Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Binary and Ternary Kloosterman sums Kseniya Garaschuk University of Victoria July 22, 2010
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Binary Kloosterman sums 1 Melas codes and caps 2 Highly nonlinear functions 3 Ternary Kloosterman sums 4
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums The trace mapping finite field of order p m , p is prime F p m . . . F ∗ p m := F p m \ { 0 } Tr : F p m → F p . . . trace mapping given by: m − 1 � x p i = x + x p + · · · + x p m − 1 . Tr ( x ) = i = 0
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums General Kloosterman map Definition The Kloosterman map is the mapping K : F p m → R defined by � ω Tr ( x − 1 + ax ) , K ( a ) := x ∈ F ∗ pm where ω = e 2 π i / p . Spectrum of binary Kloosterman sums ⇑ (Lachaud and Wolfmann) ⇓ Number of points on elliptic curves
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums General Kloosterman map Definition The Kloosterman map is the mapping K : F p m → R defined by � ω Tr ( x − 1 + ax ) , K ( a ) := x ∈ F ∗ pm where ω = e 2 π i / p . Spectrum of binary Kloosterman sums ⇑ (Lachaud and Wolfmann) ⇓ Number of points on elliptic curves
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Binary Kloosterman curves Theorem (Lachaud, Wolfmann) An ordinary elliptic curve E over F 2 m can be transformed into one of the Kloosterman curves: ax + 1 a : y 2 + y K + = x , ax + 1 a : y 2 + y K − = x + τ, where a , τ ∈ F 2 m , Tr ( τ ) = 1 . Theorem (Lachaud, Wolfmann) a = 2 m + 1 ± K ( a ) . Let a ∈ F 2 m . Then # K ±
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: cross-correlation functions Consider two binary sequences with period 2 m − 1, u ( t ) = Tr ( α t ) and v ( t ) = u (− t ) . The cross-correlation function between u ( t ) and v ( t ) is defined by 2 m − 2 � � (− 1 ) u ( t + a )+ v ( t ) = (− 1 ) Tr ( x − 1 + ax ) = K ( a ) . C t ( a ) = t = 0 x ∈ F ∗ 2 m Problem: determine the values and the number of occurrences of each value taken on by C t ( a ) .
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: cross-correlation functions Consider two binary sequences with period 2 m − 1, u ( t ) = Tr ( α t ) and v ( t ) = u (− t ) . The cross-correlation function between u ( t ) and v ( t ) is defined by 2 m − 2 � � (− 1 ) u ( t + a )+ v ( t ) = (− 1 ) Tr ( x − 1 + ax ) = K ( a ) . C t ( a ) = t = 0 x ∈ F ∗ 2 m Problem: determine the values and the number of occurrences of each value taken on by C t ( a ) .
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: cross-correlation functions Consider two binary sequences with period 2 m − 1, u ( t ) = Tr ( α t ) and v ( t ) = u (− t ) . The cross-correlation function between u ( t ) and v ( t ) is defined by 2 m − 2 � � (− 1 ) u ( t + a )+ v ( t ) = (− 1 ) Tr ( x − 1 + ax ) = K ( a ) . C t ( a ) = t = 0 x ∈ F ∗ 2 m Problem: determine the values and the number of occurrences of each value taken on by C t ( a ) .
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: cross-correlation functions Consider two binary sequences with period 2 m − 1, u ( t ) = Tr ( α t ) and v ( t ) = u (− t ) . The cross-correlation function between u ( t ) and v ( t ) is defined by 2 m − 2 � � (− 1 ) u ( t + a )+ v ( t ) = (− 1 ) Tr ( x − 1 + ax ) = K ( a ) . C t ( a ) = t = 0 x ∈ F ∗ 2 m Problem: determine the values and the number of occurrences of each value taken on by C t ( a ) .
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: K ( a ) = − 1 Open problem: describe elements a ∈ F 2 m for which K ( a ) = − 1. Theorem (Lachaud, Wolfmann) The set of K ( a ) , a ∈ F ∗ 2 m is the set of all the integers s ≡ − 1 ( mod 4 ) in the range [− 2 m / 2 + 1 , 2 m / 2 + 1 ] . Hence there are some a ∈ F 2 m for which K ( a ) = − 1, but their number is still unknown. Partial results could narrow down the search field.
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: K ( a ) = − 1 Open problem: describe elements a ∈ F 2 m for which K ( a ) = − 1. Theorem (Lachaud, Wolfmann) The set of K ( a ) , a ∈ F ∗ 2 m is the set of all the integers s ≡ − 1 ( mod 4 ) in the range [− 2 m / 2 + 1 , 2 m / 2 + 1 ] . Hence there are some a ∈ F 2 m for which K ( a ) = − 1, but their number is still unknown. Partial results could narrow down the search field.
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Applications of Kloosterman sums: K ( a ) = − 1 Open problem: describe elements a ∈ F 2 m for which K ( a ) = − 1. Theorem (Lachaud, Wolfmann) The set of K ( a ) , a ∈ F ∗ 2 m is the set of all the integers s ≡ − 1 ( mod 4 ) in the range [− 2 m / 2 + 1 , 2 m / 2 + 1 ] . Hence there are some a ∈ F 2 m for which K ( a ) = − 1, but their number is still unknown. Partial results could narrow down the search field.
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Elliptic curve E t Let t ∈ F 2 m , t �∈ { 0 , 1 } , and consider the elliptic curve y 2 + xy = x 3 + a 2 x 2 + ( t 8 + t 6 ) , E t : where a 2 = Tr ( t ) . Later we will show that E t arises naturally in the problem of counting coset leaders for the Melas code.
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Elliptic curve E t Let t ∈ F 2 m , t �∈ { 0 , 1 } , and consider the elliptic curve y 2 + xy = x 3 + a 2 x 2 + ( t 8 + t 6 ) , E t : where a 2 = Tr ( t ) . Later we will show that E t arises naturally in the problem of counting coset leaders for the Melas code.
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums ⇒ a = t 4 + t 3 3 | K ( a ) ⇐ Theorem Let m ≥ 3 be odd and let a ∈ F ∗ 2 m . Then K ( a ) is divisible by 3 if and only if a = t 4 + t 3 for some t ∈ F 2 m . “ ⇐ ” (Proved first by Helleseth and Zinoviev, 1999) Due to Lachaud and Wolfmann we get � 2 m + 1 + K ( t 4 + t 3 ) if Tr ( t ) = 0 , # E t = 2 m + 1 − K ( t 4 + t 3 ) if Tr ( t ) = 1 . We find a point on E t of order 6, hence 6 | # E t . Since 3 | ( 2 m + 1 ) , we get 3 | K ( t 4 + t 3 ) . (We will later see a more combinatorial proof of 6 | # E t )
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums ⇒ a = t 4 + t 3 3 | K ( a ) ⇐ Theorem Let m ≥ 3 be odd and let a ∈ F ∗ 2 m . Then K ( a ) is divisible by 3 if and only if a = t 4 + t 3 for some t ∈ F 2 m . “ ⇐ ” (Proved first by Helleseth and Zinoviev, 1999) Due to Lachaud and Wolfmann we get � 2 m + 1 + K ( t 4 + t 3 ) if Tr ( t ) = 0 , # E t = 2 m + 1 − K ( t 4 + t 3 ) if Tr ( t ) = 1 . We find a point on E t of order 6, hence 6 | # E t . Since 3 | ( 2 m + 1 ) , we get 3 | K ( t 4 + t 3 ) . (We will later see a more combinatorial proof of 6 | # E t )
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums ⇒ a = t 4 + t 3 3 | K ( a ) ⇐ “ ⇒ ” Charpin, Helleseth and Zinoviev (2007): 3 | K ( a ) ⇔ Tr ( a 1 / 3 ) = 0 Tr ( a 1 / 3 ) = 0 ⇔ a = t 4 + t 3 In fact, we can generalize the last equivalence.
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Characterization for Tr ( a 1 / ( 2 k − 1 ) ) = 0 Theorem Let m > 1 and let k be such that gcd ( 2 k − 1 , 2 m − 1 ) = 1 . Then for each a ∈ F 2 m we have Tr ( a 1 / ( 2 k − 1 ) ) = 0 if and only if a = t 2 k + t 2 k − 1 for some t ∈ F 2 m . (The case k = 1 is a well-known fact.)
Outline Binary Kloosterman sums Melas codes and caps Highly nonlinear functions Ternary Kloosterman sums Binary linear codes Definition A binary linear [ n , k , d ] -code C is a k -dimensional linear subspace of F n 2 such that any two different elements of the code are at Hamming distance at least d . Definition H is called a parity check matrix for a linear code C if ⇒ Hx T = 0 . Then Hx T is called the syndrome of x . x ∈ C ⇐ Definition A coset leader for a coset D of C is an element of D with the smallest Hamming weight. The weight of a coset is the weight of its coset leader(s).
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