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RESEARCH PROGRAM

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Scientific and education field classification
International Patent Classification
  • PHYSICS
    • EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
      • CODING OR CIPHERING APPARATUS FOR CRYPTOGRAPHIC OR OTHER PURPOSES INVOLVING THE NEED FOR SECRECY (secret transmission H04K; arrangements for secret telegraphic communication H04L9/00)
      • EDUCATIONAL OR DEMONSTRATION APPLIANCES; APPLIANCES FOR TEACHING, OR COMMUNICATING WITH, THE BLIND, DEAF OR MUTE; MODELS; PLANETARIA; GLOBES; MAPS; DIAGRAMS (devices for psychotechnics or for testing reaction times A61B5/16; games, sports, amusements A63; projectors, projector screens G03B)
Geographical classification
Bibliografia
[A] Avitabile M., Some loop algebras of Hamiltonian Lie algebras, PhD thesis, 1999

[AJ] Avitabile M., Jurman G. Diamonds in thin Lie algebras. Boll. UMI Sez. B Artic. Ric. Mat. (8) 4 (2001), n. 3, 597-608

[AMAZ] M. Avitabile, S. Mattarei, Thin loop algebras of Albert-Zassenhaus Lie algebras, 27 pages, submitted

[AM05] M. Avitabile, S. Mattarei, Some thin Lie algebras with diamonds of finite and infinite type, J. Algebra 293 (2005), n. 1, 34-64

[AMd] M. Avitabile, S. Mattarei, Diamonds of finite type in thin Lie algebras, 19 pages, submitted
arXiv:math.RA/0511256.

[BKK] Benkart G., Kostrikin A., Kuznetsov M., Finite-dimensional simple Lie algebras with a nonsingular derivation. J. Algebra 171 (1995), n. 3, 894-916

[C97] Caranti A., Presenting the graded Lie algebra associated to the Nottingham group. J. Algebra 198 (1997), n. 1, 266-289

[C98] Caranti A., Thin groups of prime-power order and thin Lie algebras: an addendum. Quart. J. Math. Oxford Ser. (2) 49 (1998), n. 196, 445-450

[C99] A. Caranti, Loop algebras of Zassenhaus algebras in characteristic three, Israel J. Math. 110 (1999), 61-73

[CC] Camina A., Camina R., Pro-p-groups of finite width, Comm. Algebra 29 (2001), n. 4, 1583-1593

[CDV] Caranti A., Dalla Volta F., The round functions of cryptosystem PGM generate the symmetric group. (English. English summary) Des. Codes Cryptogr. 38 (2006), n. 1, 147-155

[CDVS] Caranti A., Dalla Volta F., Sala M., Abelian regular subgroups of the affine group and radical rings, Publ. Math. Debrecen, to appear

[CHZ] Charpin P., Helleseth T., Zinoviev V., Propagation characteristics of x->1/x and Kloosterman sums, Finite Fields Appl., to appear

[CM99] Caranti A., Mattarei S., Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class. J. Austral. Math. Soc. Ser. A 67 (1999), n. 2, 157-184

[CM04] A. Caranti, S. Mattarei, Nottingham Lie algebras with diamonds of finite type, Internat. J. Algebra Comp. 14 (2004), 35-67

[CM05] A. Caranti, S. Mattarei, Gradings of non-graded Hamiltonian Lie algebras, J. Austral. Math. Soc., Series A 79 (2005), n. 3, 399-440

[CMN] Caranti A., Mattarei S., Newman M., Graded Lie algebras of maximal class. Trans. Amer. Math. Soc. 349 (1997), n. 10, 4021-4051

[CMNS] Caranti A., Mattarei S., Newman M., Scoppola C. M., Thin groups of prime-power order and thin Lie algebras. Quart. J. Math. Oxford Ser. (2) 47 (1996), n. 187 , 279-296

[CN] Caranti A., Newman M., Graded Lie algebras of maximal class II. J. Algebra 229 (2000), n. 2, 750-784

[DP] Di Pietro C., Wreath products and modular Lie algebras, PhD Thesis (Univ. dell'Aquila), 2005

[DR02] Daemen J., Rijmen V., The design of Rijndael. AES-the advanced encryption standard. Information Security and Cryptography. Springer-Verlag, Berlin, 2002

[DR06] Daemen J., Rijmen V., Two-Round AES Differentials, IACR Preprint, 2006

[E04] Ershov M., New just-infinite pro-p groups of finite width and subgroups of the Nottingham group, J. Algebra 275 (2004) 419–449

[E05] Ershov M., The Nottingham group is finitely presented. J. London Math. Soc. (2) 71 (2005), n. 2, 362-378

[EM] Egizii Di Marco M., Norm and conjugacy classes of normalizers in finite p-groups, PhD Thesis (Univ. dell'Aquila), 2005

[GGSZ] Goldstein D., Guralnick R., Small L., Zelmanov E., Inversion invariant additive subgroups of division rings, Pacific J. Math., to appear

[GLSTbams] Gavioli N., Legarreta L., Sica C., Tota M., On the number of conjugacy classes of normalizers in a finite p-group, Bull. Austral. Math. Soc., to appear

[GM] Gavioli N., Monti V., Ideally constrained Lie algebras. J. Algebra 253 (2002), n. 1, 31-49

[GMS] Gavioli N., Monti V., Scoppola C. M., Just infinite periodic Lie algebras, in Proc. Gainesville Conf. On Finite Groups, March 6-12, 2003, Chat Y. Ho, P. Tiep, A. Turull, ed., De Gruyter (2004), 73-86

[GMSjgt] Gavioli N., Monti V., Scoppola C. M., Soluble normally constrained pro-p-groups, J. Group theory, to appear

[HNV] G. Havas, M. Newman, M. Vaughan-Lee, A nilpotent quotient algorithm for graded Lie rings, J. Symbolic Comput., 9, 653-664, 1990

[Hua] Hua Loo-Keng, Some properties of a sfield. Proc. Nat. Acad. Sci. U. S. A. 35, (1949). 533-537

[J] Jacobson N., A note on automorphisms and derivations of Lie algebras. Proc. Amer. Math. Soc. 6, (1955). 281-283

[KLP] Klaas G., Leedham-Green C., Plesken W., Linear pro-p-groups of finite width, LNM 1674 (1997) Springer-Verlag, Berlin

[KRS] Kaliski B., Rivest R., Sherman A., Is the data encryption standard a group? (Results of cycling experiments on DES). J. Cryptology 1 (1988), n. 1, 3-36

[K] Khukhro E., Nilpotent groups and their automorphisms, Expositions in Math., 8 de Gruyter, Berlin, 1993

[L] Leedham-Green C., The structure of finite p-groups, J. London Math. Soc (2) 50, 49-67

[LLL] A. Lenstra, H. Lenstra, L. Lovasz, Factoring Polynomials with Rational Coefficients. Math. Ann. 261, 515-534, 1982

[LM] Leedham-Green C., McKay S., The structure of Groups of Prime Power Order, London Math. Soc. Monographs, New Series, 27 (2000) Oxford Science Pub.

[LN] Leedham-Green C., Newman M., Space groups and groups of prime-power order. I. Arch. Math. (Basel) 35 (1980), n. 3, 193-202

[LR] La Haye R., Rhemtulla A., Groups with a bounded number of conjugacy classes of non-normal subgroups. J. Algebra 214 (1999), n. 1, 41-63

[M99] Mattarei S., Some thin pro-p-groups. J. Algebra 220 (1999), n. 1, 56-72

[M02] S. Mattarei, The orders of nonsingular derivations of modular Lie algebras, Israel J. Math. 132 (2002), 265-275.

[M06] Mattarei, S. Inverse-closed additive subgroups of fields, arXiv math.RA/0511538, Israel J. Math, to appear

[Mch] S. Mattarei, Constituents of graded Lie algebras of maximal class and chain lengths of thin Lie algebras, in preparation.

[Md] S. Mattarei, Deforming thin Lie algebras, in preparation.

[Mff] Mattarei S., On a bound of Garcia and Voloch for the number of points of a Fermat curve over a prime field, 4 pages, Finite Fields Appl., to appear

[Mfer] S. Mattarei, Fermat curves over finite fields and characters of nonabelian groups, in preparation.

[Mo] S. Mattarei, The orders of nonsingular derivations of modular Lie algebras of odd characteristic, in preparation.

[Mev] S. Mattarei, The orders of nonsingular derivations of modular Lie algebras of characteristic two, Israel J. Math., to appear arXiv:math.RA/0602668.

[Mp] Monti V., Periodic just infinite pro-p-groups, preprint

[Mphd] T. Meskanen. On the NTRU Cryptosystem. PhD thesis, Jun 2005, TorkU center for Computer Science (TUCS)

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Keywords
PRO-P-GROUPS, GRADED LIE ALGEBRAS, PERIODICITY, DERIVATIONS, CRYPTOGRAPHY

Groups, Lie Algebras, Criptography

Università degli Studi de L'Aquila
Abstract
In our previous work we have started the classification project for just infinite pro-p-groups, and we have shown that a similar project of classification makes sense also for just infinite dimensional modular graded Lie algebras. The classification of a crucial class of graded Lie algebras, i.e. the thin Lie algebras, has been carried on by one of our units with continuous progress.
As a consequence of our growing interaction with other international leading groups of researchers, and of the natural evolution of our research interests, we have recently opened entirely new perspectives for these lines of research.

At the same time our group has acquired a new researcher (de Graaf). This acquisition reinforces the line of research that refers to the study of Lie algebras, and stresses the computational aspect. Note that computational experiments have always had an important role in our work. Indeed, we use the evidence of computer constructions of algebraic objects as a guideline for our theoretical research work. At the same time, in some occasions we have produced original contributions to existing packages for algebraic computation. We also intend to emphasize this second aspect, with a particular attention to parallel computing (collaboration with the University of St. Andrews for production of GAP libraries).

Both units have started studying problems in cryptography. Looking at the more recent literature on the subject, it appears >>>

Principal Investigator
Carlo Maria Scoppola Università degli Studi de L'AQUILA
Research Objectives
The aim of this research program is twofold.
On the one hand, we want to boost our activity on the research topics we studied in the last few years, i.e. just infinite pro-p-groups and graded Lie algebras, in the light of the new ideas and perspectives recently opened as an effect of the expansion of our network of international collaborations.
On the other hand we have extended our research interests, obtaining results on new topics, like algebraic questions related to cryptography, and computation in Lie algebras.
We want to reinforce our commitment on these new research lines.

Timescale
24 months
National and international background
(This item is necessarily the join of the two corresponding items on forms B. For the convenience of the reader, we mark each part of this item with the number of the Unit).

The construction of a graded Lie ring associated to a filtration of a pro-p-group (the direct sum of the quotients of the filtration endowed with the Lie structure that is inherited from the commutation structure of the group) dates back to classical work of Magnus, Zassenhaus and Lazard.

In our past work we have applied this construction frequently. Our main interest, when we started some years ago, was centered on the class of thin (pro-p)-groups, i.e. (topologically) 2-generated (pro-)p-groups in which every (open) normal subgroup lies, in the lattice of (open) normal subgroups, between two consecutive terms of the lower central series. In [CMNS], [CMN], [M99], [GMSjgt] thin (pro-)p-groups were studied.

Thinness can also be defined as a property of the associated Lie ring, which indeed turns out to be a Lie algebra over a finite field, graded over the positive integers and generated by its component of degree 1.

JUST INFINITE PRO-p-GROUPS AND GRADED LIE ALGEBRAS (I)

Some time after the completion of the proof of the well known coclass conjectures (see [S94a] and [L]), a more refined classification project for pro-p-groups was proposed ([KLP], and then, with more details, [LM]). This project focuses on just infinite (JI for >>>