Three-Bit Monomial Hyperovals

Transcription

1 Timothy Vis University of Colorado Denver Rocky Mountain Discrete Math Days 2008

2 Definition of a Hyperoval Definition In a projective plane of order n, a set of n + 1-points, no three on a line, is called an oval. A set of n + 2-points, no three on a line, is called a hyperoval. Ovals always exist in Desarguesian planes PG (2, q). Theorem If n is odd, hyperovals cannot exist. If n is even, every oval can be uniquely extended to a hyperoval.

3 Describing Hyperovals With Polynomials The points (0, 0, 1), (0, 1, 0), (1, 0, 0), and (1, 1, 1) are called the fundamental quadrangle. Every hyperoval in a Desarguesian plane is equivalent to one containing the fundamental quadrangle. Every hyperoval can then be described as the points D (f ) = {(1, x, f (x)) : x GF (q)} {(0, 1, 0), (0, 0, 1)} where f (x) is a polynomial satisfying certain conditions. Definition A polynomial f (x) such that D (f ) is a hyperoval is called an o-polynomial.

4 The Known Classification of hyperovals remains an open and difficult problem. It is interesting to restrict to the case in which f is the monomial x k. We then write D (k) for D (f ). Four families of monomial hyperovals are currently known in PG ( 2, 2 h). Translation hyperovals: D ( 2 i) where (i, h) = 1 Segre hyperovals: D (6) where h is odd Glynn hyperovals: D (σ + γ) where σ 2 = γ 4 = 2 and h is odd Glynn hyperovals: D (3σ + 4) where h is odd

5 Domination Among the Integers Definition We define a partial order on integers by a b if and only if every term in the binary expansion of a appears in the binary expansion of b. Example 5 13 because 5 = and 13 = On the other hand, 7 13, because 7 =

6 Glynn s Criterion: A Classification Tool Theorem (Glynn s Criterion) D (k) is a hyperoval in PG ( 2, 2 h) if and only if for all 1 d 2 h 2, d kd, where kd is reduced modulo 2 h 1. Example Let k = 2 q 0m + 2 q 1m + 2 q 2m and let h = qm. d = q 0m d = q 1m d = q 2m d = kd = Since d kd, D (k) is not a hyperoval in PG ( 2, 2 h).

7 Bitwise Classification Results Every known monomial hyperoval is equivalent to either D ( 2 i 0), D ( 2 i i 1), or D ( 2 i i i 2). Theorem (Segre, 1957) A hyperoval of the form D ( 2 i 0) is a translation hyperoval. Theorem (Cherowitzo, Storme, 1998) A hyperoval of the form D ( 2 i i 1) is a translation hyperoval, Segre hyperoval, or Glynn hyperoval. Conjecture A hyperoval of the form D ( 2 i i i 2) is a translation hyperoval, Segre hyperoval, or Glynn hyperoval.

8 Divisibility Restrictions on D ( 2 i 0 + 2i 1 + ) 2i 2 Lemma (V.) If i 0, i 1, i 2, and h have a common divisor greater than one, D ( 2 i i i 2) is not a hyperoval in PG ( 2, 2 h ). Lemma (V.) If each of i 0, i 1, and i 2 has a common divisor with h, there exists some m dividing (WLOG) i 0 and h but neither i 1 nor i 2. Lemma (V.) If i 1 and i 2 are not both congruent to 1 mod m, then D ( 2 i i i 2) is not a hyperoval in PG ( 2, 2 h ).

9 Conditions on a Three-Bit Hyperoval Lemma (V.) If D ( 2 i i i ( 2) is a hyperoval in PG 2, 2 h ) then either 1 (i 1, h) = (i 2, h) 1 mod (i 0, h) 2 i a i b 1 mod (i c, h) for all choices of a, b, and c. 3 (i a, h) = 1 for some choice of a. Theorem (V.) If (i 1, h) = (i 2, h) (i0, h), then D ( 2 i i i 2) is not a hyperoval in PG ( 2, 2 h).

10 The Tool of Reduction Definition If c h and i 0, i 1, and i 2 have distinct residues mod c, then the reduction of D ( 2 i i i ( 2) in PG 2, 2 h ) ( ) with respect to c is D 2 i i i 2 in PG (2, 2 c ), where i a is the residue of i a mod c. Theorem (V.) If D (k) is a hyperoval in PG ( 2, 2 h), any reduction of D (k) is also a hyperoval.

11 An Example of Reduction Example Let k = and let h = 21. A reduction of D (k) is described by k = and c = 7. d = d = d = d = k d = d = d = d = d = kd =

12 Consequences of Reduction Lemma (V.) If D (k) has no reduction in PG ( 2, 2 h), then h has at most two prime divisors. Theorem (V.) If D (k) has no reduction in PG ( 2, 2 h), then at least one of i 0, i 1, or i 2 is relatively prime to h. This allows us to express k as α + α i + α j. Since we are merely permuting the bits of k, we may still apply Glynn s Criterion to this α-ary expansion of k.

13 Further Consequences of Reduction Theorem (V.) If i a i b 1 mod (i c, h) for all choices of a, b, and c, then if D ( 2 ia + 2 i b + 2 ic ) is a hyperoval either it is a known hyperoval or there exists some heretofore unknown hyperoval with (i a, h) = 1 for some choice of a. Thus, it remains only to classify D ( α + α i + α j) unless new hyperovals arise. To do this, we express h as mj + ni + l, where ni + l < j and l < i.

14 Conditions on α + α i + α j Theorem (V.) If D ( α + α i + α j) is a hyperoval, one of the following conditions holds: 1 n = m = 1, 2i + 1 j 2i + l 2 i = 2, l = 1 3 m = 1, j = i + l 1, n = 0 4 m = 1, l = i 1, j = 2n + 2i 2 5 m = 1, j = ni + l n = 1, j = 2i, l 0 7 m = 1, j = ni + i + l 8 n = 1, j = i + 1, l = 0 9 n = 0, j i + l 10 m = 1, n = 0, l = i 1

15 Collapsing of Cases Theorem (V.) If k has an α-ary expansion, it has a β-ary expansion which must satisfy the same conditions if D (k) is a hyperoval. Example This allows for some collapsing of cases. If m = 1 and j = ni + i + l, notice that 2j = j + ni + l + i = i. Since either i or j is relatively prime to h, j must be, and if β = α j, we have a new expansion β + β 2 + β x, which must fall into one of these cases.

16 Classification Thus Far Theorem (V.) If D (k) has an α-ary expansion that falls into any but the last two cases, it is a known hyperoval or it is not a hyperoval.

17 Some Further Work Finish up the n = 0, j i + l case. The other remaining case has been reduced to this one. Extend ideas such as reduction to arbitrary number of bits. Work towards classification results on monomial hyperovals in general.