Borovik, A.

Yalcinkaya, S.

English

We introduce a new approach in black box group theorywhich deals with black box group problems in the category of blackboxes and their morphisms. This enables us to enrich black box groupsby actions of outer automorphisms such as Frobenius maps or graphautomorphisms of simple groups of Lie type. As an application of thisnew technique, we present a number of new results, including a solutionof an old problem about constructing unipotent elements in groups ofLie type of odd characteristic

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The University of Manchester ResearchNew approaches in black box group theoryDOI:10.1007/978-3-662-44199-2_10Link to publication record in Manchester Research ExplorerCitation for published version (APA):Borovik, A., & Yalcinkaya, S. (2014). New approaches in black box group theory. In Mathematical Software { ICMS2014 { 4th International Congress, Seoul, South Korea, August 5{9, 2014. (Vol. 8592, pp. 53-58). (Lecture Notes inComputer Science). Springer Nature. https://doi.org/10.1007/978-3-662-44199-2_10Published in:Mathematical Software { ICMS 2014 { 4th International Congress, Seoul, South Korea, August 5{9, 2014.Citing this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:08. Jun. 2022New approaches in black box group theoryAlexandre Borovik1 and Şükrü Yalçınkaya21 University of Manchester, United Kingdomalexandre@borovik.net,2 İstanbul University, Turkeysukru.yalcinkaya@istanbul.edu.tr,Abstract. We introduce a new approach in black box group theorywhich deals with black box group problems in the category of blackboxes and their morphisms. This enables us to enrich black box groupsby actions of outer automorphisms such as Frobenius maps or graphautomorphisms of simple groups of Lie type. As an application of thisnew technique, we present a number of new results, including a solutionof an old problem about constructing unipotent elements in groups ofLie type of odd characteristic.Keywords: Black box groups, projective geometry, classical groups,unipotent elements1 Black box groupsA black box group X is a black box (or an oracle, or an algorithm) operating on0 − 1 strings of uniform length which encrypt elements of some finite group G.The procedures performed by a black box are specified as follows.BB1 X produces strings of fixed length l(X) encrypting random (almost) uni-formly distributed elements from G; this is done in probabilistic time poly-nomial in l(X).BB2 X computes, in probabilistic time polynomial in l(X), a string encryptingthe product of two group elements given by strings or a string encryptingthe inverse of an element given by a string.BB3 X decides, in probabilistic time polynomial in l(X), whether two strings en-crypt the same element in G—therefore identification of strings is a canonicalprojectionX π−→ G.In this situation we say that X encrypts the group G.A natural question here is to determine the isomorphism type of a black boxgroup X or, if it is known, find an isomorphism between X and its natural copy.To that end, we need additional assumptions about X, which we are keeping toa minimum by adopting an additional axiom.2 Borovik-YalçınkayaBB4 We are given a global exponent of X, that is, a natural number E such thatπ(x)E = 1 for all strings x ∈ X while computation of xE is computationallyfeasible (say, logE is polynomially bounded in terms of log |G|).Note that axioms BB1–BB4 hold, for example, in matrix groups over finitefields where we can take for E the exponent of the ambient GLn(q).In this paper, we assume BB1–BB4 and are concerned with structure re-covery of black box groups X encrypting an explicitly given group G of Lie typeover Fq, that is, with constructing, in probabilistic polynomial time in log |G|,– a black box field K encrypting Fq, and– a morphism Ψ : G(K)→ X.Unlike the constructive recognition algorithms of black box groups [7–11], weshall note here that we are not using a discrete logarithm oracle or an SL2(q)-oracle, see [4] for a detailed discussion of the hierarchy of black box group prob-lems.2 Morphisms and automorphismsLet X and Y be two black box groups encrypting the groups G and H, respec-tively. We say that a map ζ, which assigns strings from X to Y, is a morphismof black box groups if– the map ζ is computable in probabilistic time polynomial in l(X) and l(Y);and– there is an abstract homomorphism φ : G → H such that the followingdiagram is commutative:Xζ−→ YπX ↓ ↓ πYGφ−→ Hwhere πX and πY are the canonical projections of X and Y onto G and H,respectively.In this case we say that a morphism ζ encrypts the homomorphism φ. Observethat replacing a given generating set of a black box group X by a more suitableone means that we construct a new black box Y and work with the correspondingmorphism Y → X.The first result based on this new philosophy is “amalgamation of local au-tomorphisms”:Theorem 1. [4, Theorem 5.1] Let X be a black box group encrypting a groupG. Assume that G contains subgroups G1, . . . , Gl invariant under an automor-phism α ∈ AutG and that these subgroups are encrypted in X as black boxes Xi,i = 1, . . . , l, supplied with morphismsφi : Xi −→ XiNew approaches in black box group theory 3which encrypt restrictions α |Gi of α on Gi. Assume also that 〈Gi, i = 1, . . . , l〉 =G. Then we can construct, in time polynomial in l(X), a morphism φ : X −→ Xwhich encrypts α.This theorem can be applied, for example, to groups of Lie type and systemsof root SL2-subgroups corresponding to the nodes in the associated Dynkin di-agrams. That way, we construct the following automorphisms of groups of Lietype.(1) Frobenius maps on groups of Lie type of odd characteristic [4];(2) Graph automorphisms of SLn(q), Dn(q) (including the triality of D4(q)),F4(q), and E6(q) (for odd q) [6].Interestingly, construction of graph automorphisms in black box groups of Lietype of odd characteristic does not use information about the underlying field.Further manipulation with morphisms between root SL2(q)-subgroups yields, forexample, the following (field-independent) black box embeddings constructed intime polynomial in log q and n:– SUn(q) ↪→ SLn(q2);– G2(q) ↪→ SO7(q) ↪→ SO+8 (q) ↪→ SL8(q);– 3D4(q) ↪→ SO+8 (q) ↪→ SL8(q);These embeddings are implemented in GAP for various fields but notably weconstruct the embedding SU3(p) ↪→ SL3(p2) for the 60 digit primep = 622288097498926496141095869268883999563096063592498055290461.Notice that the size of SL3(p2) is bigger than 10960.Another very important corollary of Theorem 1 is that if the action of aninvolutive automorphism a of G is known on some a-invariant subgroups of Ggenerating G, then we can transfer the action of a on these subgroups to wholegroup G. We call this process a reification of a. More precisely, we haveTheorem 2. [4, Theorem 7.1] Let X be a black box group encrypting a finitegroup G. Assume that G admits an involutive automorphism a ∈ AutG andcontains a-invariant subgroups H1, . . . ,Hn where a either inverts or centralizeseach Hi.Assume also that we are given black boxes Y1, . . . ,Yn encrypting subgroupsH1, . . . ,Hn. Then we can construct, in polynomial time,– a black box for the structure {Y, α }, where Y encrypts H = 〈H1, . . . ,Hn〉and α encrypts the restriction of a |H of a to H;– a black box subgroup Z encrypting Ω1(Z(CH(a))), the subgroup generated byinvolutions from Z(CH(a));– if, in addition, the automorphism a ∈ G and H = G then α is induced byone of the involutions in Z.4 Borovik-YalçınkayaAn immediate application of Theorem 2 is that we can append a diagonalautomorphism d of PSL2(q) to a black box group X encrypting PSL2(q) to obtaina black box group Y = X o 〈δ〉 encrypting PGL2(q), where δ encrypts d, see[5] for details. This construction plays a crucial role in the proof of Theorem 3below.In addition, if a is an inner involutive automorphism in a group G of small2-rank, after reification it can be identified with a string in X.It turns out that construction of an involution in black box groups encryptingPGL2(2k) by Kantor and Kassabov [12] is a special case of Theorem 2, see [4] forfurther discussion. Moreover, the construction of a black box projective plane isbased on reification of involutions.3 PSL2(q): structure recovery and unipotent elementsThis is our principal result.Theorem 3. [5] Let Y be a black box group encrypting PSL2(Fq) for q = pk ofknown odd characteristic p. Then we construct, in probabilistic time polynomialin log q,– a black box group X encrypting PGL2(Fq) and an effective embeddingY ↪→ X;– a black box field K of order q, and– polynomial in log q time isomorphismsY↓PGL2(Fq)  PGL2(K)  X −→ SO3(K)where Fq is the standard explicitly given field of order q.Construction of unipotent elements in X is an automatic corollary, but canbe actually done at early stages of the proof of this theorem.Our approach to the proof is recovery, within X, of geometric structuresarising from the adjoint representation of the group PGL2(q) on its Lie algebrasl2 seen as an inner product space with respect to its Killing form—this explainsappearance of the morphismX −→ SO3(K)in the statement of Theorem 3.Our proof in [5] starts by exploiting the fact that the set of involutions inX is the set I = P r Q of regular points in projective plane P over sl2 with aquadric Q (coming from the Killing form), and the points in Q are the BorelNew approaches in black box group theory 5subgroups in X. There are also two types of lines in P: regular and parabolic.The regular lines are the polar images of regular pointsπ(t) = {x ∈ I | [t, x] = 1 and t 6= x}. (1)The parabolic lines correspond to Borel subgroups B in X and consist of invo-lutions inverting a maximal unipotent subgroup U of B, together with U itselfseen as a point in P.It turns out that the set I is a finite symmetric space with the conjugationoperation ◦, for s, t ∈ I, s ◦ t = ts, forming a finite field analogue of the realhyperbolic (Lobachevsky) plane viewed as a symmetric space. The black boxfield K is built by applying the Hilbert’s coordinatization on this Lobachevskyplane I. The analysis of the action of X on I produces the morphismX −→ SO3(K).Constructing the black box field by coordinatizing the Lobachevsky plane enablesus to construct arbitrary elements in P with specified coordinates. In particu-lar, we can construct unipotent elements in X encrypting PGL2(q), which areprecisely the points on the quadric Q.4 Toolbox in Lobachevsky planeIt is shown in [5] that the following procedures are performed in time polynomialin log q inside the Lobachevsky plane constructed in X. So we construct a blackbox that(a) produces uniformly distributed points from I;(b) checks the equality of points;(c) checks collinearity of triples of points;(d) for any two points s, t ∈ I, computes the half turn of t around s, which wedenote by s ◦ t;(e) for any involution t ∈ I, produces uniformly distributed regular points inthe polar image of t:$(t) = { s ∈ I | s ◦ t = t and s 6= t };(f) for any two distinct points s, t ∈ I, produces uniformly distributed regularpoints on the line s ∨ t through s and t;(g) for a regular line through two distinct points s and t, constructs its pole,which is the involution commuting with both s and t;(h) for any two distinct lines k and l, finds its intersection point k ∧ l or, if thelines k and l do not intersect in I and therefore their intersection point zbelongs to Q, computes the unipotent element.(i) for a point s ∈ I, computes the polar projectionξs : I r { s } −→ π(s)x 7→ π(x) ∧ (s ∨ x);6 Borovik-Yalçınkaya(j) for any two points s, t ∈ I conjugate under the action of X, finds r ∈ I suchthat r ◦ s = t;(k) represents any element of X as a product of two involutions from X.As an example, we show how we draw the line passing through two distinctpoints s, t ∈ I as in item (f). For an involution x ∈ X denote by Tx the maximaltorus in CX(x).If z = st is a unipotent element then 〈zTs〉s is a parabolic line. Otherwiseobserve that it suffices to construct the involution j := j(s, t) which commuteswith both s and t. Indeed, the line passing through s and t is the coset Tjwwhere w is an involution inverting Tj , see Equation (1). If z = st has evenorder, then j is the involution in 〈z〉 which can be constructed by using squareand multiply method. However, if z has odd order, then we can not construct jimmediately but we know its action on X:– j centralizes 〈z〉,– j inverts every element in the torus Ts.Hence, j(s, t) can be reified from these two conditions by using Theorem 2.References1. A. V. Borovik and Ş. Yalçınkaya, Construction of Curtis-Phan-Tits system for blackbox classical groups, Available at arXiv:1008.2823v1.2. A. V. Borovik and Ş. Yalçınkaya, Steinberg presentations of black box classical groupsin small characteristics, Available at arXiv:1302.3059v1.3. A. V. Borovik and Ş. Yalçınkaya, Fifty shades of black, Available at arXiv:1308.2487.4. A. V. Borovik and Ş. Yalçınkaya, Black box, white arrow, Available atarXiv:1404.7700.5. A. V. Borovik and Ş. Yalçınkaya, Revelations and reifications: Adjoint representa-tions of black box groups PSL2(q), in preparation.6. A. V. Borovik and Ş. Yalçınkaya, Subgroup structure and automorphisms of blackbox classical groups, in preparation.7. P. A. Brooksbank, A constructive recognition algorithm for the matrix group Ω(d, q),Groups and Computation III (W. M. Kantor and Á. Seress, eds.), Ohio State Univ.Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 79–93.8. P. A. Brooksbank, Fast constructive recognition of black-box unitary groups, LMSJ. Comput. Math. 6 (2003), 162–197.9. P. A. Brooksbank, Fast constructive recognition of black box symplectic groups, J.Algebra 320 (2008), no. 2, 885–909.10. P. A. Brooksbank and W. M. Kantor, On constructive recognition of a black boxPSL(d, q), Groups and Computation III (W. M. Kantor and Á. Seress, eds.), OhioState Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 95–111.11. P. A. Brooksbank and W. M. Kantor, Fast constructive recognition of black boxorthogonal groups, J. Algebra 300 (2006), no. 1, 256–288.12. W. M. Kantor and M. Kassabov, Black box groups PGL(2, 2e), arXiv:1309.3715v2.13. Ş. Yalçınkaya, Black box groups, Turkish J. Math. 31 (2007), no. suppl., 171–210.

New approaches in black box group theory

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New approaches in black box group theory

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