Some Recent Progress in the Applications of Niho Exponents Nian Li - PowerPoint PPT Presentation

Some Recent Progress in the Applications of Niho Exponents Nian Li Faculty of Mathematics and Statistics Hubei University Wuhan, China July 5, 2017 Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 1 / 35

Outline Niho Exponents 1 Cross Correlation Functions of Niho Type 2 Bent Functions From Niho Exponents 3 Cyclic Codes with Niho Type Zeros 4 Permutation Polynomials From Niho Exponents 5 Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 2 / 35

Definition Let p be a prime, n = 2 m a positive integer and q = p m . Let F q denote the finite field with q elements. Niho Exponent A positive integer d is called a Niho exponent (with respect to F q 2 ) if there exists some 0 ≤ j ≤ n − 1 such that p j d ≡ (mod q − 1) Normalized form: j = 0 , i.e., d = ( q − 1) s + 1 . Generalized form: d ≡ ∆ (mod q − 1) for some integer ∆ . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 3 / 35

Cross Correlation Between an m -sequence and Its Decimation Sequence The determination of the cross correlation between an m -sequence and its d -decimation sequence is a classic research problem. Basic Notations: Tr( · ) is the trace function from F q to F p . α is a primitive element of F q . ω is a p -th primitive root of unity. s ( t ) = Tr( α t ) is an m -sequence of period q − 1 . s ( dt ) = Tr( α dt ) is the d -decimation sequences of s ( t ) . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 4 / 35

Correlation Function Correlation Function The periodic cross correlation function C d ( τ ) between the sequences s ( t ) and s ( dt ) is defined for τ = 0 , 1 , 2 , · · · , q − 2 by q − 2 � � w s ( t + τ ) − s ( dt ) = w Tr( α τ x − x d ) − 1 . C d ( τ ) = t =0 x ∈ F q Main Research Problems Find decimation d such that C d ( τ ) takes few values. Determine the value distribution of C d ( τ ) . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 5 / 35

Correlation Function Known 3-valued Correlation Function C d ( τ ) over F 2 n No. d -Decimation Condition Remarks 2 k + 1 1 n/ gcd( n, k ) odd Gold, 1968 2 2 k − 2 k + 1 2 n/ gcd( n, k ) odd Kasami, 1971 2 n/ 2 − 2 ( n +2) / 4 + 1 3 n ≡ 2 (mod 4) Cusick et al., 1996 2 n/ 2+1 + 3 4 n ≡ 2 (mod 4) Cusick et al., 1996 2 ( n − 1) / 2 + 3 5 n odd Canteaut et al., 2000 2 ( n − 1) / 2 + 2 ( n − 1) / 4 − 1 6 n ≡ 1 (mod 4) Hollmann et al., 2001 2 ( n − 1) / 2 + 2 (3 n − 1) / 4 − 1 7 n ≡ 3 (mod 4) Hollmann et al., 2001 Remarks: (1) No. 5 is the Welch’s conjecture; (2) Nos. 6 and 7 are the Niho’s conjectures Open Problem Show that the table contains all decimations with 3-valued correlation function. Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 6 / 35

Correlation Function Known 3-valued Correlation Function C d ( τ ) over F p n No. d -Decimation Condition Remarks ( p 2 k + 1) / 2 1 n/ gcd( n, k ) odd Trachtenberg, 1970 p 2 k − p k + 1 2 n/ gcd( n, k ) odd Trachtenberg, 1970 2 · 3 ( n − 1) / 2 + 1 3 n/ gcd( n, k ) odd Dobbertin et al., 2001 2 · 3 k + 1 4 n | 4 k + 1 , n odd Katz and Langevin, 2015 Remarks: (1) Nos. 1 and 2 are due to Helleseth for even n ; (2) The result obtained by Xia et al. (IEEE IT 60(11), 2014) is covered by No. 1 Open Problems Show that the table contains all decimations with 3-valued correlation function for p > 3 . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 7 / 35

Correlation Function Known 4-valued Correlation Function C d ( τ ) over F 2 n No. d -Decimation Condition Remarks 2 n/ 2+1 − 1 1 n ≡ 0 (mod 4) Niho, 1972 (2 n/ 2 + 1)(2 n/ 4 − 1) + 2 2 n ≡ 0 (mod 4) Niho, 1972 2 ( n/ 2+1) r − 1 3 n ≡ 0 (mod 4) Dobbertin, 1998 2 r − 1 2 n +2 s +1 − 2 n/ 2+1 − 1 4 n ≡ 0 (mod 4) Helleseth et al., 2005 2 s − 1 (2 n/ 2 − 1) 2 r 5 2 r ± 1 + 1 n ≡ 0 (mod 4) Dobbertin et al., 2006 Remarks: (1) All are the Niho type decimations; (2) No. 5 covers previous four cases. Conjecture (Dobbertin, Helleseth et al., 2006) No. 5 covers all 4-valued cross correlation for Niho type decimation. Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 8 / 35

Correlation Function Known 4-valued Correlation Function C d ( τ ) over F p n No. d -Decimation Condition Remarks 2 · p n/ 2 − 1 p n/ 2 �≡ 2 (mod 3) 1 Helleseth, 1976 3 k + 1 2 n = 3 k, k odd Zhang et al., 2013 3 2 k + 2 3 n = 3 k, k odd Zhang et al., 2013 Remarks: (1) No. 1 is a Niho type decimation; (2) Nos. 2 and 3 are due to Zhang et al. if gcd( k, 3) = 1 and due to Xia et al. if gcd( k, 3) = 3 . Open Problem Find new 4-valued C d ( τ ) for any prime p . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 9 / 35

Correlation Function Known 5 or 6-valued Correlation Function C d ( τ ) over F 2 n No. d -Decimation Condition Remarks 2 n/ 2 + 3 1 n ≡ 0 (mod 2) Helleseth, 1976 2 n/ 2 − 2 n/ 4 + 1 2 n ≡ 0 (mod 8) Helleseth, 1976 2 n − 1 + 2 i 3 n ≡ 0 (mod 2) Helleseth, 1976 3 2 n/ 2 + 2 n/ 4 + 1 4 n ≡ 0 (mod 4) Dobbertin, 1998 Remarks: (1) No. 1 was conjectured by Niho; (2) No. 3 is of Niho type if n/ 2 is odd. Open Problem (Dobbertin, Helleseth et al., 2006) Determine the cross correlation distribution of C d ( τ ) for the Niho type decimation d = 3 · (2 n/ 2 − 1) + 1 . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 10 / 35

Correlation Function Known 5 or 6-valued Correlation Function C d ( τ ) over F p n No. d -Decimation Condition Remarks ( p n − 1) / 2 + p i p n ≡ 1 (mod 4) 1 Helleseth, 1976 ( p n − 1) / 3 + p i 2 p ≡ 2 (mod 3) Helleseth, 1976 p n/ 2 − p n/ 4 + 1 p n/ 4 �≡ 2 (mod 3) 3 Helleseth, 1976 3 k + 1 4 n = 3 k, k even Zhang et al., 2013 3 2 k + 2 5 n = 3 k, k even Zhang et al., 2013 Remarks: (1) No. 1 is of Niho type if n/ 2 is odd; (2) Nos. 4 and 5 are due to Zhang et al. if gcd( k, 3) = 1 and due to Xia et al. if gcd( k, 3) = 3 . Open Problem (Dobbertin, Helleseth and Martinsen, 1999) Determine the cross correlation distribution of C d ( τ ) for the Niho type decimation d = 3 · (3 n/ 2 − 1) + 1 . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 11 / 35

Correlation Function: Recent Results Let k be a positive integer and N k denote the number of solutions to x 1 + x 2 + · · · + x k = 0 , x d 1 + x d 2 + · · · + x d = 0 . k Question: How to determine the values of N k ? Open Problem (Dobbertin, Helleseth et al., 2006) Determine the cross correlation distribution of C d ( τ ) for the Niho type decimation d = 3 · (2 n/ 2 − 1) + 1 . Solved! (surprising connection with the Zetterberg code) by Xia, L., Zeng and Helleseth 2016 (IEEE IT, 62(12), 2016) Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 12 / 35

Correlation Function: Recent Results Open Problem (Dobbertin, Helleseth and Martinsen, 1999) Determine the cross correlation distribution of C d ( τ ) for the Niho type decimation d = 3 · (3 n/ 2 − 1) + 1 . Solved! by Xia, L., Zeng and Helleseth 2017 (it is available on arXiv). Future Work Determine the cross correlation distribution of C d ( τ ) for the Niho type decimation d = 3 · ( p n/ 2 − 1) + 1 for p > 3 . This case is much more complicated! Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 13 / 35

Bent Functions From Niho Exponents Bent functions have significant applications in cryptography and coding theory. Walsh Transform Let f ( x ) be a function from F 2 n to F 2 . The Walsh transform of f ( x ) is defined by � � ( − 1) f ( x )+Tr( λx ) , λ ∈ F 2 n . f ( λ ) = x ∈ F 2 n Bent Function f ( λ ) | = 2 n/ 2 for any A function f ( x ) from F 2 n to F 2 is called Bent if | � λ ∈ F 2 n . Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 14 / 35

Bent Functions From Niho Exponents Problem Description Let f ( x ) be a function from F 2 n to F 2 defined by 2 n − 2 � Tr( a i x i ) , a i ∈ F 2 n . f ( x ) = i =1 Then how to choose a i and i such that f ( x ) is Bent? Remarks Known infinite classes of Boolean Bent functions: 1 Monomial Bent: only 5 classes 2 Binomial Bent: only about 6 classes 3 Polynomial form: quadratic form, Dillon type and Niho type Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 15 / 35

Constructions of Bent Functions of Niho Type Known Constructions of Niho Bent Functions Table: Known Niho Bent Functions No. Class of Functions Authors Year 1 ( ax (2 m − 1) 1 Tr n 2 +1 ) 1 – – 1 ( ax (2 m − 1) 1 2 +1 + bx (2 m − 1)3+1 ) Tr n 2 Dobbertin et al. 2006 2 +1 + bx (2 m − 1) 1 1 ( ax (2 m − 1) 1 Tr n 4 +1 ) 3 Dobbertin et al. 2006 2 +1 + bx (2 m − 1) 1 1 ( ax (2 m − 1) 1 Tr n 6 +1 ) 4 Dobbertin et al. 2006 2 r − 1 − 1 � 2 +1 + 1 ( ax (2 m − 1) 1 x (2 m − 1) i Tr n 2 r +1 ) 5 Leander, Kholosha 2006 i =1 Remarks: (1) No. 1 is trivial; (2) No. 3 is covered by No. 5 Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 16 / 35

Niho Type Bent Functions: Some Recent Results Let n = 2 m , p be a prime and q = p m . Define p r − 1 � 1 ( ax ( ip m − r +1)( q − 1)+1 ) Tr n f ( x ) = i =1 Theorem (L., Helleseth, Kholosha and Tang, 2013) 1 f ( x ) is Bent if p = 2 and gcd( r, m ) = 1 (4-valued otherwise), and it is equivalent to the Leander-Kholosha’s Bent functions. 2 The proof (based on quadratic form) is self-contained and much simpler than the original one (by using Dickson polynomials and complicated techniques over finite fields). Nian Li Problems From Niho Exponents BFA-2017, OS, Norway 17 / 35