Lucas Sequences, Permutation Polynomials, and Inverse Polynomials - PowerPoint PPT Presentation

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 Lucas Sequences 1 Fibonacci numbers, Lucas numbers Lucas sequences Dickson polynomials Generalized Lucas Sequences Permutation polynomials (PP) over finite fields 2 Introduction of permutation polynomials Permutation binomials and sequences Inverse Polynomials 3 Compositional inverse polynomial of a PP Inverse polynomials of permutation binomials Summary 4 logo

Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Outline Lucas Sequences 1 Fibonacci numbers, Lucas numbers Lucas sequences Dickson polynomials Generalized Lucas Sequences Permutation polynomials (PP) over finite fields 2 Introduction of permutation polynomials Permutation binomials and sequences Inverse Polynomials 3 Compositional inverse polynomial of a PP Inverse polynomials of permutation binomials Summary 4 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 , · · · F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 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 , · · · L 0 = 2, L 1 = 1, L n = L n − 1 + L n − 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 △ = P 2 − 4 Q be a nonsquare. Fibonacci type U 0 ( P , Q ) = 0, U 1 ( P , Q ) = 1, U n ( P , Q ) = PU n − 1 ( P , Q ) − QU n − 2 ( P , Q ) for n ≥ 2. Lucas type V 0 ( P , Q ) = 2, V 1 ( P , Q ) = P , V n ( P , Q ) = PV n − 1 ( P , Q ) − QV n − 2 ( P , Q ) for n ≥ 2. U n ( 1 , − 1 ) - Fibonacci numbers V n ( 1 , − 1 ) - Lucas numbers U n ( 2 , − 1 ) - Pell numbers V n ( 2 , − 1 ) - Pell-Lucas numbers logo U n ( 1 , − 2 ) - Jacobsthal numbers

Lucas Sequences Permutation polynomials (PP) over finite fields Inverse Polynomials Summary Lucas sequences basic properties characteristic equation: x 2 − Px + Q = 0. a = P + √△ and b = P −√△ ∈ Q [ √△ ] 2 2 U n ( P , Q ) = a n − b n a − b . V n ( P , Q ) = a n + b n . 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., ( e A , d A ) and ( e B , d B ) . c ≡ m e B ( mod n ) Alice − → Bob c d B ≡ ( m e B ) d B ≡ m e B d B ≡ 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 = ( p 2 − 1 )( q 2 − 1 ) . gcd ( e , k ) = 1 and ed ≡ 1 ( mod k ) . V e B ( m , 1 ) Alice − → Bob V d ( V e ( m , 1 ) , 1 ) ≡ V de ( 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 ⌊ n / 2 ⌋ n � n − j � S n = α n + β n = � ( − 1 ) j ( αβ ) j ( α + β ) n − 2 j . n − j j j = 0 ⌊ n / 2 ⌋ � n − j � n � ( − a ) j x n − 2 j , n ≥ 1 . D n ( x , a ) = n − j j j = 0 α, a ) = α n + a n D n ( α + a α 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 ⌊ n / 2 ⌋ n � n − j � S n = α n + β n = � ( − 1 ) j ( αβ ) j ( α + β ) n − 2 j . n − j j j = 0 ⌊ n / 2 ⌋ � n − j � n � ( − a ) j x n − 2 j , n ≥ 1 . D n ( x , a ) = n − j j j = 0 α, a ) = α n + a n D n ( α + a α 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 ⌊ n / 2 ⌋ n � n − j � S n = α n + β n = � ( − 1 ) j ( αβ ) j ( α + β ) n − 2 j . n − j j j = 0 ⌊ n / 2 ⌋ � n − j � n � ( − a ) j x n − 2 j , n ≥ 1 . D n ( x , a ) = n − j j j = 0 α, a ) = α n + a n D n ( α + a α 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 α n + β n = ( α + β )( α n − 1 + β n − 1 ) − ( αβ )( α n − 2 + β n − 2 ) . D n ( x , a ) = xD n − 1 ( x , a ) − aD n − 2 ( x , a ) . D 0 ( x , a ) = 2 , D 1 ( x , a ) = x . D n ( P , a ) = V n ( 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 α n + β n = ( α + β )( α n − 1 + β n − 1 ) − ( αβ )( α n − 2 + β n − 2 ) . D n ( x , a ) = xD n − 1 ( x , a ) − aD n − 2 ( x , a ) . D 0 ( x , a ) = 2 , D 1 ( x , a ) = x . D n ( P , a ) = V n ( 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 α n + β n = ( α + β )( α n − 1 + β n − 1 ) − ( αβ )( α n − 2 + β n − 2 ) . D n ( x , a ) = xD n − 1 ( x , a ) − aD n − 2 ( x , a ) . D 0 ( x , a ) = 2 , D 1 ( x , a ) = x . D n ( P , a ) = V n ( 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 α n + β n = ( α + β )( α n − 1 + β n − 1 ) − ( αβ )( α n − 2 + β n − 2 ) . D n ( x , a ) = xD n − 1 ( x , a ) − aD n − 2 ( x , a ) . D 0 ( x , a ) = 2 , D 1 ( x , a ) = x . D n ( P , a ) = V n ( 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 ⌊ n / 2 ⌋ � n − j � � ( − a ) j x n − 2 j . E n ( x , a ) = j j = 0 E 0 ( x , a ) = 1, E 1 ( x , a ) = x , E n ( x , a ) = xE n − 1 ( x , a ) − aE n − 2 ( x , a ) E n ( x , a ) = α n + 1 − β n + 1 α and α 2 � = a . for x = α + β and β = a α − β Moreover, E n ( ± 2 √ a , a ) = ( n + 1 )( ±√ a ) n . E n ( P , Q ) = U n + 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 ⌊ n / 2 ⌋ � n − j � � ( − a ) j x n − 2 j . E n ( x , a ) = j j = 0 E 0 ( x , a ) = 1, E 1 ( x , a ) = x , E n ( x , a ) = xE n − 1 ( x , a ) − aE n − 2 ( x , a ) E n ( x , a ) = α n + 1 − β n + 1 α and α 2 � = a . for x = α + β and β = a α − β Moreover, E n ( ± 2 √ a , a ) = ( n + 1 )( ±√ a ) n . E n ( P , Q ) = U n + 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 ⌊ n / 2 ⌋ � n − j � � ( − a ) j x n − 2 j . E n ( x , a ) = j j = 0 E 0 ( x , a ) = 1, E 1 ( x , a ) = x , E n ( x , a ) = xE n − 1 ( x , a ) − aE n − 2 ( x , a ) E n ( x , a ) = α n + 1 − β n + 1 α and α 2 � = a . for x = α + β and β = a α − β Moreover, E n ( ± 2 √ a , a ) = ( n + 1 )( ±√ a ) n . E n ( P , Q ) = U n + 1 ( P , Q ) . logo