacobsthal numbers Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Lucas sequences basic properties characteristic equation: x2 − Px + Q = 0. p p P+ 4 P− 4 p a = 2 and b = 2 2 Q[ 4] an−bn Un(P; Q) = a−b : n n Vn(P; Q) = a + b . logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Lucas sequences Applications RSA n = pq, p and q are distinct primes. k = (p − 1)(q − 1). gcd(e; k) = 1 and ed ≡ 1 (mod k). Here e is called a public key and d a private key. Each party has a pair of keys, i.e., (eA; dA) and (eB; dB). c ≡ meB (mod n) Alice −! Bob cdB ≡ (meB )dB ≡ meBdB ≡ m (mod n): logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Lucas sequences Applications LUC n = pq, p and q are distinct primes. k = (p2 − 1)(q2 − 1). gcd(e; k) = 1 and ed ≡ 1 (mod k). VeB (m; 1) Alice −! Bob Vd (Ve(m; 1); 1) ≡ Vde(m; 1) ≡ m (mod n): logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n bn=2c X n n − j S = αn + βn = (−1)j (αβ)j (α + β)n−2j : n n − j j j=0 bn=2c X n n − j D (x; a) = (−a)j xn−2j ; n ≥ 1: n n − j j j=0 a an D (α + ; a) = αn + : n α αn logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n bn=2c X n n − j S = αn + βn = (−1)j (αβ)j (α + β)n−2j : n n − j j j=0 bn=2c X n n − j D (x; a) = (−a)j xn−2j ; n ≥ 1: n n − j j j=0 a an D (α + ; a) = αn + : n α αn logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n bn=2c X n n − j S = αn + βn = (−1)j (αβ)j (α + β)n−2j : n n − j j j=0 bn=2c X n n − j D (x; a) = (−a)j xn−2j ; n ≥ 1: n n − j j j=0 a an D (α + ; a) = αn + : n α αn logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n αn + βn = (α + β)(αn−1 + βn−1) − (αβ)(αn−2 + βn−2): Dn(x; a) = xDn−1(x; a) − aDn−2(x; a): D0(x; a) = 2; D1(x; a) = x: Dn(P; a) = Vn(P; a). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n αn + βn = (α + β)(αn−1 + βn−1) − (αβ)(αn−2 + βn−2): Dn(x; a) = xDn−1(x; a) − aDn−2(x; a): D0(x; a) = 2; D1(x; a) = x: Dn(P; a) = Vn(P; a). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n αn + βn = (α + β)(αn−1 + βn−1) − (αβ)(αn−2 + βn−2): Dn(x; a) = xDn−1(x; a) − aDn−2(x; a): D0(x; a) = 2; D1(x; a) = x: Dn(P; a) = Vn(P; a). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of the first kind of degree n αn + βn = (α + β)(αn−1 + βn−1) − (αβ)(αn−2 + βn−2): Dn(x; a) = xDn−1(x; a) − aDn−2(x; a): D0(x; a) = 2; D1(x; a) = x: Dn(P; a) = Vn(P; a). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of second kind of degree n bn=2c X n − j E (x; a) = (−a)j xn−2j : n j j=0 E0(x; a) = 1, E1(x; a) = x, En(x; a) = xEn−1(x; a) − aEn−2(x; a) n+1 n+1 E (x; a) = α −β for x = α + β and β = a and α2 6= a. n α−βp p α n Moreover, En(±2 a; a) = (n + 1)(± a) . En(P; Q) = Un+1(P; Q). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of second kind of degree n bn=2c X n − j E (x; a) = (−a)j xn−2j : n j j=0 E0(x; a) = 1, E1(x; a) = x, En(x; a) = xEn−1(x; a) − aEn−2(x; a) n+1 n+1 E (x; a) = α −β for x = α + β and β = a and α2 6= a. n α−βp p α n Moreover, En(±2 a; a) = (n + 1)(± a) . En(P; Q) = Un+1(P; Q). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of second kind of degree n bn=2c X n − j E (x; a) = (−a)j xn−2j : n j j=0 E0(x; a) = 1, E1(x; a) = x, En(x; a) = xEn−1(x; a) − aEn−2(x; a) n+1 n+1 E (x; a) = α −β for x = α + β and β = a and α2 6= a. n α−βp p α n Moreover, En(±2 a; a) = (n + 1)(± a) . En(P; Q) = Un+1(P; Q). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Dickson polynomials Dickson polynomials Dickson polynomials of second kind of degree n bn=2c X n − j E (x; a) = (−a)j xn−2j : n j j=0 E0(x; a) = 1, E1(x; a) = x, En(x; a) = xEn−1(x; a) − aEn−2(x; a) n+1 n+1 E (x; a) = α −β for x = α + β and β = a and α2 6= a. n α−βp p α n Moreover, En(±2 a; a) = (n + 1)(± a) . En(P; Q) = Un+1(P; Q). logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Generalized Lucas Sequences Vn(1; −1) Lucas numbers Vn(1; −1) p p 1+ 5 n 1− 5 n 2; 1; 3; 4; 7; ··· =) Vn(1; −1) = ( 2 ) + ( 2 ) . p 1 + 5 π − π π a = = 2 cos( ) = e 5 + e 5 : 2 5 p 1 − 5 3π − 3π 3π b = = 2 cos( ) = e 5 + e 5 : 2 5 Let η be a primitive 10th root of unity. Then a = η + η−1 and b = η3 + η−3: Hence −1 n 3 −3 n Vn(1; −1) = (η + η ) + (η + η ) : logo Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Generalized Lucas Sequences Definition Generalized Lucas sequence (Akbary, W., 2006) For any odd integer ` = 2k + 1 ≥ 3 and η be a fixed primitive 2`th root of unity.