Constructing Commutative Semifields of Square Order Advances in Mathematics of Communications, 10(2), 2016, p291-306 Stephen M. Gagola III & Joanne L. Hall Department of Mathematics, Miami University, USA School of Science, RMIT Information Security Seminar 10 March 2017 J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 1 / 31
Information Security Seminar Series Alternating Fridays Cryptography Cyber Security Discrete Mathematics Information Theory Theory of telecomunications Upcoming Seminars 24th March: Stephen Davis 7 April: Jessica Leibig J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 2 / 31
Outline Semifields and Planar Functions 1 Projection Construction for Semifields and Planar Functions 2 New Semifields 3 J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 3 / 31
Outline Semifields and Planar Functions 1 Projection Construction for Semifields and Planar Functions 2 New Semifields 3 J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 4 / 31
Semi fields Semi fields A semi field is a set S with two binary operations + and × such that ♣ ( S , +) is an abelian group ♥ ( S \ { 0 } , × ) has an identity and division is always possible. ♦ The left and right distribution laws hold. ♣ Abelian group: + is well behaved + commutativity x+y=y+x + associativity (x+y)+z=x+(y+z) + identity x+0=0+x=x + inverses x+ (-x)=0 Familiar Abelian Groups ( Z , +) , ( Z n , +) , ( R , +) , ( R \ { 0 } , × ) J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 5 / 31
Semi fields ♥ ( S \ { 0 } , × ) has an identity, x × 1 = 1 × x = x . Division is always possible: for each a , b ∈ S there is a unique x , y ∈ S such that a × x = b y × a = b . and If ( S \ { 0 } , × ) is also an abelian group, then S is a field. Familiar Fields: ( R , + , × ) , ( C , + , × ) , ( F q , + × ) . The set of m × m invertible matrices is a semi field. ( Z , + , × ) is not a semifield, eg. 5 × x = 3 has no solution in integers. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 6 / 31
Semi field ♦ The left and right distribution laws hold. ( a + b ) × c = ( a × c ) + ( b × c ) + and × behave well together J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 7 / 31
Planar functions Planar Function Let F be a field. A function f : F → F is called a planar function if for every a ∈ F ∗ the function ∆ f , a : x �→ f ( x + a ) − f ( x ) is a permutation. Example 1 f ( x ) = x 2 on R . ∆ f , a ( x ) = ( x + a ) 2 − x 2 = 2 ax − a 2 This is a linear function, which is a permutation of R Example 2 g ( x ) = x 10 + x 6 − x 2 on F 3 4 . ∆ f , a ( x ) = ( x + a ) 1 0 + ( x + a ) 6 − ( x + a ) 2 − x 1 0 − x 6 + x 2 = a 10 + a 9 x + a 6 + 2 a 3 x 3 − a 2 + ax 9 − 2 ax Which is a permutation on F 4 3 J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 8 / 31
Planar functions Example 2 g ( x ) = x 10 + x 6 − x 2 on F 3 4 . Rewrite the powers as sums of powers of the characteristic of the field g ( x ) = x 3 2 + 3 0 + x 3 1 + 3 1 + x 3 0 + 3 0 Dembowski-Ostrom Polynoimial A polynomial f ( x ) ∈ F p r [ x ] is a Dembowski-Ostrom polynomial if it has the shape k a ij x p i + p j . � f ( x ) = i , j = 0 Almost all known planar functions on finite fields are Dembowski-Ostrom polynomials. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 9 / 31
Equivalence of Semi fields and Planar functions Theorem 1 [Coulter and Henderson 2008] A planar function which is a Dembowski-Ostrom polynomial is equivalent to a commutative semi field Let p be an odd prime, and let f ( x ) be a polynomial on ( F p r , + , × ) Then an operation may be defined by x ⋆ y = f ( x + y ) − f ( x ) − f ( y ) Then the structure S = ( F p r , + , ⋆ ) is a commutative semi-field if and only if f ( x ) is a Dembowski-Ostrom planar polynomial. Furthermore EA equivalence of planar functions corresponds with isotopism ( ≃ ) of commutative semi fields. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 10 / 31
Applications of commutative semi fields and planar functions Finite Geometry Hadamard matrices Cryptography Quantum Information Theory Error correcting codes Signal sets for wireless transmission J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 11 / 31
Known Planar Functions and commutative semifields x 2 , Galois Fields x p k + 1 on F p r such that r / gcd ( r , k ) is odd, Alberts Twisted Semi Fields [Dembowski & Ostrom, 1968] x 2 + j ( x − x p r ) 2 ( β − β p r ) 2 − β 2 ( x − x p r ) 2 ( β − β p r ) 2 on F p 2 r where j is a non-square and beta is non-zero. Dickson Semi Fields [Dickson 1906, Bundunghyn & Helleseth 2008] x p r + 1 + ω ( β x p s + 1 + β p r x ( p s + 1 ) p r ) on F p 2 r where ω p r = − ω , there is no a ∈ F ∗ p 2 r such that a p r = − a and a p s = − a and β p r − 1 is not contained in the subgroup of order p r + 1 / gcd ( p r + 1 , p s + 1 ) . Projection Semi Fields [Bierbrauer, 2009] . . . J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 12 / 31
More known planar functions and commutative semifields x 10 ± x 6 − x 2 on F 3 r [Coulter & Mathews, 1997] x 2 + x 90 on F 3 5 (Weng, 2007) x ( 3 k + 1 ) / 2 on F 3 r where k is odd and gcd ( k , r ) = 1 [Coulter & Mathews,1997)]No Semifield Not Dembowski-Ostrom Generalised Budaghyan-Helleseth semifields [Bierbrauer, 2011] Zhou Pott Semifield [Zhout & Pott, 2013] . . . J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 13 / 31
Equivalences Planar polynomials Π and Π ′ with Π ′ ( x ) = ( L 1 ◦ Π ◦ L 2 ) ( x ) + L 3 ( x ) are said to be EA-equivalent if L 1 ( x ) , L 2 ( x ) , L 3 ( x ) are affine polynomials and L 1 ( x ) , L 2 ( x ) are bijections Two presemifields S 1 = ( S , + , ∗ ) and S 2 = ( S , + , ◦ ) are isotopic if there exist three linearized permutation polynomials L 1 , L 2 , L 3 over S such that L 1 ( x ) ◦ L 2 ( y ) = L 3 ( x ∗ y ) for any x , y ∈ S . Theorem [Coulter & Henderson 2008] EA-equivalent Dembowski-Ostrom planar polynomials construct isotopic commutative semifields. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 14 / 31
Outline Semifields and Planar Functions 1 Projection Construction for Semifields and Planar Functions 2 New Semifields 3 J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 15 / 31
Lets find some new planar functions and semifields. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 16 / 31
Projection Construction Theorem 3 [Gagola & Hall, 2016] Let g ( x ) and h ( x ) be planar Dembowski-Ostrom monomials over F p 2 r . If either g ( x ) = x 2 or h ( x ) = x 2 , then f ( x ) = g ( x ) + g ( x ) p r + h ( x ) − h ( x ) p r is planar over F p 2 r . The corresponding semifield multiplication is x ⋆ y = f ( x + y ) − f ( x ) − f ( y ) J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 17 / 31
Theorem 4 [Gagola & Hall, 2016] Let h ( x ) = x 2 ∈ F p 2 r [ x ] , g ( x ) = x p k + 1 be a planar function on F p 2 r , and let f ( x ) = g ( x ) + g ( x ) p r + h ( x ) − h ( x ) p r . Then the semifield associated with f ( x ) is isotopic to the Zhou-Pott commutative semifield . Proof. The Zhou-Pott semifield has multiplication x ◦ y = ( x 1 + x 2 ω ) ◦ ( y 1 + y 2 ω ) Let L 1 and L 2 be linearized permutation polynomials over F p 2 r L 1 ( x ) = 3 x − x p r 2 x p r . L 2 ( x ) = 3 2 x − 1 L 1 ( x 1 + x 2 ω ) ◦ L 2 ( y 1 + y 2 ω ) = ( x 1 + x 2 ω ) ⋆ ( y 1 + y 2 ω ) J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 18 / 31
Theorem 5 [Gagola & Hall, 2016] Let g ( x ) = x 2 ∈ F p 2 r [ x ] , h ( x ) = x p k + p i be a planar function on F p 2 r and let f ( x ) = g ( x ) + g ( x ) p r + h ( x ) − h ( x ) p r . If either r is even or p ≡ 1 ( mod 4 ) , then f ( x ) is equivalent to a generalized Budaghyan-Helleseth planar function. Theorem 6 [Gagola & Hall, 2016] Let g ( x ) = x 2 ∈ F p 2 r [ x ] , h ( x ) = x p k + 1 a planar function on F p 2 r and let f ( x ) = g ( x ) + g ( x ) p r + h ( x ) − h ( x ) p r . If r is odd and p ≡ 3 ( mod 4 ) , then the semifield associated with f ( x ) is isotopic to a Zhou-Pott semifield. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 19 / 31
All semifields constructed using the projection construction with the planar functions x 2 and x p k + 1 are known. J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 20 / 31
Outline Semifields and Planar Functions 1 Projection Construction for Semifields and Planar Functions 2 New Semifields 3 J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 21 / 31
Computer Generated Planar functions/semifields The following are examples of planar functions of the form x p i + 1 + x ( p i + 1 ) p r + x p s + 1 − x ( p s + 1 ) p r ∈ F p 2 r [ x ] . F1 x 6 + x ( 6 ) 5 3 + x 26 − x ( 26 ) 5 3 ∈ F 5 6 [ x ] F2 x 8 + x ( 8 ) 7 3 + x 50 − x ( 50 ) 7 3 ∈ F 7 6 [ x ] F3 x 10 + x ( 10 ) 3 6 + x 82 − x ( 82 ) 3 6 ∈ F 3 12 [ x ] F4 x 10 + x ( 10 ) 3 6 + x 28 − x ( 28 ) 3 6 ∈ F 3 12 [ x ] Obtained via computer search using GAP . J. Hall, S.Gagola (RMIT, Miami) Semifields 10 Mar 2017 22 / 31
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