THE AFFINITY OF A PERMUTATION OF A FINITE VECTOR SPACE

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1 DEPARTMENT OF MATHEMATICS TECHNICAL REPORT THE AFFINITY OF A PERMUTATION OF A FINITE VECTOR SPACE DR. W. EDWIN CLARK DR. XIANG-DONG HOU DR. ALEC MIHAILOVS JULY 004 No TENNESSEE TECHNOLOGICAL UNIVERSITY Cooeville, TN 38505

2 THE AFFINITY OF A PERMUTATION OF A FINITE VECTOR SPACE W. EDWIN CLARK, XIANG-DONG HOU*, AND ALEC MIHAILOVS Abstract. For a permutatio f of a -dimesioal vector space V over a fiite field of order we let -affiity(f) deote the umber of -flats X of V such that f(x) is also a -flat. By -spectrum(, ) we mea the set of itegers -affiity(f) where f rus through all permutatios of V. The problem of the complete determiatio of -spectrum(, ) seems very difficult except for small or special values of the parameters. However, we are able to establish that 0 -spectrum(, ) i the followig cases: (i) 3 ad 1 1; (ii) =, 3 1; (iii) =, =, 3 odd. The maximum of -affiity(f) is, of course, obtaied whe f is ay semi-affie mappig. We cojecture that the ext to largest value of -affiity(f) is whe f is a traspositio ad we are able to prove this whe =, =, 3 ad whe 3, = 1,. 1. Itroductio It is a classical result, see, e.g., Sapper ad Troyer [9], that if V is a - dimesioal vector space over a field F such that ad F 3 the a bijectio f : V V which taes 1-flats to 1-flats is a semi-affie mappig, that is, there is a automorphism σ of F, a additive automorphism g : V V ad a vector b V such that g(αx) = σ(α)g(x) for all x V, α F ad f(x) = g(x) + b for all x V. We remar that if the automorphism σ is the idetity the g is just a o-sigular liear mappig ad f is said to be affie. This will be the case whe F has o o-trivial automorphisms. The above result is ot true whe F =. I this case, a 1-flat i V is just a two elemet subset, hece every permutatio of V taes all 1-flats to 1-flats. However, the above result has a easy aalog for the case F = : A permutatio of V which taes every -flat to a -flat must be affie (cf. [5]). Let F be the fiite field with elemets ad let F be the -dimesioal vector space over F. I this paper, we are cocered with permutatios of F. Let Per(F ) deote the group of all permutatios of F. Recall that a -flat (or - dimesioal affie subspace) X i F is a coset U + x of a -dimesioal subspace U of F. Defiitio 1.1. For f Per(F ) ad 0 we defie -affiity(f) to be the umber of -flats X i F such that f(x) is a -flat. We defie -coaffiity(f) to be the umber of -flats X i F such that f(x) is ot a -flat. Key words ad phrases. affie, almost perfect oliear, fiite field, flat, geeral affie group, permutatio, semi-affie group, vector space. * Partially supported by NSA grat MDA

3 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS It is well ow that the umber of -dimesioal subspaces of F the -biomial coefficiet [ ] = ( 1)( 1 1)... ( +1 1) ( 1)( 1 1)... ( 1 1) ad the umber of -flats i F is give by [ ] It follows that. ] is give by -affiity(f) + -coaffiity(f) = [ for all permutatios f of F ad all 0. The cases = 0 ad = are trivial ad we shall igore them. Defiitio 1.. For itegers 0 ad prime power, we defie -spectrum(, ) to be the set of values -affiity(f) for all f Per(F ). The preset paper is a cotiuatio of the secod author s wor [5]. I [5], the otio of -affiity of permutatios of F was implicitly itroduced ad permutatios of F with -affiity 0 were studied. We poit out that a permutatio f Per(F ) with -affiity(f) = 0 is a almost perfect oliear (APN) permutatio. APN permutatios arose i cryptography as a meas to resist the differetial cryptaalysis [, 8]. APN permutatios of F are ow to exist for odd 3 ([, 7]) ad ot to exist for = 4 ([5]). Their existece for eve 6 is a ope uestio. For recet wor o APN permutatios ad related topics, we refer the reader to [1,, 3, 4, 5]. However, we must remid the reader that this paper is ot a respose to ay problem from cryptography. Rather, it is a pure mathematical exploratio. Our primary iterest is the set -spectrum(, ). I particular, we would lie to ow if 0 -spectrum(, ) ad what the secod largest umber i -spectrum(, ) is. (The largest umber i -spectrum(, ) is, of course, [ ].) I Sectio, we show that with few exceptios, 0 -spectrum(, ). The result of Sectio relies o a ieuality ivolvig -biomial coefficiets whose proof is give i Sectio 3. Hou [5] showed that -spectrum(4,) is {5 0,, 4 6, 8, 30, 3, 36, 38, 44, 48, 5, 56, 76, 84, 140} where a b deotes all itegers from a to b. More examples of -spectra are give i Sectio 4. I Sectio 5, we determie ( 1)-spectrum(, ) completely. These examples ad results led to the cojecture that the ext to largest -affiity is that of a traspositio. We compute the -affiity T (,, ) of a traspositio i Per(F ) i Sectio 6. We call this cojecture The Threshold Cojecture sice it says that if -affiity(f) > T (,, ) the f taes every -flat to a -flat. We prove that the cojecture holds for =, =, 3 i Sectio 7 ad for >, = 1, i Sectio 8.. Whe -affiity(f) = 0 It should be oted that there appears to be o clear relatioship betwee - affiity(f) ad l-affiity(f). For example, there are permutatios f 1, f, f 3, f 4 i

4 AFFINITY OF A PERMUTATION 3 Per(F 3 3) such that 1-affiity(f 1 ) = 1 ad -affiity(f 1 ) = 0 1-affiity(f ) = 0 ad -affiity(f ) = 1 1-affiity(f 3 ) = 0 ad -affiity(f 3 ) = 0 1-affiity(f 4 ) = 1 ad -affiity(f 4 ) = 1 I the followig theorem, we see that with few exceptios there is a permutatio f Per(F ) such that simultaeously -affiity(f) = 0 for all 1 1. Theorem.1. (i) If = ad 3 is odd, there exists f Per(F ) such that -affiity(f) = 0. (ii) If = ad 4, there exists f Per(F ) such that -affiity(f) = 0 for all 3 1. (iii) If 4 ad, there exists f Per(F ) such that -affiity(f) = 0 for all 1 1. (iv) If = 3 ad 3, there exists f Per(F 3 ) such that -affiity(f) = 0 for all 1. (v) If = 3 ad, there exists f Per(F 3 ) such that 1-affiity(f) = 0. The proof of Theorem.1 is spread out i parts i the rest of this sectio. Part (i) of Theorem.1 is well ow. (See [3, 4] for several families of permutatios of F m+1 with -affiity 0.) It also follows from the followig example i which we compute the -affiity of the permutatio f of F ( = F ) defied by f(x) = x. We remar that this permutatio has bee discussed by Nyberg [7] i terms of differetial uiformity ad that our computatio is slightly differet from that of [7]. Example.. Idetify F with F. Defie f Per(F ) by f(x) = x for x F. Note that { 0 if x = 0, f(x) = 1 x if x 0. We claim that { 0 if is odd, -affiity(f) = 1 3 if is eve. Suppose X F is some -flat such that f(x) is also a -flat. The we ca write X = {x, y, z, w} where x + y + z + w = 0. Suppose first that x, y, z, ad w are all ozero. The we have f(x) = { 1 x, 1 y, 1 z, 1 w } ad 1 x + 1 y + 1 z + 1 w = 0. It follows that w = x + y + z ad 0 = 1 x + 1 y + 1 z + 1 x + y + z (x + y)(x + z)(y + z) =. xyz(x + y + z) It follows that x = y or x = z or y = z which cotradicts the assumptio that X is a -flat. Thus without loss of geerality we may assume that w = 0. The X = {0, x, y, x + y}, f(x) = {0, 1 x, 1 y, 1 x+y } ad 1 x + 1 y + 1 x + y = 0.

5 4 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS This is euivalet to (.1) ( y x ) + ( y x) + 1 = 0. Hece y x is a root of the irreducible polyomial g(t) = t + t + 1 F [t]. Therefore F ( y x ) is a subfield of F with [F ( y x ) : F ] =. It follows that is eve. So if is odd, o such x ad y exist ad -affiity(f) = 0. O the other had, if is eve, g(t) has two roots β ad 1 + β i F ad K = {0, 1, β, 1 + β} is the uiue subfield of order 4 i F. It follows from (.1) that y x = β or 1 + β. I both cases, X = {0, x, y, x + y} = {0, x, xβ, x(1 + β)} = xk = {0} xk where K is the multiplicative group of K ad xk is a coset of the subgroup K i F. There are ( 1)/3 cosets of K i F ad therefore the same umber of - flats of the form xk. Sice f(xk) = 1 x K, it follows that -affiity(f) = ( 1)/3, as claimed. Note that whe = 4, = 5 is the miimum -affiity of permutatios of F 4 ([5]). The proof of parts (ii) (iv) of Theorem.1 relies o the followig theorem whose proof will be give i Sectio 3. Theorem.3. Let, m, be itegers such that > m ad, m satisfy oe of the followig coditios: The (.) (a) =, m = 3, (b) = 3, m =, (c) 4, m = 1. 1 =m ( ) [ ]! ( )! <!. Proof of Theorem.1 (ii) (iv). Let Φ deote the set of all f Per(F ) such that -affiity(f) 1. Let X be ay fixed -flat i F ad defie S X = {f Per(F ) : f(x) = X}. Note that sice X = we have S X =! ( )!. The group of ivertible affie trasformatios of F, i.e., the geeral affie group AGL(, F ), acts trasitively o the set of all -flats of F. Hece for every -flat W, there exists α W AGL(, F ) such that α W (W ) = X. Let f Φ ad assume that f(w ) = Z where W ad Z are -flats. The α Z f α 1 S X, i.e., f α 1 Z S X α W. Sice there are [ ( Φ [ ] ] W -flats i F it follows that ) [ ] S X = ( )! ( )!.

6 AFFINITY OF A PERMUTATION 5 Hece if ad m satisfy oe of the coditios i Theorem.3 ad > m, we have 1 =m Φ 1 =m ( ) [ ]! ( )! <!. Thus there exists f Per(F ) such that f / 1 =m Φ. Note that ieuality (.) does ot cover the case = 3 ad m = 1, that is, part (v) of Theorem.1. This case is dealt with as a corollary to the followig lemma. Lemma.4. Let F be ay field. If the permutatios f : F F ad g : F m F m each have 1-affiity 0 the the permutatio f g : F F m F F m has 1-affiity 0. Proof. Assume to the cotrary that there exists a 1-flat X i F F m such that (f g)(x) is also a flat. Let π 1 : F F m F ad π : F F m F m be the projectios. The either π 1 (X) is a 1-flat i F or π (X) is a 1-flat i F m. Without loss of geerality, assume that the former is the case. Thus f ( π 1 (X) ) ( ) = π 1 (f g)(x) is a 1-flat i F, which is impossible sice 1-affiity(f) = 0. Corollary.5. If, there exists f Per(F 3 ) such that 1-affiity(f) = 0. Proof. By Lemma.4, it suffices to show that F 3 ad F 3 3 have permutatios of 1- affiity 0. Such permutatios are easily foud through a computer search. For F 3, label the elemets (0, 0), (0, 1),..., (, ) with 0, 1,..., 8. A desirable permutatio f is give by ( f(0),, f(8) ) = (0, 1, 8,, 3, 4, 5, 6, 7). For F 3 3, we label the elemets (0, 0, 0), (0, 0, 1),..., (,, ) with 0, 1,..., 6. A desirable permutatio f i this case is give by ( f(0),, f(6) ) = (0, 1, 4,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 5, 15, 16, 17, 6, 18, 19, 3, 0, 1, ). 3. Ieualities betwee Biomial ad -Biomial Coefficiets I this sectio we assume that i,, m,, are itegers. Lemma 3.1. For >, > 1 ad =, >, [ ] [ ] 1 1 (3.1) ( ) < ( ) 1. Proof. Ieuality (3.1) is euivalet to (3.) ( 1) ( 1)... ( ( 1)) < ( ) ( )... ( ( 1)), which ca be further rewritte as 1 i= ( i) 1 ( 1) < 1 1 i ( i) i 1 ( 1).

7 6 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS Clearly, ( i) i= Thus it suffices to show that (3.3) Ieuality (3.3) follows from 1 i i 1 ( i). 1 ( 1) < ( 1). ( )( ( 1)) ( 1)( ( 1)) = ( 1)( ( ) + ) > 0. Lemma 3.. For 4, 1, or = 3,, or =, 3, [ ] + 1 (3.4) ( ) < Proof. The left had side of (3.4) euals I this product, ad for every 1 i 4, (3.5) 3 ( 1) ( )( +1 + ) < 1 i +1 i 1 1 i=1 i +1 i 1. (To see (3.5), ote that sice, we have +1 i 1 +1 i.) Therefore, it suffices to show that 6 ( 1) ( )( +1 + ) 1 4. Let It suffices to show that f(, ) = ( 1) ( ( 1) + 1 )( ( 1) + ). f(, ) 6. The fuctio f(, ) is icreasig with respect to for fixed > 1. For ad or = 1 ad 4, we have d [ ( 1) + 1 ] d = ( 1) ( ) 3 3 > 0

8 AFFINITY OF A PERMUTATION 7 ad d [ ( 1) + ] d = ( 1) ( ) Hece f(, ) is icreasig with respect to for ad i the above rage. Thus, for 4 ad 1, f(, ) f(4, 1) = > 6; for = 3 ad, f(, ) f(3, ) = > 6; for = ad 4, f(, ) f(, 4) = > 6. For = ad = 3, (3.4) is verified directly: [ ] 4 3 ( ) 4 = 5 86 < 1 3 = Corollary 3.3. For 4, > 1, or = 3, >, or =, > 3, [ ] (3.6) 1 ( ) <. ( )( )+ Proof. Applyig Lemma 3.1 to the left had side of (3.6) 1 times ad applyig Lemma 3. after that, we get [ ] [ ] ( ) < ( 1)( ) < = 1 ( 1)( ) 1 ( )( )+. ( ) +1 1 Lemma 3.4. For 4, 1, or = 3,, or =, 3, (3.7) 0. Proof. Let g(, ) deote the left had side of (3.7). For fixed 1, g(, ) is icreasig with respect to. Also ote that g = l > 4,

9 8 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS which is o-egative i the described rage for ad. So, g(, ) is icreasig with respect to i such rage. Thus, for 4 ad 1 for = 3 ad, for = ad 3, g(, ) g(4, 1) = 0; g(, ) g(3, ) = 3 > 0; g(, ) g(, 3) = 0. Proof of Theorem.3. Let ad m satisfy oe of the coditios (a) (c) i Theorem.3 ad let > m. By Corollary 3.3 ad Lemma 3.4, for m <, we have [ ] Therefore, ( ) 1 ( ) < 1 ( )( )+. [ ] 1 =m ( ) which proves the theorem. ( ) < 1 =m 1 < =1 1 = 1 1 1, 4. Examples of -Spectra for Small Values of,, Recall that for give,,, -spectrum(, ) = {-affiity(f) : f Per(F )}. Here we give some examples. I all cases a b deotes all itegers from a to b. Oly a few of the spectra (as idicate below) are proved to be complete. The other examples are spectra obtaied by radom searches. I such cases it is possible but ot certai that some values are missig, hece the results are called partial spectra. The full spectrum for = 1, =, = 3: {0 4, 6, 1} The full spectrum for = 1, =, = 4: {0 6, 8, 1, 0} A partial spectrum for = 1, =, = 5: {0 1, 14, 15, 0, 30} A partial spectrum for = 1, =, = 7: {0, 5 7, 10 1, 14 18, 0 30, 3, 35, 4, 56} A partial spectrum for = 1, = 3, = 3: {0 64, 66, 70, 71, 73, 75, 81, 93, 117} The full spectrum for =, = 3, = : {0,, 6, 14}

10 AFFINITY OF A PERMUTATION 9 A partial spectrum for =, = 3, = 3: {0 9, 11 13, 15, 1, 39} The full spectrum for =, = 4, = : {5 0,, 4 6, 8, 30, 3, 36, 38, 44, 48, 5, 56, 76, 84, 140} A partial spectrum for =, = 5, = : {0, 9 416, 418, 40, 4, 44, 46 48, , 434, , 44, , 454, , 464, 466, , 474, 476, 480, 48, 484, 486, 488, 490, 49, 496, 500, 504, 506, 508, 51, , , 50, 56 58, 530, 53, 536, 540, 548, 550, 55, 554, 556, 560, 564, 568, 576, 600, , 608, 618, 60, 640, 648, 664, 704, 706, 78, 73, 736, 79, 80, 960, 140} A partial spectrum for =, = 6, = : {1, 8 513, 5134, , , , 5386, 5388, 5390, 539, 5394, , 540, , , , 5470, , 548, , 5488, 5490, 549, 5496, 5498, 5500, 550, 5504, 5506, 5508, 5510, 551, 5514, 5516, 550, 55, 554, 556, 558, 5530, 553, 5534, 5536, 5538, 5540, 5544, 5548, , , 5558, , 5564, , , 5574, 5576, 5578, , 5584, 5586, , , , , , 5738, , , , 5786, , 579, 5794, 5796, 580, 58, 584, 5830, 583, 5834, 5836, 5840, 5844, 5849, , , 5868, 5874, 5876, 5878, 5880, 588, 5884, 5886, 5888, 5890, 589, 5896, 5898, 5900, 5904, 5908, 591, 5936, 5940, 5944, 5948, 595, 5960, 5974, 5976, 5978, 5984, 5986, 5988, 5990, 5994, 6000, 601, 600, 6030, 603, 6034, , , 6086, 6088, 6090, 609, 6096, , , 6138, 6140, 614, 6144, 6146, 6148, 615, 6160, , , 6186, , 619, 6194, 60, 608, 68, 630, 63, 634, 636, 638, 640, 64, 644, 648, 656, 660, 664, 686, 688, 690, 696, 698, 6316, 6336, 6340, 635, 6360, 6384, 6444, 6448, 645, 6460, , 648, 6484, 6488, 6496, , , , 6536, 6538, 6540, 6544, , 6586, 6588, 6590, 659, 6594, 6596, 6640, 664, 6644, 6648, 6650, 6656, 6756, 6764, 6768, 683, 6955, , , 6986, 6988, 699, 7036, 7038, 7040, 704, 7044, 7048, 7056, 715, 7156, 7160, 7384, 7461, 7490, 749, 755, 7994, 805, 8056, 8176, 8556, 9176, 10416} Observatios: (1) From Theorem.1, 0 -spectrum(, ) for 1 1 uless = 1, = or =, = ad is eve. () Near the begiig i each example spectrum there is a log seuece of cosecutive values. For = ad =, there seems to be a gap precedig the first o-zero affiity. For > the limited experimetal data shows o such gaps. This suggests that for =, if there is oe -flat that is carried to a -flat the there are a certai umber of other -flats that must also be carried to -flats. (3) Precedig the largest value i -spectrum(,) there appears to a gap of size [ 1 ]. Note that this gap may be cosidered a threshold i the ] [ ] 1, the f taes all -flats sese that if -affiity(f) > [ to -flats.

11 10 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS 5. ( 1)-spectrum(, ) I this sectio, we will determie ( 1)-spectrum(, ), which is the set of all ( 1)-affiities of permutatios of F. The stadard dot product of a, b F is deoted by a, b. Every ( 1)-flat i F is uiuely of the form H(a, ɛ) := {x F : a, x = ɛ} for some a F \ {0} ad ɛ F. Let f Per(F ). If for some a F \ {0} ad some ɛ F, f ( H(a, ɛ) ) is a ( 1)-flat, say f ( H(a, ɛ) ) = H(b, δ) for some b F \ {0} ad δ F, we must have f ( H(a, 1 + ɛ) ) = H(b, 1 + δ). Therefore, for each such a ad b, there exists φ Per(F ) such that (5.1) f ( H(a, t) ) = H ( b, φ(t) ) for all t F. Lemma 5.1. Let f Per(F ) ad let V f = {0} { a F \ {0} : f ( H(a, 0) ) is a ( 1)-flat }. The V f is a subspace of F Proof. For a 1, a V f, we prove that a 1 + a V f. We may assume that a 1 0, a 0, ad a 1 a. By (5.1), there exist b i F \ {0} ad φ i Per(F ), i = 1,, such that f ( H(a i, t) ) = H ( b i, φ i (t) ) for all t F. Clearly, b 1 b. Sice F has oly two permutatios, t t or t t + 1, we see that φ 1 +φ is a costat, say ɛ. For ay x H(a 1 +a, 0), let t = a 1, x = a, x. The x H(a i, t), hece f(x) H ( b i, φ i (t) ). It follows that b 1 + b, f(x) = φ 1 (t) + φ (t) = ɛ, i.e., f(x) H(b 1 +b, ɛ). Thus we have proved that f ( H(a 1 +a, 0) ) = H(b 1 +b, ɛ), which implies that a 1 + a V f. Theorem 5.. Let >. The ( 1)-spectrum(, ) = { i : 1 i + 1}. Proof. For each f Per(F ), by Lemma 5.1, we have ( 1)-affiity(f) = V f \ {0} = dim V f +1 { i : 1 i + 1}. It remais to show that for each 1 i + 1, there exists f Per(F ) with ( 1)-affiity(f) = i. We prove this claim by iductio o. For = 3, the claim was established by computer as metioed i Sectio 4. Assume > 3. If i = 1, the claim follows from Theorem.1. Thus we will assume i +1. By the iductio hypothesis, there exists g Per(F 1 ) such that ( )-affiity(g) = i 1. Defie f Per(F ) by f(c, x) = (c, g(x)), c F, x F 1. Clearly, {i} F 1, i = 0, 1, are mapped ito flats by f. Let X F be ay ( 1)-flat other tha {i} F 1, i = 0, 1, such that f(x) is a flat. Write X ( {i} F 1 ) = {i} Ui, i = 0, 1., where U i F 1 is a ( )-flat ad U 0 = U 1 or F 1 \ U 1. The (5.) X = ( {0} U 0 ) ( {1} U1 ).

12 AFFINITY OF A PERMUTATION 11 Sice ( {i} g((u i ) = f X ( {i} F 1 ) ) = f(x) ( {i} F 1 ) is a ( )-flat, g(u i ) is a ( )-flat. O the other had, give ay ( )-flats U 0, U 1 F 1 such that U 0 = U 1 or F 1 \ U 1 ad g(u i ), i = 0, 1, are ( )-flats, both X (i (5.)) ad f(x) are ( 1)-flats i F. Therefore, ( 1)-affiity(f) = + (( )-affiity(g) ) The proof is ow complete. = + ( i 1 ) = i. 6. Threshold Cojecture for -Affiity I may cases it appears that the ext to largest -affiity is the -affiity of a traspositio. We calculate this value i the followig lemma. I this case it is more coveiet to compute the -coaffiity. Lemma 6.1. Let f Per(F ) be ay traspositio. The [ ] 1 -coaffiity(f) = ad ( ( )( ) [ ] 1) 1 -affiity(f) = Proof. Assume that f iterchages x ad y, where x, y F are distict. If U is a -flat, the f(u) is ot a -flat if ad oly if U cotais exactly oe of x ad y. The umber of -flats i F cotaiig a fixed poit is [ ] ; the umber of -flats i F cotaiig two fixed poits is [ ] 1 1. Hece ( [ ] [ ] ) [ ] 1 1 -coaffiity(f) = =. 1 It follows that [ ] [ ] 1 -affiity(f) = ( ( )( ) [ ] 1) 1 = It is atural to call the ext largest -affiity a threshold for affiity ad we mae the followig cojecture: Cojecture 6. (Threshold Cojecture). Let 1 1 ad f Per(F ). If [ ] 1 -coaffiity(f) <,..

13 1 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS the -coaffiity(f) = 0, i.e., f AGL(, F ). Euivaletly, if ( ( )( ) [ ] 1) 1 -affiity(f) > the -affiity(f) = [ ]. That is, the ext to largest -affiity is that of a traspositio. This cojecture is supported by the examples i Sectio 4 ad the result i Sectio 5. More importatly it is supported by the proof for =, =, > i Sectio 7, ad the proof for >, = 1, > 1 i Sectio 8., 7. Proof of the Threshold Cojecture for =, = Recall that a -flat i F is simply a 4-elemet subset {x 1, x, x 3, x 4 } such that x 1 + x + x 3 + x 4 = 0. For f Per(F ) ad a -flat X F, f(x) is a -flat if ad oly if f is affie o X. For the proof i this sectio, the reader s familiarity with the Fourier trasformatio of boolea fuctios will be helpful. We first itroduce the ecessary otatio. The set of all fuctios from F to F is deoted by P. Every fuctio i P is uiuely represeted by a polyomial i F [X 1,..., X ] whose degree i each X i is at most 1. Namely, P = F [X 1,..., X ]/ X 1 X 1,..., X X. For each g P, put g = g 1 (1). The Fourier trasform of g P is the fuctio ĝ : F C defied by ĝ(a) = ( 1) g(x)+ a,x, a F, x F where a, x is the stadard dot product i F. Clearly, (7.1) ĝ(a) = g + a,. Note that for, g + a, g (mod ), hece ĝ(a) g (mod 4). It is well ow (also straightforward to prove) that (ĝ(a) ) (7.) = a F ad (7.3) (ĝ(a) ) [ 4 = ( 1) g(x+a)+g(x)]. a F a F x F (Euatio (7.) is the Parseval idetity; euatio (7.3) is a relatio betwee the Fourier trasform ad the covolutio of the fuctio. Cf. [6].) If A ( ĝ(a) ) B

14 AFFINITY OF A PERMUTATION 13 for all a F, from (7.), we have ( B A ) [ (ĝ(a) ) A + B Thus a F ] = (ĝ(a) ) 4 (A + B) a F = a F a F (ĝ(a) ) 4 (A + B) + ( A + B (ĝ(a) ) 4 (A + B) AB. (ĝ(a) ) ( + A + B ) ). a F Combiig the above with (7.3), we have [ (7.4) ( 1) g(x+a)+g(x)] (A + B) AB. a F x F The euality i (7.4) holds if ad oly if ( ĝ(a) ) = A or B for all a F. Lemma 7.1. Let g P with deg g. The [ (7.5) ( 1) g(x+a)+g(x)] + ( 1)( 4). a F x F The euality holds if ad oly if g + h = 1 for some h P with deg h 1. Proof. Sice deg g, ĝ(a) ± for all a F. Case 1. g is eve. By (7.1), 0 ĝ(a) 4 for all a F. Hece 0 ( ĝ(a) ) ( 4) for all a F. By (7.4), [ ( 1) g(x+a)+g(x)] ( 4) < + ( 1)( 4). a F x F Case. g is odd, By (7.1), ĝ(a) for all a F. Hece By (7.4), ( ĝ(a) ) ( ) for all a F. [ ( 1) g(x+a)+g(x)] [ + ( ) ] ( ) a F x F = + ( 1)( 4). The euality holds if ad oly if ( ĝ(a) ) = or ( ) for all a F. First assume that the euality i (7.5) holds. Sice a F (ĝ(a) ) =, for at least oe a F, ( ĝ(a) ) = ( ). By (7.1), ( g + a, ) = ( ).

15 14 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS Thus g + a, = 1 or 1. Let { a, if g + a, = 1, h = a, + 1 if g + a, = 1. The g + h = 1, as claimed. Now assume that g + h = 1 for some h P with deg h 1. The for every a F, g + a, = 1 ± 1, or 1, or 1. It follows from (7.1) that ĝ(a) = ± or ±( ). Hece ( ĝ(a) ) = or ( ) for all a F. Therefore the euality holds i (7.5). Theorem 7.. Let 3 ad f Per(F ) \ AGL(, F ). The -coaffiity(f) 8 3 ( 1 1)( 1). The euality holds if ad oly if f AGL(, F ) τ AGL(, F ) where τ Per(F ) is ay traspositio. Corollary 7.3. The Threshold Cojecture (Cojecture 6.) holds for =, =, >. Proof of Theorem 7.. Case 1. f(x + d) + f(x) = costat for some d F \ {0}. Without loss of geerality, assume d = (1, 0,..., 0). Write f = (f 1,..., f ) where f i : F F. The f i = c i X 1 + g i (X,..., X ) for all 1 i, where c i F ad g i P 1. Note that (c 1,..., c ) 0 sice otherwise, f is idepedet of X 1 ad caot be a permutatio of F. By composig a suitable elemet of GL(, F ) to the left of f, we may assume (c 1,..., c ) = (1, 0,..., 0). Thus f = ( X 1 + g 1 (X,..., X ), g (X,..., X ),..., g (X,..., X ) ). It follows that g = (g,..., g ) is a permutatio of F 1. Case 1.1. g / AGL( 1, F ). Usig iductio, we may assume -coaffiity(g) 8 3 ( 1)( 3 1). Note that if g is ot affie o a -flat {y 1, y, y 3, y 4 } i F 1, f is ot affie o the -flat {(x i, y i ) : i i 4} where x x 4 = 0. Hece -coaffiity(f) 8 (-coaffiity(g) ) > 8 3 ( 1 1)( 1). Case 1.. g AGL( 1, F ). The deg g 1. We may assume g = id. I this case, the -flats o which f is ot affie are precisely {(x i, y i ) : 1 i 4} where {y 1,..., y 4 } is a -flat i F 1 such that 4 i=1 g 1(y i ) = 1 ad x 1,..., x 4 F with 4 i=1 x i = 0. Defie G : (F 1 ) 3 F (y, a, b) g 1 (y) + g 1 (y + a) + g 1 (y + b) + g 1 (y + a + b)

16 AFFINITY OF A PERMUTATION 15 The We have G = 1 Therefore, = 1 = 1 [ 3( 1) -coaffiity(f) = 8 4! G = 1 3 G. y,a,b F 1 [ 3( 1) a F 1 [ 3( 1) a F 1 ( 1) G(y,a,b)] y,b F 1 ( y F 1 ( 1) g1(y)+g1(y+a)+g1(y+b)+g1(y+a+b)] ( 1) g1(y)+g1(y+a)) ] 1 [ 3( 1) [ ( 1) + ( 1 1)( 1 4) ]] (by Lemma 7.1) = 3 ( 1 1)( 1). -coaffiity(f) = 1 3 G 8 3 ( 1 1)( 1). If the euality holds i the above, the the euality i (7.5) holds with g 1 i place of g. By Lemma 7.1, there exists h P 1 such that g 1 + h = 1. Usig a liear trasformatio, we may replace g 1 with g 1 +h. Thus we may assume g 1 = 1. The clearly, f = ( ) X 1 + g 1 (X,..., X ), X,..., X is a traspositio. Case. f(x + d) + f(x) costat for all d F \ {0}. For each d F \ {0}, let (d) = { {x, x + d} : x F } ( F where ( ) F deotes the set of all -elemet subsets of F. Deote {f(x) : X (d)} by f( (d)) (a abuse of otatio for the coveiece). By the assumptio, f ( (d) ) (c) for every c F \ {0}. Sice the subsets i (d) ad (c) form partitios of F, we have (7.6) f ( (d) ) (c) 1. We claim that we ca partitio F \ {0} ito A ad B such that f ( (d) ) ( (a)), f ( (d) ) a A ( (b)). b B Note that (a), a F \ {0} form a partitio of ( ) ( ) F. If f (d) (a) 1 for all a F \ {0}, choose a 1, a F \ {0} distict such that ( ) f (d) (ai ) = 1, i = ( 1,. ) The A = {a 1, a }, B = F \ {0, a 1, a } have the desired property. If f (d) (a) for some a F \ {0}, let A = {a} ad B = F \ {0, a}. By (7.6), we have f ( (d) ) ( (b)) = 1 ( ) f (d) (a). b B ),

17 16 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS Hece A ad B also have the desired property. Therefore, amog the -flats which are a uio of two elemets i (d), there are at least ( 1 ) o which f is ot affie. Sice this statemet is true for all d F \ {0}, it follows that -coaffiity(f) ( 1 ) ( 1) 3 > 8 3 ( 1 1)( 1). For the remaider of this sectio, we compute the umber of permutatios f Per(F ) with -coaffiity(f) = [ ] 1 = 8 3 ( 1 1)( 1). Lemma 7.4. Let 3 ad choose a F \ {0}. Let τ Per(F ) be the traspositio which permutes 0 ad a ad let f AGL(, F ). (i) If 4, the τ f τ AGL(, F ) if ad oly if f({0, a}) = {0, a}. (ii) If = 3, the τ f τ AGL(, F ) if ad oly if f(a) + f(0) = a. Proof. (i) ( ) I fact, τ f τ = f. ( ) Assume to the cotrary that f({0, a}) {0, a}. Without loss of geerality, assume f 1 (0) / {0, a}. Sice 4, there is a -flat A i F which cotais f 1 (0) but does cotai 0, a ad f 1 (a). Write A = {f 1 (0), b, b 3, b 4 } where b i / {0, a}, f(b i ) / {0, a} ad f(b ) + f(b 3 ) + f(b 4 ) = 0. We the have (τ f τ)(a) = τ ( f(a) ) = τ ( {0, f(b ), f(b 3 ), f(b 4 )} ) = {a, f(b ), f(b 3 ), f(b 4 )}, which is ot a -flat. This is a cotradictio. (ii) ( ) Without loss of geerality, assume a = (1, 0, 0). The where (7.7) g(x, x 3 ) = Sice f(a) + f(0) = a, we have τ(x 1, x, x 3 ) = (x 1 + g(x, x 3 ), x, x 3 ) { 1 if (x, x 3 ) = 0, 0 if (x, x 3 ) 0. f(x) = xa + b, where b F 3, A GL(3, F ) ad aa = a, i.e., [ ] 1 0 A =, B GL(, F c B ), c F. Let b = (b, b ) where b F ad b F. The (τ f τ)(x 1, x, x 3 ) = (τ f)(x 1 + g(x, x ), x, x 3 ) ( ) = τ (x 1 + g(x, x ), x, x 3 )A + b = τ (x 1 + g(x, x 3 ) + (x, x 3 )c + b, (x, x 3 )B + b ) ( = x 1 + g(x, x 3 ) + (x, x 3 )c + b + g ( (x, x 3 )B + b ), (x, x 3 )B + b ).

18 AFFINITY OF A PERMUTATION 17 By (7.7), g ( (x, x 3 )B + b ) = g ( (x, x 3 ) + b B 1). Thus g(x, x 3 ) + g ( (x, x 3 )B + b ) = g(x, x 3 ) + g ( (x, x 3 ) + b B 1) has degree 1 sice deg g. Therefore τ f τ AGL(3, F ). ( ) Assume to the cotrary that f(a) + f(0) a. The f ( {0, a} ) {0, a}. Without loss of geerality, we may assume f 1 (0) / {0, a}. By the proof of ( ) of (i), it suffices to show that there is a -flat A i F 3 which cotais f 1 (0) but ot 0, a ad f 1 (a). By the assumptio, the set {0, a, f 1 (0), f 1 (a)} is ot -flat of F 3, hece it either has oly three distict elemets or is a frame of F 3. (A set of +1 elemets i a affie space is called frame if their affie spa is a -flat.) To see the existece of a desirable -flat A, first ote that { (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) } is a frame of F 3 ad { (x 1, x, x 3 ) : x 1 +x +x 3 = 0 } is a -flat which cotais exactly oe of the elemets i the frame. Usig a suitable affie trasformatio, we see that for ay frame {ɛ 0,..., ɛ 3 } of F 3, there is a -flat A such that A {ɛ 0,..., ɛ 3 } = {ɛ 0 }. Corollary 7.5. I the otatio of Lemma 7.4, we have (7.8) ( τ AGL(, F ) τ ) AGL(, F ) = Proof. Let f AGL(, F ) be give by f(x) = xa + b, { GL( 1, F ) if 4, 5 GL(, F ) if = 3. where A GL(, F ) ad b F. By Lemma 7.4, whe 4, f τ AGL(, F ) τ if ad oly if aa = a ad b = 0 or a; whe = 3, f τ AGL(, F ) τ if ad oly if aa = a. Euatio (7.8) follows immediately. Corollary 7.6. Assume 3. The umber of permutatios f Per(F ) with -coaffiity(f) = [ ] 1 is give by { 1 ( +3 ) ( 1) 1 j=1 (j 1) if 4, if = 3. Proof. Let τ Per(F ) be ay traspositio. f Per(F ) with -coaffiity(f) = [ ] 1 is By Theorem 7., the umber of AGL(, F ) AGL(, F ) τ AGL(, F ) = ( τ AGL(, F ) τ ) AGL(, F ). The result follows immediately from the above corollary. 8. Proof of the Threshold Cojecture for = 1, > The Threshold Cojecture for = 1, > is proved i two steps: first = the 3. For =, the cases = 3 ad 4 reuire differet treatmets. The group of all ivertible semi-affie trasformatios of F is deoted by AΓL(, F ). For ay two distict poits x, y F, xy deotes the lie through x ad y i F. Theorem 8.1. Let 4 ad f Per(F ) \ AΓL(, F ). The [ ] 1 1-coaffiity(f) =. 1

19 18 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS Proof. Amog the + 1 parallel classes of lies i F, we first assume that at most oe parallel class has the property that all lies i the class are mapped to lies by f. I each of the remaiig parallel classes, there are at least lies which are ot mapped to lies by f. (Sice the lies i a parallel class form a partitio of F, it caot be the case that exactly oe lie i a parallel class is ot mapped to a lie.) Therefore 1-coaffiity(f). Now assume that there are two parallel classes of lies i F such that all lies i the two parallel classes are mapped to lies by f. By composig suitable liear trasformatios to both sides of f, we may assume that f maps all horizotal lies to horizotal lies ad all vertical lies to vertical lies. (A horizotal lie i F is a lie with directio vector (1, 0); a vertical lie i F is a lie with directio vector (0, 1).) Assume that for every z F, there is a lie through z which is ot mapped to a lie by f. The there are at least two lies through z which are ot mapped to lies. Sice each lie cotais poits, we have 1-coaffiity(f) =. Therefore, we may assume that there exists z F such that all lies through z are mapped to lies by f. Usig suitable affie trasformatios, we may further assume that (i) f maps all horizotal (vertical) lies to horizotal (vertical) lies, (ii) f(0, 0) = (0, 0), f(1, 0) = (1, 0), f(0, 1) = (0, 1), (iii) all lies through (0, 0) are mapped to lies. Let f(x, 0) = (φ(x), 0) x F, f(0, y) = (0, ψ(y)) y F, where φ ad ψ are permutatios of F with φ(0) = ψ(0) = 0 ad φ(1) = ψ(1) = 1. For ay (x, y) F, it is the itersectio of the vertical lie through (x, 0) ad the horizotal lie through (0, y). Hece f(x, y) is the itersectio of the vertical lie through (φ(x), 0) ad the horizotal lie through (0, ψ(y)), i.e., f(x, y) = ( φ(x), ψ(y) ). Sice f(1, 1) = (1, 1), by (iii), the lie {(x, x) : x F } is mapped to itself. Hece φ = ψ. Let F. By (iii), f ( {(x, x) : x F } ) = {(φ(x), φ(x)) : x F } is a lie. Hece φ(x) φ(x) = g() x F \ {0}, where g() F is a fuctio of. Settig x = 1, we have g() = φ(). Thus (8.1) φ(x) = φ()φ(x) for all x, F. Assume that for every a F \ {0}, all lies through (a, 0) which are either horizotal or vertical are ot mapped to lies. The we have 1-coaffiity(f) ( 1)( + 1 ) >, sice 4. Therefore, we may assume that there exist a F \ {0} ad a lie L through (a, 0) such that L is either horizotal or vertical ad f(l) is a lie. Let (0, b) L,

20 AFFINITY OF A PERMUTATION 19 where b F \{0}. The f(l) is the lie through (φ(a), 0) ad (0, φ(b)). (See Figure 1.) For each x F, the itersectio of L ad the vertical lie through (x, 0) is ( b x, a (x a)) ; the itersectio of f(l) ad the vertical lie through (φ(x), 0) is ( φ(x), φ(b) ( ) ) φ(x) φ(a). φ(a) By (i), we have φ ( b a (x a)) = φ(b) ( ) φ(x) φ(a). φ(a) Usig (8.1), we obtai (8.) φ(a x) = φ(a) φ(x). For ay b, x F, by (8.1) ad (8.), (8.3) φ(ba bx) = φ(b)φ(a x) = φ(b) ( φ(a) φ(x) ) = φ(ba) φ(bx). Combiig (8.1) ad (8.3), φ is a automorphism of F. Hece f AΓL(, F ), which is a cotradictio. b φ(b) (x, b a (x a)) (φ(x), φ(b) φ(a) (φ(x) φ(a))) x a L φ(x) φ(a) f(l) Figure 1. Proof of Theorem 8.1 Theorem 8.. Let f Per(F 3) \ AGL(, F 3 ). The 1-coaffiity(f) 6. The euality holds if ad oly if f AGL(, F 3 ) τ AGL(, F 3 ), where τ Per(F 3 ) is ay traspositio. Proof. Case 1. There do ot exist two oparallel lies i F 3 which are mapped ito lies by f. The 1-coaffiity(f) 3 ( ) = 9 > 6. Case. There are two oparallel lies L 1 ad L i F 3 which are mapped ito lies. Note that f is affie o each of these two lies. (This a special property of F 3.) Therefore, through suitable affie trasformatios, we may assume that L 1 = F 3 {0}, L = {0} F 3 ad that f L1 = id, f L = id. Case.1. f moves exactly two elemets i F 3 \ (L 1 L ) = {1, 1} {1, 1}. f is a traspositio ad 1-coaffiity(f) = 6.

21 0 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS Case.. f moves exactly three elemets i {1, 1} {1, 1}, say, f = (a 1, a, a 3 ) where {1, 1} {1, 1} = {a 1, a, a 3, a 4 }. For each a {1, 1} {1, 1}, let L a be the uiue lie through a such that L a (L 1 L ) =. The f(l ai ) (1 i 3) is ot a lie sice a i is ot fixed by f but the other two poits o L ai are. Note that the third poit o the lie a 4 a i (1 i 3) is o L 1 L. Thus f(a 4 a i ) (1 i 3) is ot a lie. We also claim that for 1 i < j 3, f(a i a j ) is ot a lie. Otherwise, without loss of geerality, assume that f(a 1 a ) is a lie. a 1 a must itersect L 1 L, say, at a 0. Note that a 0, a f(a 1 a ). Hece f(a 1 a ) = a 0 a = a 0 a 1. Thus a, a 3 f(a 1 a ) = a 0 a 1, which is impossible. Therefore, ( ) 4 1-coaffiity(f) 3 + = 9 > 6. Case.3. f = (a 1, a, a 3, a 4 ), where {1, 1} {1, 1} = {a 1, a, a 3, a 4 }. By the argumet i Cases., f(l ai ) (1 i 4) ad f(a 1 a ), f(a a 3 ), f(a 3 a 4 ), f(a 4 a 1 ) are ot lies. Hece 1-coaffiity(f) 8 > 6. Case.4. f = (a 1, a )(a 3, a 4 ), where {1, 1} {1, 1} = {a 1, a, a 3, a 4 }. Assume a 1 = (1, 1). If a = (1, 1), let g GL(, F 3 ) be give by g(x, y) = (x, y). The it is easy to see that g f is the traspositio which moves (0, 1) ad (0, 1). If a = ( 1, 1), let h GL(, F 3 ) be give by h(x, y) = ( x, y). The h f is the traspositio which moves (1, 0) ad ( 1, 0). If a = ( 1, 1), the f(l ai ) (1 i 4) ad f(a 1 a 3 ), f(a 1 a 4 ), f(a a 3 ), f(a a 4 ) are ot lies. Hece 1-coaffiity(f) 8 > 6. We ow tur to the proof of the Threshold Cojecture with = 1, > ad 3. Lemma 8.3. Let m ad let f : F F m be a oe-to-oe mappig which is ot semi-affie. Let 1-coaffiity(f) deote the umber of lies L i F such that f(l) is ot a lie i F m. The 1-coaffiity(f) 1 1. Proof. Use iductio o. First assume =. Case 1. There do ot exist two oparallel lies i F which are ot mapped ito lie by f. The 1-coaffiity(f) ( + 1 1) > + 1. Case. There are two oparallel lies L 1 ad L i F such that f(l 1 ), f(l ) are lies i F m. Sice f(l 1 ) ad f(l ) are itersectig lies i F m, their affie spa i F m is a -flat which, without loss of geerality, is assumed to be F {0}. If for every lie L 3 i F such that L i L j (1 i < j 3) are 3 distict poits f(l 3 ) is ot a lie, the 1-coaffiity(f) ( 1) + 1.

22 AFFINITY OF A PERMUTATION 1 So, we assume that there exists a lie L 3 i F such that L i L j (1 i < j 3) are 3 distict poits ad f(l 3 ) is a lie i F m. Sice f(l 1 ) f(l ) F {0}, it follows that f(l 3 ) F {0}. If f(f ) F {0}, by Theorems 8.1 ad 8., 1-coaffiity(f) > + 1. So we assume that f(f ) F {0}. Let a F \(L 1 L L 3 ) such that f(a) / F {0}. If L is a lie through a such that L (L 1 L L 3 ), the f(l) is ot a lie sice two poits o L (belogig to L (L 1 L L 3 )) are mapped to F {0} by f but a is ot mapped to F {0}. There are oly 3 lies L i (i = 1,, 3) i F such that L i (L 1 L L 3 ) < : the lies through L j L ad parallel to L i where {i, j, } = {1,, 3}. If there are two poits a 1, a F \ (L 1 L L 3 ) such that f(a i ) / F {0}, i = 1,, the for ay lie L i F through a 1 or a with L L i, i = 1,, 3, f(l) is ot a lie. Hece 1-coaffiity(f) ( + 1) If there is exactly oe poit a F \ (L 1 L L 3 ) such that f(a) / F {0}, the every lie i F through a is ot a lie i F m. Thus, 1-coaffiity(f) + 1. Now assume 3. Case 1. For ay two oparallel hyperplaes H 1 ad H i F, f is ot semiaffie o at least oe of H 1 ad H. The f is ot semi-affie o at least ( 1 1 1) = ( 1 1) 1 hyperplaes i F. By the iductio hypothesis, at least these hyperplaes are ot mapped ito lies. Sice each lie i F hyperplaes, we have (8.4) 1-coaffiity(f) ( 1 1) = ( 1 1) 1 lies o each of > 1 1. lies i Case. There are two oparallel hyperplaes H 1 ad H i F such that f is semi-affie o both H 1 ad H. Sice f(h 1 ) ad f(h ) are ( 1)-flats i F m whose itersectio is a ( )-flat, their affie spa i F m is a -flat which, without loss of geerality, is assumed to be F {0}. Through a suitable semi-affie trasformatio, we may assume that (8.5) f(x) = (x, 0) for all x H 1. Sice f(x) = (x, 0) for all x H 1 H ad sice dim(h 1 H ) 1, f H must be affie. The it is clear that through a additioal affie trasformatio, we may assume that i additio to (8.5), (8.6) f(x) = (x, 0) for all x H. Thus we have (8.7) f(x) = (x, 0) for all x H 1 H.

23 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS Case.1. For every hyperplae H 3 i F such that H i H j (1 i < j 3) are 3 distict ( )-flats, f is ot semi-affie o H 3. By the iductio hypothesis, at least H 3 is ( 1 1 lies o H 3 are ot mapped ito lies. The umber of such hyperplaes ) ( 1). By the same argumet for (8.4), we have (8.8) 1-coaffiity(f) ( 1 1 ) ( 1) > 1 1. Case.. There exists a hyperplae H 3 i F such that H i H j (1 i < j 3) are 3 distict ( )-flats ad f is semi-affie o H 3. By (8.7), f(x) = (x, 0) for all x (H 3 H 1 ) (H 3 H ). Sice H 3 H 1 ad H 3 H affiely spa H 3, we have f(x) = (x, 0) for all x H 3. Thus f(x) = (x, 0) for all x H 1 H H 3. Sice f is ot semi-affie, f(a) (a, 0) for some a F \ (H 1 H H 3 ). If L is a lie through a such that L (H 1 H H 3 ) ad f(a) / L {0}, the f(l) is ot a lie. (Let b 1, b L (H 1 H H 3 ). The f(b i ) = (b i, 0) L {0}, but f(a) / L {0}.) We claim that umber of lies L i F through a with L (H 1 H H 3 ) 1 is (8.9) I fact, we may assume H 1 = {0} F F F 3, H = F {0} F F 3, H 3 = F F {0} F 3. Let a = (a (1), a (), a (3),... ) where a (1), a (), a (3) F \ {0} ad let L be a lie through a with directio vector v = (v (1), v (), v (3),... ). The L (H 1 H H 3 ) 1 if ad oly if oe of the followig is true: (i) at least two of v (1), v (), v (3) are 0; (ii) v () = 0 ad (v (i), v (j) ) = t(a (i), a (j) ) for some t F \ {0} where {i, j, } = {1,, 3}; (iii) (v (1), v (), v (3) ) = t(a (1), a (), a (3) ) for some t F \ {0}. Formula (8.9) follows from these coditios. Therefore, the umber of lies L i F through a such that L (H 1 H H 3 ) ad f(a) / L {0} is at least (8.10) = 3 ( + 6) 1. 1

24 AFFINITY OF A PERMUTATION 3 First assume that there are at least 4 poits a 1,..., a 4 F \ (H 1 H H 3 ) such that f(a i ) (a i, 0). From the above, 1-coaffiity(f) 4 [ 3 ( + 6) 1 ] ( ) 4 = 4 3 ( + 6) 10 > 1 1. Next, assume that there are exactly s poits a 1,..., a s F \ (H 1 H H 3 ), where s = or 3, such that f(a i ) (a i, 0), 1 i s. The for every lie L passig through exactly oe of a 1,..., a s, f(l) is ot a lie. Hece 1-coaffiity(f) s 1 1 ( ) s > 1 1. Fially, assume that there is exactly oe poit a F \ (H 1 H H 3 ) such that f(a) (a, 0). The the lies i F which are ot mapped ito lies by f are precisely the oes passig through a. Thus, 1-coaffiity(f) = 1 1. Theorem 8.4. Let 3 ad f Per(F ) \ AΓL(, F ). The [ ] 1 1-coaffiity(f) = ( 1 1). 1 1 The euality holds if ad oly if f AΓL(, F ) τ AΓL(, F ), where τ Per(F ) is ay traspositio. Proof. The argumets i this proof are very similar to those i the proof of Lemma 8.3. Case 1. For ay two oparallel hyperplaes H 1 ad H i F, f is ot semiaffie o at least oe of H 1 ad H. By Lemma 8.3 ad (8.4), 1-coaffiity(f) ( 1 1) 1 > ( 1 1). 1 Case. There are two oparallel hyperplaes H 1 ad H i F such that f is semi-affie o both H 1 ad H. By the proof of Lemma 8.3, we may assume that f H1 = id, f H = id. Case.1. For every hyperplae H 3 i F such that H i H j (1 i < j 3) are 3 distict ( )-flats, f is ot semi-affie o H 3. By (8.8), we have ( 1 ) 1-coaffiity(f) 1 ( 1) > ( 1 1). 1 Case.. There exists a hyperplae H 3 i F such that H i H j (1 i < j 3) are 3 distict ( )-flats ad f is semi-affie o H 3. By the same argumet i Case. of the proof of Lemma 8.3, we have f(x) = x for all x H 1 H H 3.

25 4 W. EDWIN CLARK, XIANG-DONG HOU, AND ALEC MIHAILOVS First assume that there are at least 5 elemets a 1,..., a 5 F \ (H 1 H H 3 ) such that f(a i ) a i, 1 i 5. By (8.10), we have (8.11) 1-coaffiity(f) 5 [ 3 ( + 6) 1 ] ( ) 5 = 5 3 ( + 6) 15. Whe 4, we have 5 3 ( + 6) 15 > ( 1 1). 1 Whe = 3, ay lie L i F 3 with L (H 1 H H 3 ) caot pass through more tha oe of a 1,..., a 5 sice L = 3. Thus (8.11) ca be improved to 1-coaffiity(f) 5 [ 3 ( + 6) 1 ] > ( 1 1). 1 Next, assume that there are exactly s elemets a 1,..., a s F \ (H 1 H H 3 ), where s = 3 or 4, such that f(a i ) a i, 1 i s. The for every lie L passig through exactly oe of a 1,..., a s, f(l) is ot a lie. Hece 1-coaffiity(f) s 1 1 ( ) s = s 1 [ (s 1) + s ] > ( 1 1). 1 Fially, assume that there are exactly elemets a 1, a F \ (H 1 H H 3 ) such that f(a i ) a i, i = 1,. The f is a traspositio. Combiig Theorems 8.1, 8. ad 8.4, we have the followig corollary. Corollary 8.5. The Threshold Cojecture holds for 3, = 1 ad > 1. Remar. Theorems 8. ad 8.4 state that for = 3, = 1, or > 3, = 1, 3, 1-coaffiity(f) = [ ] 1 1 oly whe f AΓL(, F ) τ AΓL(, F ) for some traspositio τ Per(F ). It is a ope uestio whether the same holds for > 3, = 1 ad =. Acowledgmet The proof of Theorem 8.1 is based o a method by Professor W-X. Ma. Refereces [1] T. Beth ad C. Dig, O almost perfect oliear permutatios, Eurocrypt 93 (Lofthus, 1993) Lecture Notes i Computer Sciece 765, 65 76, Spriger, [] C. Carlet, P. Charpi, V. Zioview, Codes, bet fuctios ad permutatios suitable for DES-lie cryptosystems, Desigs, Codes ad Cryptography, 15 (1998), [3] H. Dobberti, Almost perfect oliear power fuctios o GF( ): the Niho case, Iform. Comput. 151 (1999), [4] H. Dobberti, Almost perfect oliear power fuctios o GF( ): a ew case for divisible by 5, Fiite Fields ad Applicatios (Augsburg, 1999), , Spriger, Berli, 001. [5] X. Hou, Affiity of permutatios of F, Discrete Appl. Math., to appear. [6] P. Lagevi, O the orphas ad coverig radius of the Reed-Muller codes, Applied Algebra, Algebraic Algorithms ad Error-correctig Codes (New Orleas, LA, 1991), Lecture Notes i Computer Sciece 539, 34 40, Spriger, Berli, [7] K. Nyberg, Differetially uiform mappigs for cryptography, Eurocrypt 93 (Lofthus, 1993) Lecture Notes i Computer Sciece 765, 55 64, Spriger, [8] K. Nyberg, S-boxes ad roud fuctios with cotrollable liearity ad differetial uiformity, Fast Software Ecryptio (Leuve, 1994), Lecture Notes i Computer Sciece 1008, , Spriger, Berli, 1995.

26 AFFINITY OF A PERMUTATION 5 [9] E. Sapper ad R. J. Troyer, Metric Affie Geometry, Academic Press, New Yor ad Lodo Departmet of Mathematics, Uiversity of South Florida, Tampa, FL address: eclar@math.usf.edu Departmet of Mathematics, Uiversity of South Florida, Tampa, FL address: xhou@tarsi.math.usf.edu Departmet of Mathematics, Teessee Techological Uiversity, Cooeville, TN address: alec@mihailovs.com