Twin bent functions and Clifford algebras Paul Leopardi Mathematical Sciences Institute, Australian National University. For presentation at ADTHM 2014, Lethbridge. 8 July 2014
Acknowledgements Richard Brent, Padraig Ó Catháin, Bill Martin, Judy-anne Osborn. National Computational Infrastructure. Australian Mathematical Sciences Institute. Australian National University.
Introduction Restricted amicability/anti-amicability graphs Let m be the graph whose vertices are the n 2 = 4 m canonical basis matrices of the real representation of the Clifford algebra R m,m, with each edge having one of two colours, red and blue: Matrices A j and A k are connected by a red edge if they have disjoint support and are anti-amicable. Matrices A j and A k are connected by a blue edge if they have disjoint support and are amicable. Otherwise there is no edge between A j and A k. We call m the restricted amicability / anti-amicability graph of the Clifford algebra R m,m. (L 2014)
Introduction Results Theorem 1 (L 2014) The graph of anti-amicability of the canonical basis matrices of the neutral Clifford algebra R m,m is strongly regular with parameters (ν, k, λ = µ) = (4 m, 2 2m 1 2 m 1, 2 2m 2 2 m 1 ). Theorem 2 The graph of amicability with disjoint support of the canonical basis matrices of the neutral Clifford algebra R m,m is also strongly regular with the same parameters as those in Theorem 1.
Introduction The graphs from Theorem 1 and Theorem 2 are the red and the blue subgraphs of m. They are isomorphic. 2
Introduction Overview What led to this investigation? Key concepts. Constructions. Proof of Theorem 2. Conclusion and open question.
Introduction Motivation 854 16234 16277 22268 23027 26063 27199 38336 39472 42508 43267 49258 49301 22019 27328 38207 43516 64598 64681 22796 25904 39631 42739 53146 62374 53093 62297 857 16229 16282 22259 23036 26048 27184 38351 39487 42499 43276 49253 49306 22028 27343 38192 43507 64601 64678 22787 25919 39616 42748 53141 62377 53098 62294 869 16217 16294 22223 22847 26108 27379 38156 39427 42688 43312 49241 49318 22976 25859 39676 42559 64613 64666 22064 27148 38387 43471 53161 62314 53078 62357 874 16214 16297 22208 22832 26099 27388 38147 39436 42703 43327 49238 49321 22991 25868 39667 42544 64618 64661 22079 27139 38396 43456 53158 62309 53081 62362 917 922 934 937 1334 14074 14837 24428 24467 25519 27743 37792 40016 41068 41107 50698 51461 27808 37727 20579 44956 62918 64201 25424 40111 20636 44899 51706 64054 50933 62777 1337 14069 14842 24419 24476 25504 27728 37807 40031 41059 41116 50693 51466 27823 37712 20588 44947 62921 64198 25439 40096 20627 44908 51701 64057 50938 62774 1379 13999 14687 24377 24518 25594 27893 37642 39941 41017 41158 50848 51536 27658 37877 20534 45001 62828 64156 25349 40186 20681 44854 50783 64099 51631 62867 1388 13984 14672 24374 24521 25589 27898 37637 39946 41014 41161 50863 51551 27653 37882 20537 44998 62819 64147 25354 40181 20678 44857 50768 64108 51616 62876 1427 13919 14767 1436 13904 14752 1478 13834 14597 1481 13829 14602 1589 13817 15094 21407 23663 28508 28579 36956 37027 41872 44128 50441 51718 23696 41839 24659 40876 63173 63946 21344 44191 24748 40787 50678 63034 51961 63797 1594 13814 15097 21392 23648 28499 28588 36947 37036 41887 44143 50438 51721 23711 41824 24668 40867 63178 63941 21359 44176 24739 40796 50681 63029 51958 63802 1619 13727 14959 21497 23798 28474 28613 36922 37061 41737 44038 50576 51808 23561 41974 24629 40906 63068 63916 21254 44281 24778 40757 50543 63139 51871 63827 1628 13712 14944 21494 23801 28469 28618 36917 37066 41734 44041 50591 51823 23558 41977 24634 40901 63059 63907 21257 44278 24773 40762 50528 63148 51856 63836 1699 13679 15007 1708 13664 14992 1733 13577 14854 1738 13574 14857 2357 2362 2387 2396 2467 2476 2501 2506 2614 2617 2659 2668 2707 2716 2758 2761 3158 12394 12437 3161 12389 12442 3173 12377 12454 3178 12374 12457 3221 3226 3238 3241 4679 11899 11908 18413 18658 29918 31534 34001 35617 46877 47122 53627 53636 18194 31697 33838 47341 60743 60856 18461 29729 35806 47074 56971 58039 56948 57928 4680 11892 11915 18402 18669 29905 31521 34014 35630 46866 47133 53620 53643 18205 31710 33825 47330 60744 60855 18450 29742 35793 47085 56964 58040 56955 57927 4724 11848 11959 18398 18478 29933 31714 33821 35602 47057 47137 53576 53687 18641 29714 35821 46894 60788 60811 18209 31517 34018 47326 57016 57979 56903 57988 4731 11847 11960 18385 18465 29922 31725 33810 35613 47070 47150 53575 53688 18654 29725 35810 46881 60795 60804 18222 31506 34029 47313 57015 57972 56904 57995 4740 4747 4791 4792 5159 10219 10468 20093 20098 29374 32078 33457 36161 45437 45442 55067 55316 32177 33358 16754 48781 58583 60376 29249 36286 16781 48754 55531 60199 55268 58408 5160 10212 10475 20082 20109 29361 32065 33470 36174 45426 45453 55060 55323 32190 33345 16765 48770 58584 60375 29262 36273 16770 48765 55524 60200 55275 58407 5234 10174 10318 20008 20183 29419 32228 33307 36116 45352 45527 55217 55361 32027 33508 16679 48856 58493 60301 29204 36331 16856 48679 55118 60274 55486 58498 5245 10161 10305 20007 20184 29412 32235 33300 36123 45351 45528 55230 55374 32020 33515 16680 48855 58482 60290 29211 36324 16855 48680 55105 60285 55473 58509 5250 10062 10430 5261 10049 10417 5335 10011 10260 5336 10004 10267 5924 9448 11239 17038 19838 32333 32434 33101 33202 45697 48497 54296 56087 19841 45694 28994 36541 59348 59611 17009 48526 29117 36418 54503 59179 56296 59428 5931 9447 11240 17025 19825 32322 32445 33090 33213 45710 48510 54295 56088 19854 45681 29005 36530 59355 59604 17022 48513 29106 36429 54504 59172 56295 59435 5954 9358 11134 17128 19943 32299 32468 33067 33236 45592 48407 54401 56177 19736 45799 28964 36571 59213 59581 16919 48616 29147 36388 54398 59314 56206 59458 5965 9345 11121 17127 19944 32292 32475 33060 33243 45591 48408 54414 56190 19735 45800 28971 36564 59202 59570 16920 48615 29140 36395 54385 59325 56193 59469 6066 9342 11150 6077 9329 11137 6100 9240 11031 6107 9239 11032 6180 6187 6210 6221 6322 6333 6356 6363 6951 6952 7026 7037 7042 7053 7127 7128 7495 8571 8580 7496 8564 8587 7540 8520 8631 7547 8519 8632 7556 7563 7607 7608 Anti-amicability of 4 4 Hadamard matrices: 24 components. (L 2014)
Key concepts A long history and a deep literature Difference sets. Bruck (1955), Hall (1956), Menon (1960, 1962), Mann (1965), Turyn (1965), Baumert (1969), Dembowski (1969), McFarlane (1973), Dillon (1974), Kantor (1975, 1985), Ma (1994),... Bent functions. Dillon (1974), Rothaus (1976), Canteaut et al. (2001), Canteaut and Charpin (2003), Dempwolff (2006), Tokareva (2011),... Strongly regular graphs. Brouwer, Cohen and Neumaier (1989), Ma (1994), Bernasconi and Codenotti (1999), Bernasconi, Codenotti and VanderKam (2001)...
Key concepts Difference sets The k-element set D is a (v, k, λ, n) difference set in an abelian group G of order v if for every non-zero element g in G, the equation g = d i d j has exactly λ solutions (d i, d j ) with d i, d j in D. The parameter n := k λ. (Dillon 1974).
Key concepts Hadamard difference sets A (v, k, λ, n) difference set with v = 4n is called a Hadamard difference set. Lemma 3 (Menon 1962) A Hadamard difference set has parameters of the form (v, k, λ, n) = (4N 2, 2N 2 N, N 2 N, N 2 ) or (4N 2, 2N 2 + N, N 2 + N, N 2 ). (Menon 1962, Dillon 1974).
Key concepts Hadamard transforms H m, the Sylvester Hadamard matrix of order 2 m, is defined by [ ] 1 1 H 1 := ; H 1 m := H m 1 H 1, for m > 1. For a boolean function f : Z m 2 Z 2, define the vector [f] by [f] = [( 1) f(0), ( 1) f(1),..., ( 1) f(2m 1) ] T, where f(i) uses the binary expansion of i. The Hadamard transform of f is the vector H m [f]. (Dillon 1974)
Key concepts Bent functions The Boolean function f : Z m 2 Z 2 is bent if its Hadamard transform has constant magnitude: H m [f] = C[1,..., 1] T for some constant C. Each bent function f on Z m 2 has a dual function f given by (H m [f]) i =: 2 m/2 ( 1) f(i). (Dillon 1974, Tokareva 2011)
Key concepts Bent functions and Hadamard difference sets Lemma 4 (Dillon 1974, Theorem 6.2.2) The Boolean function f : Z m 2 Z 2 is bent if and only if the set f 1 (1) is a Hadamard difference set. Lemma 5 (Dillon 1974, Remark 6.2.4) Bent functions exist on Z m 2 only when m is even. (Dillon 1974)
Key concepts Strongly regular graphs A simple graph Γ of order v is strongly regular with parameters (v, k, λ, µ) if each vertex has degree k, each adjacent pair of vertices has λ common neighbours, and each nonadjacent pair of vertices has µ common neighbours. (Brouwer, Cohen and Neumaier 1989)
Key concepts Bent functions and strongly regular graphs The Cayley graph of a binary function f : Z m 2 Z 2 is the undirected graph with adjacency matrix F given by F i,j = f(g i + g j ), for some ordering (g 1, g 2,...) of Z m 2. Lemma 6 (Bernasconi and Codenotti 1999, Lemma 12) The Cayley graph of a bent function on Z2 m is a strongly regular graph with λ = µ. Lemma 7 (Bernasconi, Codenotti and VanderKam 2001, Theorem 3) Bent functions are the only binary functions on Z m 2 whose Cayley graph is a strongly regular graph with λ = µ.
Constructions The groups G 1,1 and Z 2 2 The 2 2 orthogonal matrices [ ] [.. 1 e 1 :=, e 1. 2 := 1. ] generate the group G 1,1 of order 8, an extension of Z 2 by Z 2 2, with Z 2 {I, I}, and cosets 0 00 {±I}, 1 01 {±e 1 }, 2 10 {±e 2 }, 3 11 {±e 1 e 2 }. (L 2005)
Constructions The groups G m,m and Z 2m 2 For m > 1, the group G m,m of order 2 2m+1 consists of matrices of the form g 1 g m 1 with g 1 in G 1,1 and g m 1 in G m 1,m 1. This group is an extension of Z 2 {±I} by Z 2m 2 : 0 00... 00 {±I}, 1 00... 01 {±I (m 1) (2) e 1 }, 2 00... 10 {±I (m 1) (2) e 2 },... 2 2m 1 11... 11 {±(e 1 e 2 ) m }. (L 2005)
Constructions Canonical basis matrices of R m,m A canonical ordered basis of the matrix representation of the Clifford algebra R m,m is given by an ordered transversal of Z 2 {±I} in Z 2m 2. For example, (I, e 1, e 2, e 1 e 2 ) is one such ordered basis. We define a function γ m : Z 2 2m G m,m to choose the corresponding canonical basis matrix for R m,m for some transversal, and use binary expansion to get a function on Z 2m 2. For example, γ 1 (1) = γ 1 (01) := e 1. (L 2005)
Constructions The sign function s 1 on Z 4 and Z 2 2 We use the function γ 1 to define the sign function s 1 : { 1 γ 1 (i) 2 = I s 1 (i) := 0 γ 1 (i) 2 = I, for all i in Z 2 2. Using our vector notation, we see that [s 1 ] = [1, 1, 1, 1] T. (L 2014)
Constructions The sign function s m on Z 2 2m and Z 2m 2 We use the function γ m to define the sign function s m : { 1 γ m (i) 2 = I s m (i) := 0 γ m (i) 2 = I, for all i in Z 2m 2. (L 2014)
Constructions Properties of the sign function s m If we define : Z 2 Z 2m 2 2 Z 2m 2 as concatenation, e.g.. 01 1111 := 011111, it is easy to verify that s m (i 1 i m 1 ) = s 1 (i 1 ) + s m 1 (i m 1 ) for all i 1 in Z 2 and i m 1 in Z 2m 2 2, and therefore [s m ] = [s 1 ] [s m 1 ]. Also, since each γ m (i) is orthogonal, s m (i) = 1 if and only if γ m (i) is skew. (L 2014)
Constructions The symmetry function t m on Z 2 2m and Z 2m 2 For i in Z 2 2 : t 1 (i) := { 1 if γ 1 (i) = e 2, 0 otherwise. For i in Z 2m 2 2 : t m (00 i) := t m 1 (i), t m (01 i) := s m 1 (i), t m (10 i) := s m 1 (i) + 1, t m (11 i) := t m 1 (i). where denotes concatenation.
Constructions Properties of the symmetry function t m It is easy to verify that t m (i) = 1 if and only if γ m (i) is symmetric but not diagonal. This can be checked directly for t 1. For m > 1 it results from properties of the Kronecker product: (A B) T = A T B T. A B is diagonal if and only if both A and B are diagonal.
Theorem 2 Proof of Theorem 2: t m is bent Lemma 8 (Tokareva, 2011 Theorem 1; Canteaut and Charpin, 2003 Theorem 2; Canteaut et al., 2001, Theorem V.4) If a binary function f on Z 2m 2 can be decomposed into four functions f 0, f 1, f 2, f 3 on Z 2m 2 2 as f(00 i) =: f 0 (i), f(01 i) =: f 1 (i), f(10 i) =: f 2 (i), f(11 i) =: f 3 (i), where all four functions are bent, with dual functions such that f 0 + f 1 + f 2 + f 3 = 1, then f is bent.
Theorem 2 Proof of Theorem 2: t m is bent In Lemma 8, set f 0 = f 3 := t m 1, f 1 = s m 1, f 2 = s m 1 + 1. Clearly, f 0 = f 3. Also, f 2 = f 1 + 1, since H m 1 [f 2 ] = H m 1 [f 1 ]. Therefore f 0 + f 1 + f 2 + f 3 = 1. Thus, these four functions satisfy the premise of Lemma 8, as long as both s m 1 and t m 1 are bent. I have already shown that s m is bent for all m. It is easy to show that t 1 is bent, directly from its definition. Therefore t m is bent.
Conclusion The graphs from Theorem 1 and Theorem 2 are the red and the blue subgraphs of m. They are isomorphic. 2
Conclusion Open question For which m is there an isomorphism of m that swaps all red and blue edges? Isomorphisms have been constructed for m = 1, 2, 3 so far. (L 2014)
Conclusion References [1] Anne Canteaut, Claude Carlet, Pascale Charpin, and Caroline Fontaine. On cryptographic properties of the cosets of R(1, m). IEEE Transactions on Information Theory, 47.4 (2001): 1494-1513. [2] Anne Canteaut and Pascale Charpin. Decomposing bent functions. IEEE Transactions on Information Theory, 49.8 (2003): 2004-2019. [3] Natalia Tokareva. On the number of bent functions from iterative constructions: lower bounds and hypotheses. Adv. in Mathematics of Communications (AMC) 5.4 (2011): 609-621.