THE FIRST NTU - VIASM WORKSHOP ON DISCRETE MATHEMATICS PROGRAM AND ABSTRACTS

Transcription

1 THE FIRST NTU - VIASM WORKSHOP ON DISCRETE MATHEMATICS PROGRAM AND ABSTRACTS HANOI 2014

2 The first NTU-VIASM Workshop on Discrete Mathematics The first joint workshop on Discrete Mathematics is organized by the Nanyang Technological University (NTU) and the Vietnam Institute for Advanced Study in Mathematics (VIASM). The workshop is going to bring together and provide a platform for the researchers of NTU and Vietnam in all aspects of Discrete Mathematics and related areas. The purpose is to promote the mutual relation between NTU and VIASM and to eventually create collaborations between Singaporean and Vietnamese researchers. The program is composed of three keynote of experts in Cryptography and Graph theory. Other contributed talks from Singaporean and Vietnamese researchers will focus on new trends and developments in the domain of Discrete Mathematics, such as: coding, enumerative combinatorics, graph theory, algebraic combinatorics, algorithms, discrete optimization, modeling and simulation. Place: VIASM, 7th Floor, Ta Quang Buu Library, 1 Dai Co Viet, Hanoi, Vietnam Time: December 27-30, 2014 Organizing and Scientific Committee: Nguyen Huu Du, Phan Thi Ha Duong, San Ling, Huaxiong Wang, Chaoping Xing. Local Organizing Committees: Tran Thi Bich Diep, Ngo Thien Nga, Nguyen Ngoc Tuan. Sponsoring Organizations: Vietnam Institute for Advanced Study in Mathematics Nanyang Technological University Vietnam Academy of Science and Technology. 2

3 CONTENT Introduction 2 Part 1: Program 4 Part 2: Abstracts 9 A. Keynote Speakers 9 B. Invited Speakers 12 Part 3: List of Participants 27 3

4 Program The First NTU-VIASM Workshop on Discrete Mathematics December 27 Morning Session 07:30 08:30 Registration 08:30 09:00 Opening Ceremony Chair: Ling San 09:00 09:50 Keynote Speaker: Phong Quang Nguyen The Algorithmic Revolution in Geometry of Numbers 09:50 10:15 Ngo Dac Tan On Vertex Disjoint Cycles of Different Lengths in Digraphs 10:15 10:45 Coffee break Chair: Ngo Viet Trung 10:45 11:10 Lee Troy The Cover Number of A Matrix and Its Algorithmic Applications 11:10 11:35 Tran Thi Thu Huong The Duality of Critical and Super-stable Configurations of Chip Firing Games on Directed Graphs 11:35 12:00 Chen Ning Optimal Competitive Auctions 12:00 14:00 Lunch 4

5 December 27 Afternoon Session Chair: Hoeteck Wee 14:00 14:25 Tran Vinh Linh On The Recovering Threshold of The Stochastic Block Model 14:25 14:50 Li Mengling Two-sided Many-to-many Matching with Ties 14:50 15:15 Dao Thi Thu Ha Time-Parallel Simulation for Stochastic Automata Networks and Stochastic Process Algebra 15:15 15:45 Coffee break Chair: Nguyen Huu Du 15:45 16:10 Do Duy Hieu The Structure of Some Special Classes of Graphs and Application 16:10 16:35 Tran Dan Thu Identities for Well-known Inequalities in Combinatorics 16:35 17:00 Lim Kay Jin Partitions and The Representation Theory of Symmetric Groups 5

6 December 28 Morning Session Chair: Phan Thi Ha Duong 09:00 09:05 Keynote Speaker: Hoeteck Wee Cryptography, Encryption and Big Data 09:50 10:15 Wu Hongjun Convolutional Code and The Design of Light-weight Authenticated Cipher 10:15 10:45 Coffee break Chair: Wang Huaxiong 10:45 11:10 Le Tuan Hoa A Survey on Some Problems on Combinatorial Commutative Algebra 11:10 11:35 Klauck Hartmut Two Results about Quantum Messages 11:35 12:00 Bui Thu Lam Discovery of Pathways in Protein-protein Interaction Networks: A Multi-objective Approach Using Evolutionary Algorithms 12:00 12:25 Vu Van Khu Permutation Codes and Its Application in Flash Memories 12:25 14:00 Lunch December 28 afternoon 14:00 17:00 City Tour (Hanoi) December 28 evening 18:30 21:00 Banquet 6

7 December 29 Morning Session Chair: Xing Chaoping 09:00 09:5 Keynote Speaker: Moshe Rosenfeld Geometric Graphs 09:50 10:15 Ngo Viet Trung Associated primes of powers of edge ideals and ear decompositions of graphs 10:15 10:45 Coffee break Chair: Phong Quang Nguyen 10:45 11:10 Tran Nam Trung A Characterization of Triangle-free Gorenstein graphs 11:10 11:35 Nguyen Khoa Improved Zero-knowledge Protocol for the ISIS Problem and Applications 11:35 12:00 Tan Ming Ming Construction of Hadamard Groups and Matrices 12:00 14:00 Lunch 7

8 December 29 Afternoon Session Chair: Ngo Dac Tan 14:00 14:25 Wu Guohua Nonhemimaximal Sets and Degrees 14:25 14:50 Vu Duc Minh Using Graph Algorithms to Define Steps That Satisfy Constraints in Optimization Problems 14:50 15:15 Ng Keng Meng Diamond Embedding and Minimal Pairs in The Computably Enumerable Truth Table Degrees 15:15 15:45 Coffee break Chair: Moshe Rosenfeld 15:45 16:10 Nguyen Ngoc Doanh A Discrete Model for Competition in Multi-zone Environment 16:10 16:35 Yamaleev Mars On A Classification of 2-c.e. Turing Degrees 16:35 17:00 Ho Thi Phuong Nhi Partitions and Ranks of Partitions December 30: Halong Bay Tour 8

9 A. KEYNOTE SPEAKERS The Algorithmic Revolution in Geometry of Numbers Phong Quang NGUYEN Inria, France and Tsinghua University, China Geometry of numbers is the branch of number theory which studies Euclidean lattices, which are discrete subgroups of R n. In the past 30 years, there has been significant progress in the study of algorithmic questions in geometry of numbers, which has led to many new applications such as in cryptography. This research has used or revisited many classical mathematical results. In this talk, we survey the connection between algorithmic and mathematical aspects of geometry of numbers. Our starting point is Hermite s constant, which is related to the densest lattice packing. Biography: Phong Quang Nguyen is a senior researcher (directeur de recherche) at INRIA (France) and guest professor at Tsinghua University, Institute for Advanced Study (China). He is also the European director of LIAMA, the Sino-European joint lab in Computer Science. His main research interests are cryptanalysis, algorithmic number theory and real-world cryptography. He has published over sixty articles in journals or peer-reviewed international conferences. He has been invited speaker at more than thirty international conferences/workshops (including the 30th EUROCRYPT conference in 2011) and ten summer schools. He received the Cor Baayen European Award in 2001 and the Best Paper Award at the 25th EUROCRYPT conference in Since 2006, he has been associate editor of the Journal of Cryptology and the Journal of Mathematical Cryptology. He regularly participates to the program committee of the three flagship conferences in cryptology: CRYPTO, EUROCRYPT and ASIACRYPT. And he was Program Co-Chair of EUROCRYPT 2014, EUROCRYPT 2013 and PKC Phong Quang is an alumni of the cole normale suprieure de Lyon ( ). He obtained his Ph.D. in 1999 at the Computer science department of the cole normale suprieure in Paris, and his Habilitation diriger des recherches in 2007 from Universit Paris 7. 9

10 Geometric Graphs Moshe ROSENFELD University of Washington Tacoma A geometric graph G(V, D) is a graph whose vertices V M where M is a metric space and D R +. Two vertices u, v V are connected by an edge if u v D. Probably the most famous geometric graph is G(R 2, {1}), the unit-distance graph. In this talk, I will discuss some examples of geometric graphs and highlight related open problems. In particular, I will discuss the recently introduced odd-distance graph G(R 2, {1, 3, 5,...}) with particular attention to properties of infinite graphs inherited from its finite subgraphs. Biography: Professor Moshe Rosenfeld received his Ph. D. in Mathematics from Hebrew University of Jerusalem From 1971 until 1986, he has been Senior lecturer, Associate Professor and Full Professor at the Mathematics and Computer Science, Ben Gurion University of the Negev (Israel). From 1986 until 2008, he has been Professor at Pacific Lutheran University ( ) and University of Washington, Tacoma ( ) and is now Professor Emeritus of University of Washington, Tacoma. Moshe Rosenfeld has published more than 60 research publications in combinatorics and computer science, including convex and discrete geometry, system theory. He has collaborated with more than 30 authors including Paul Erdos. Moshe Rosenfeld has given many courses in Vietnam National University and has received many awards including Vietnam ministry of education merit award. 10

11 Cryptography, Encryption and Big Data Hoeteck WEE Ecole Normale Suprieure We live in an era of Big Data, wherein a deluge of data is being generated, collected, and stored all around us. In order to protect this data, we need to encrypt it. This raises a fundamentally new challenge in cryptography: Can we encrypt data while enabling finegrained access control and selective computation, as is necessary to protect big, complex data? In this talk, I will present my work on functional encryption which addresses this challenge. Biography: Hoeteck Wee is a researcher at CNRS and ENS in Paris. He obtained his Ph.D. from UC Berkeley and his B.Sc. from MIT, both in Computer Science. He received the US NSF Career Award in 2010, a Humboldt Research Fellowship in 2012, as well as a Google Faculty Research Award and the French JCJC Young Researcher Grant in Hoeteck s research addresses new cryptographic challenges posed by Big Data and the Internet. 11

12 B. INVITED SPEAKERS Discovery of Pathways in Protein-protein Interaction Networks: A Multi-objective Approach Using Evolutionary Algorithms Bui Thu Lam and Nguyen Hoai Anh Faculty of Information Technology Le Quy Don Technical University, Hanoi, Vietnam Lam.bui07@gmail.com; nguyenhoaianh@yahoo.com Knowledge of protein-protein interactions, which are usually given in forms of undirected networks or graphs is essential in understanding cell activities and evolution. Previous findings show that orienting protein-protein interactions can improve pathway discovery. However, assigning orientation for protein interactions is a combinatorial optimization problem which has been proved to be NP-hard, making it critical to develop efficient algorithms. Our work introduces multi-objective evolutionary algorithms to deal with two criteria at the same time: maximizing the total weight of the satisfied paths as well as the number of standard pathways in the interaction network. We first studied the mathematical model of the interaction network orientation problem with multi-objectivity characteristics. Based on such a model, we designed a multi-objective evolutionary algorithm to find the approximated Pareto solutions for the problem. Optimal Competitive Auctions Chen Ning Nanyang Technological University ningc@ntu.edu.sg Selling digital-goods with the objective of maximizing revenue is a classic auction design problem. The current best known design was a 3.12 competitive auction, and the best known lower bound is In this talk, we show an optimal competitive auction with a competitive ratio matching the lower bound. This solves a long-standing open problem in digital goods auctions. 12

13 Time-Parallel Simulation for Stochastic Automata Networks and Stochastic Process Algebra T.H. Dao Thi, J.M. Fourneau, and F. Quessette PRiSM, Univ. Versailles St Quentin, UMR CNRS 8144, Versailles France Time Parallel Simulation (TPS) is the construction of the time-slices of a sample-path on a set of parallel processors (see [11] chap. 6 and references therein). TPS has a potential to massive parallelism as the number of logical processes is only limited by the number of time intervals which is a direct consequence of the time granularity and the simulation length. Stochastic Automata Networks (SAN in the following) and some stochastic process algebra (like PEPA) allow the construction of extremely large Markov chains which are difficult to analyze due to their size. Here, we show how we can use TPS to solve efficiently some models based on SAN or PEPA. The approach uses some graph the- oretical properties which can be checked easily on a SAN or a PEPA model. The quantitative results are obtained by a TPS based on linear recurrence equations of the daters with associative operators. 13

14 The Structure of Special Classes of Graphs and Application Do Duy Hieu Institute of Mathematics Vietnam Academy of Science and Technology Let A F q, a finite field with q elements, denote V n (A) = (A A) (A A) (A A), where the product is taken n times. D. Hart, A. Iosevich, J. Solymosi proved that if A Cq n, with a sufficiently large absolute constant C, then V n (A) = F q, where the product is taken n times. A. Balog also obtain the following result, if A C.q k, then V 2k+1 (A) = F q. Using the graph theoretic method, we obtain the following improvement, if A C.q /2 2 k, then V 2k+1 (A) = F q. We also obtain similar results in the setting of finite cyclic rings. Partitions and Ranks of Partitions Ho Thi Phuong Nhi Nayang Technological University S120056@e.ntu.edu.sg The partition function p(n), first studied by Euler, counts the number of ways of representing a positive integer n as a sum of nondecreasing positive integers. In 1944, Dyson introduced the rank of a partition, which is defined as the largest part minus the number of parts. Since then, numerous interesting results related to ranks have been discovered. One of the recent results, by Chan and Mao, states that N(m, n) N(m + 2, n), 14

15 where N(m, n) denotes the number of partitions of n with rank m. However, numerical computations also suggest that for large enough n and n m + 1, we have N(m, n) N(m + 1, n). My presentation will provide a brief introduction of partitions and ranks, as well as showing the current progress in proving the above monotonicity of ranks. Two Results about Quantum Messages Hartmut Klauck Nanyang Technological University hklauck@gmail.com We prove two results about the relationship between quantum and classical messages. Our first contribution is to show how to replace the quantum message in a one-way communication protocol by a deterministic message, establishing a new upper bound on deterministic oneway communication complexity in terms of quantum communciation. This improves previous results by Aaronson. In our second contribution we investigate the power of quantum proofs over classical proofs. We give the first example of a scenario in which quantum proofs lead to exponential savings in computing a Boolean function, for quantum verifiers. The previously only known separation (by Aaronson and Kuperberg) between the power of quantum and classical proofs is in a black-box model where the input is also quantum. Our separation is in the model of one-way communication complexity. 15

16 A survey on some problems on Combinatorial Commutative Algebra Le Tuan Hoa Institute of Mathematic Vietnam Academy of Science and Technology In this talk, I will give a review on the solution of some old and new problems on monomial ideals in order to show a deep interaction between Commutative Algebra and Combinatorics. The Cover Number of A Matrix and Its Algorithmic Applications Troy Lee Nanyang Technological University TroyLee@ntu.edu.sg We give approximation algorithms for problems like finding a Nash equilibrium and finding the densest subgraph in a graph. Our approximation algorithms are based on studying the minimum number of epsilon-sized cubes needed to cover the convex hull of the columns of a matrix. We call this quantity the cover number of the matrix. We give tight bounds on the cover number in terms of VC dimension and give efficient algorithms for enumerating elements of a cover. Joint work with Noga Alon and Adi Shraibman. 16

17 Two-sided Many-to-many Matching with Ties Mengling Li Nanyang Technological University This talk focuses on a generalized many-to-many matching problem with ties. The complications in such a problem arise from two aspects: multi-unit capacities and weak preferences. Either makes a stable outcome not necessarily Pareto efficient, resulting in efficiency loss. In such problems, a natural solution concept is Pareto stability, which ensures both pairwise stability and Pareto efficiency. Our main technical contribution is a polynomial-time algorithm that computes a Pareto stable many-to-many matching in the presence of ties. For a less generalized problem with homogeneous preferences on one side of the market, for example, course allocation with ties, we propose two new competing Pareto stable mechanisms known as the Pareto-improving draft and dictatorship mechanisms. Using actual course allocation data, our simulations show that both mechanisms can significantly improve overall efficiency and welfare of the students compared with the existing mechanism. Besides, the draft mechanism outperforms the dictatorship mechanism despite its non-strategyproofness for the students. Partitions and The Representation Theory of Symmetric Groups Lim Kay Jin Nanyang Technological University LimKJ@ntu.edu.sg Both ordinary and modular representation theories of symmetric groups are closely related to the combinatorics of partitions. We discuss how some combinatorial properties of partitions control the structure of these modules. The module structures that we are mostly interested in are vertices, complexities and cohomological varieties. 17

18 Diamond Embedding and Minimal Pairs in The Computably Enumerable Truth Table Degrees Ng Keng Meng Nanyang Technological University We present some recent and old results on minimal pairs in the computably enumerable truth table degrees. Jockusch and Mohrherr showed that the diamond lattice a,b,0,1 can be embedded in the computably enumerable truth table degrees, preserving the top and bottom elements, while a result of Lachlan shows that this is impossible if we consider the coarser structures of the weak truth table degrees and the Turing degrees. We discuss related results about the possible degrees of a and b. We also discuss how to use the determinacy of finite games to construct minimal pairs in the truth table degrees. On Vertex Disjoint Cycles of Different Lengths in Digraphs Ngo Dac Tan Institute of Mathematics Vietnam Academy of Science and Technology ndtan@math.ac.vn M.A. Henning and A. Yeo conjectured in [SIAM J. Discrete Math. 26 (2012) ] that a digraphs D contains two vertex disjoint directed cycles of different lengths if D satisfies any of the conditions listed belows: (a) D has minimum outdegree at least 4; (b) D is bipartite and 3-regular; (c) D is bipartite and has minimum outdegree at least 3; (d) D is 3-regular of sufficent large order. In this talk, we will survey recent results obtained for these conjectures. 18

19 Associated Primes of Powers of Edge Ideals and Ear Decompositions of Graphs Ngo Viet Trung Institute of Mathematics Vietnam Academy of Science and Technology We present a complete combinatorial classification of the associated primes of every fixed power of the edge ideal of a graph. This will be done by using the theory of ear decompositions. It turns out that these associated primes are characterized by certain kind of subgraphs. A Discrete Model for Multi Competing Species in Multi-zone Environment Nguyen Ngoc Doanh (joined with Phan Thi Ha Duong and Kevin Perrot) Hanoi University of Science and Technology; Institute of Mathematics, Vietnam Academy of Science and Technology doanhbondy@gmail.com; phan.haduong@gmail.com We present a discrete model for a dynamics of multi species competing for territory in multi-zone environment. Species individuals use an avoiding tactics to compete with each other. We investigate the effects of this tactics on stable configurations of the dynamics. For a particular case of three competing species in three zones, we show a characterization of the initial conditions under which the system converges to each stable configuration. 19

20 Improved Zero-knowledge Protocol for The ISIS Problem and Applications Nguyen Ta Toan Khoa Nanyang Technological University In all existing efficient proofs of knowledge of a solution to the Inhomogeneous Small Integer Solution (ISIS) problem in the infinity norm, the knowledge extractor outputs a solution vector that is only guaranteed to be soft-o(n) times longer than the witness possessed by the prover. As a consequence, in many cryptographic schemes that use these protocols as building blocks, there exists a gap between the hardness of solving the underlying ISIS problem and the hardness underlying the security reductions. In this work, we generalize Sterns protocol to obtain a zero-knowledge proof of knowledge for the ISIS problem that removes this gap. Our result yields the potential of relying on milder security assumptions for various lattice-based cryptographic constructions. As applications of our protocol, we introduce a concurrently secure identity-based identification scheme based on the worst-case hardness of the SIVP problem in general lattices with approximation factor soft-o(n 1.5 ), and an efficient zero-knowledge proof of plaintext knowledge with no gap factor for Regevs LWE-based encryption scheme. Construction of Hadamard Groups and Matrices Tan Ming Ming Nanyang Technological University MMTAN@ntu.edu.sg Let G be a group of order 4m with a normal subgroup N of order 2. A normal (2m, 2, 2m, m) relative difference set in G relative to N is a 2m-subset R of G such that every g G \ N has exactly m representations g = r 1 r2 1 with r 1, r 2 R, and no non-identity element in N can be represented this way. Such relative difference sets have been studied from various perspectives. A group G containing a normal (2m, 2, 2m, m) relative difference set is called an Hadamard group. Moreover, it is known that Hadamard groups are equivalent to cocyclic Hadamard matrices. Hence, the existence of Hadamard groups of order 4m guarantees the existence of Hadamard matrices of order 2m. We obtain the most general known construction of normal (2m, 2, 2m, m) relative difference sets, extending previously known results. The Hadamard groups that we obtain are partial semidirect products of Z 4 and abelian 20

21 groups. Our construction is recursive and uses binary as well as quaternary Golay sequences, Williamson matrices, and building sets. We also obtain some necessary conditions for the existence of normal (2m, 2, 2m, m) relative difference sets in partial semidirect products of Z 4 and abelian groups. A table of cases with m 100, for which the existence of such relative difference sets is open, is presented. We are also interested in identifying any new families of Hadamard matrices constructed from these new Hadamard groups. This is a joint work with Bernhard Schmidt. Identities for Well-known Inequalities in Combinatorics Tran Dan Thu School of Information Technology University of Science Hochiminh City tdt@hcmus.edu.vn We present our current work on establishing non-trivial identities for some well-known inequalities in combinatorics. Extremal cases of these inequalities, i.e. the cases in which it happens the equality, are also considered. Mathematics Subject Classification: 05D05. 21

22 A Characterization of Triangle-free Gorenstein Graphs Tran Nam Trung (joint work with Do Trong Hoang) Institute of Mathematics Vietnam Academy of Science and Technology Abstract: In a 1970, Plummer introduced the notion of considering graphs in which every maximal independent set has the same cardinality ; he called a graph having this property a well-covered graph. Characterize well-covered graphs is a difficult problem and the work on well-covered graphs that has appeared in the literature has focused on certain subclasses of well-covered graphs. Recently, there are well-covered graphs appeared from the algebraic point of view that are Cohen-Macaulay graphs. Let R = k[x 1,..., x n ] be a polynomial ring of n variables over the field k. Let G be a simple graph on the vertex set {x 1,..., x n }. We associate to the graph G a quadratic squarefree monomial ideal I(G) = (x i x j x i x j E(G)) R, which is called the edge ideal of G. We say that G is Cohen-Macaulay (resp. Gorenstein) if I(G) is Cohen-Macaulay (resp. Gorenstein). Note that G is well-covered whenever it is Cohen-Macaulay, and it is a wide open problem to characterize graph-theoretically the Cohen-Macaulay graphs. If we focus on Gorenstein graphs G without isolated vertices, then not only G is wellcovered but G \ x is also well-covered with α(g) = α(g \ x) for any vertex x. Such graphs form the so-called class W 2. Note that the Gorensteinness of graphs is also dependent on the characteristic of the base field k, so we cannot characterize graph-theoretically all Gorenstein graphs. In this paper, we are interested in triangle-free Gorenstein graphs. Our main result is: a triangle-free graph G is Gorenstein if and only if every non-trivial connected component of G belongs to W 2. 22

23 The Duality of Critical and Super-stable Configurations of Chip Firing Games on Directed Graphs Tran Thi Thu Huong Institute of Mathematics Vietnam Academy of Science and Technology In the talk we show a collection of firing scripts which help recognizing critical configurations of chip firing games on directed graphs. Using these firing scripts we prove a universal property of critical configurations saying that we cannot get any stable configuration from a given critical configuration by inverse firing a multi-subset of vertices. This property allows us to prove the duality of critical and super-stable configurations on directed graphs. On The Recovering Threshold of The Stochastic Block Mode Tran Vinh Linh Mathematics Department International University, National University of Vietnam at Ho Chi Minh City tvlinh@hcmiu.edu.vn Given the following model: two random d 1 regular graphs on two set of n vertices (labeled by + or -) connected by one random d 2 -regular bipartite graph without knowing the label of vertices. In this talk we will discuss the threshold for d 1 and d 2 so that we can recover all or almost all the labels with high probability by analyzing the eigenvalues of the graph. Joint work with G. Brito, I. Dumitriu and S. Ganguly. 23

24 Using Graph Algorithms To Define Steps That Satisfy Constraints in Optimization Problems Vu Duc Minh Hanoi University of Science Vietnam National University In this talk, we present some existing approaches to solve constraints in mixed linear programming problems. We show how we can model these constraints as a graph problem and show how we can solve these constraints efficiently using classic graph algorithms. Permutation Codes and Its Application in Flash Memories Vu Van Khu Nanyang Technological University VANKHU001@e.ntu.edu.sg Flash memories received a lot of attention from mathematicians, computer scientist,... as it is useful in smartphones, computers, communication systems,... Rank modulation was recently proposed as an information representation for multi-level flash memories. The error correcting codes in the space of permutations under the Cayley, Kendall tau, Ulam metrics have been studied due to applications in flash memories. We consider permutation codes under more general metrics say generalized Cayley and generalized Kendall tau metrics. As results, we construct codes under these general metrics that are larger than those previously known under more restricted metrics. 24

25 Nonhemimaximal Sets and Degrees Wu Guohua Nanyang Technological University In this talk, I will first give a brief introduction of computability theory and research focuses on degree structures, effective aspects of classical theorems in algebra and combinatorics. Some recent work (joint with Mustafa and Yamaleev) on Turing degrees of nonhemimaximal sets will be presented. Convolutional Code and The Design of Light-weight Authenticated Cipher Wu Hongjun Nanyang Technological University wuhj@ntu.edu.sg Convolutional code encodes message in a sequential way (bit-by-bit encoding) and is able to generate codes with large free distance. These two properties of convolutional codes are useful for the design of light-weight authenticated ciphers. An authenticated cipher encrypts and authenticates messages at the same time. In a light-weight authenticated cipher, hardware complexity can be greatly reduced if message is processed in a sequential way; any modification to a message should cause many differences passing through the nonlinear components of the cipher (to introduce unpredictable differences so as to detect message modification and forgery). In this talk, we introduce the ACORN light-weight authenticated cipher, which is built on the idea of convolutional code. ACORN is a candidate of the current CAESAR competition on authenticated encryption ( ). It is the first time that the idea of convolutional code is used in the design of symmetrical key ciphers. 25

26 On A Classification of 2-c.e. Turing Degrees Yamaleev Mars Nanyang Technological University YMars@ntu.edu.sg A set D ω, where ω = {0, 1, 2,... }, is computable enumerable (c.e.) if it has an effective approximation such that for each element an assumption about its membership in D is changed at most once. A set D ω is 2-c.e. if the assumption is changed at most twice. A Turing degree d is 2-c.e. if it contains a 2-c.e. set. If it also doesn t contain a c.e. set then it is a properly 2-c.e. Turing degree. The Turing degrees play one of the central roles in Computability Theory. C.e. and 2-c.e. Turing degrees are natural and important subclasses of all Turing degrees. It s known that they form upper semilattice and don t form lattice, moreover it s known that these structures are not elementarily equivalent (e.g., due to the work of B. Cooper and M. Yates on noncuppable c.e. degrees, and also M.M. Arslanov s Cupping Theorem and R. Downey s Diamond Embedding Theorem. However, there are still a lot of open natural questions about structural and model-theoretic properties of c.e. and 2-c.e. Turing degrees. In the talk we will discuss a classification of 2-c.e. Turing degrees based on their socalled Lachlans degrees. Using this classification we study a structural properties of the classes. The class of 2-c.e. isolated degrees of the classification is already widely studied and has different applications, the class of 2-c.e. exact degrees was introduced and studied by Sh.T. Ishmukhametov. Some other classes were investigated in our recent works and will be presented at the talk. 26

27 LIST OF PARTICIPANTS OF THE FIRST NTU-VIASM WORKSHOP ON DISCRETE MATHEMATICS 1. Bui Thu Lam. Le Quy Don Technical University, Hanoi, Vietnam. 2. Chee Yeow Meng. Nanyang Technological University. 3. Chen Ning. Nanyang Technological University. 4. Dang Tuan Thuong. School of Information Technology University of Scienc Hochiminh City. 5. Dao Van Duong. Central University of Construction. 6. Dao Thi Thu Ha. CNRS, France and VIASM, Vietnam. 7. Dao Tuan Anh. Hanoi University of Science and Technology. 8. Dau Son Hoang. Singapore University of Technology and Design. 9. Dau Xuan Luong. Quangninh College of Education. 10. Do Duy Hieu. Institute of Mathematics, Vietnam Academy of Science and Technology. 11. Do Minh Nam. University of Finance and Business Administration. 12. Do Xuan Thanh. Military Institute of science and technology. 13. Do Phan Thuan. Hanoi University of Science and Technology. 14. Do Dai Chi. VNU University of Science. 15. Ha Minh Lam. Institute of Mathematics, Vietnam Academy of Science and Technology. 16. Ha Binh Minh. Hanoi University of Science and Technology. 17. Ho Thi Phuong Nhi. Nanyang Technological University. 18. Hoang Manh Ha. Thu Dau Mot University. 19. Hoang Van Linh. Lang Son College of Education. 27

28 20. Hu Bingyang. Nanyang Technological University. 21. Huynh Ba Dieu. Duy Tan University. 22. Huynh Thi Thanh Binh. Hanoi University of Science and Technology. 23. Klauck Hartmut. Nanyang Technological University. 24. Le Hai Ha. Hanoi University of Science and Technology. 25. Le Quang Ham. Thieu Hoa High School. 26. Le Tuan Hoa. Institute of Mathematics, Vietnam Academy of Science and Technology. 27. Le Quang Hoa. Hanoi University of Science and Technology. 28. Le Hai Khoi. Nanyang Technological University Le Hoang Son. VNU University of Science. 30. Lee Troy. Nanyang Technological University. 31. Li Mengling. Nanyang Technological University. 32. LIM Kay Jin. Nanyang Technological University. 33. Ling San. Nanyang Technological University. 34. Moshe Rosenfeld. University of Washington Tacoma. 35. Ng Keng Meng. Nanyang Technological University. 36. Ngo Dac Tan. Institute of Mathematics, Vietnam Academy of Science and Technology. 37. Ngo Viet Trung. Institute of Mathematics. Vietnam Academy of Science and Technology. 38. Nguyen Ta Toan Khoa. Nanyang Technological University. 39. Nguyen Ngoc Anh. Hanoi National University of Education. 40. Nguyen Ngoc Doanh. Hanoi University of Science and Technology. 41. Nguyen Huu Du. Vietnam Institute for Advanced Study in Mathematics. 28

29 42. Nguyen Thanh Huyen. Vinh University. 43. Nguyen Hung Long. Vietnam University of Commerce. 44. Nguyen Hong Nam. Le Quy Don Technical University, Hanoi, Vietnam. 45. Nguyen Thanh Nhan. VNU University of Engineering and Technology. 46. Nguyen Minh Sang. Hanam College of Education. 47. Nguyen Dinh Thanh Cong. VNU University of Science. 48. Nguyen Tho Thong. VNU University of Science. 49. Nguyen Phuong Thuy. Hanoi University of Science and Technology. thuy 50. Nguyen Trong Toan. Le Quy Don Technical University, Hanoi, Vietnam. 51. Nguyen Thi Van. Water Resources University. 52. Oliver Perroud. 53. Pham Tuan Cuong. Hanoi University of Mining and Geology. 54. Pham Nhu Kien. Hanoi University of Science and Technology. 55. Pham Huy Thong. VNU University of Science. 56. Pham The Anh. Le Quy Don Technical University. 57. Phan Thi Ha Duong. Institute of Mathematics, Vietnam Academy of Science and Technology. 58. Phong Quang Nguyen. Inria, France and Tsinghua University, China. 59. Ta Anh Son. Hanoi University of Science and Technology. 60. Ta Ngoc Anh. Le Quy Don Technical University, Hanoi, Vietnam. 61. Tan Ming Ming. Nanyang Technological University. 62. Tran Thi Thu Huong. Institute of Mathematics, Vietnam Academy of Science and Technology. 63. Tran Quang Huy. Vietinbank Aviva Insurance Company. 64. Tran Vinh Linh. International University, National University of Vietnam at Hochiminh City. 29

30 65. Tran Thi Kim Oanh. Hanoi University of Science and Technology. 66. Tran Ngoc Thang. Hanoi University of Science and Technology. 67. Tran Dan Thu. School of Information Technology University of Science Hochiminh City. 68. Tran Nam Trung. Institute of Mathematics. Vietnam Academy of Science and Technology. 69. Tran Hong Yen. People Security Academy. 70. Vo Duy Tam. Thu Duc College of Technology. 71. Vu Van Khu. Nanyang Technological University. 72. Vu Duc Minh. Vietnam National University. 73. Wang Huaxiong. Nanyang Technological University. 74. Hoeteck WEE. Ecole Normale Suprieure. 75. WU Guohua. Nanyang Technological University. 76. WU Hongjun. Nanyang Technological University. 77. Xing Chaoping. Nanyang Technological University. 78. Yamaleev Mars. Nanyang Technological University. Picture: Sandpile Toppling from: moore/gallery.html 30