Almost separating and almost secure frameproof codes over q-ary alphabets

The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-015-0060-zIn this paper we discuss some variations of the notion of separating code for alphabets of arbitrary size. We show how the original definition can be relaxed in two different ways, namely almost separating and almost secure frameproof codes, yielding two different concepts. The new definitions enable us to obtain codes of higher rate, at the expense of satisfying the separating property partially. These new definitions become useful when complete separation is only required with high probability, rather than unconditionally. We also show how the codes proposed can be used to improve the rate of existing constructions of families of fingerprinting codes.Peer ReviewedPostprint (author's final draft

Constructions of almost secure frameproof codes with applications to fingerprinting schemes

The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-017-0359-zThis paper presents explicit constructions of fingerprinting codes. The proposed constructions use a class of codes called almost secure frameproof codes. An almost secure frameproof code is a relaxed version of a secure frameproof code, which in turn is the same as a separating code. This relaxed version is the object of our interest because it gives rise to fingerprinting codes of higher rate than fingerprinting codes derived from separating codes. The construction of almost secure frameproof codes discussed here is based on weakly biased arrays, a class of combinatorial objects tightly related to weakly dependent random variables.Peer ReviewedPostprint (author's final draft

On codes for traceability schemes: constructions and bounds

A traceability or fingerprinting scheme is a cryptographic scheme that facilitates the identification of the source of leaked information. In a fingerprinting setting, a distributor delivers copies of a given content to a set of authorized users. If there are dishonest members (traitors) among them, the distributor can deter plain redistribution of the content by delivering a personalized, i.e., marked, copy to each user. The set of all user marks is known as a fingerprinting code. There is, however, another threat. If several traitors collude to create a copy that is a combination of theirs, then the pirated copy generated will contain a corrupted mark, which may obstruct the identification of traitors. This dissertation is about the study and analysis of codes for their use in traceability and fingerprinting schemes, under the presence of collusion attacks. Moreover, another of the main concerns in the present work will be the design of identification algorithms that run efficiently, i.e., in polynomial time in the code length. In Chapters 1 and 2, we introduce the topic and the notation used. We also discuss some properties that characterize fingerprinting codes known under the names of separating, traceability (TA), and identifiable parent property (IPP), which will be subject of research in the present work. Chapter 3 is devoted to the study of the Kötter-Vardy algorithm to solve a variety of problems that appear in fingerprinting schemes. The concern of the chapter is restricted to schemes based on Reed-Solomon codes. By using the Kötter-Vardy algorithm as the core part of the identification processes, three different settings are approached: identification in TA codes, identification in IPP codes and identification in binary concatenated fingerprinting codes. It is also discussed how by a careful setting of a reliability matrix, i.e., the channel information, all possibly identifiable traitors can be found. In Chapter 4, we introduce a relaxed version of separating codes. Relaxing the separating property lead us to two different notions, namely, almost separating and almost secure frameproof codes. From one of the main results it is seen that the lower bounds on the asymptotical rate for almost separating and almost secure frameproof codes are greater than the currently known lower bounds for ordinary separating codes. Moreover, we also discuss how these new relaxed versions of separating codes can be used to show the existence of families of fingerprinting codes of small error, equipped with polynomial-time identification algorithms. In Chapter 5, we present explicit constructions of almost secure frameproof codes based on weakly biased arrays. We show how such arrays provide us with a natural framework to construct these codes. Putting the results obtained in this chapter together with the results from Chapter 4, shows that there exist explicit constructions of fingerprinting codes based on almost secure frameproof codes with positive rate, small error and polynomial-time identification complexity. We remark that showing the existence of such explicit constructions was one of the main objectives of the present work. Finally, in Chapter 6, we study the relationship between the separating and traceability properties of Reed-Solomon codes. It is a well-known result that a TA code is an IPP code, and that an IPP code is a separating code. The converse of these implications is in general false. However, it has been conjectured for some time that for Reed-Solomon codes all three properties are equivalent. Giving an answer to this conjecture has importance in the field of fingerprinting, because a proper characterization of these properties is directly related to an upper bound on the code rate i.e., the maximum users that a fingerprinting scheme can allocate. In this chapter we investigate the equivalence between these properties, and provide a positive answer for a large number of families of Reed-Solomon codes.Un sistema de trazabilidad o de fingerprinting es un mecanismo criptogr afi co que permite identi car el origen de informaci on que ha sido fi ltrada. En el modelo de aplicación de estos sistemas, un distribuidor entrega copias de un determinado contenido a un conjunto de usuarios autorizados. Si existen miembros deshonestos (traidores) entre ellos, el distribuidor puede disuadir que realicen una redistribuci on ingenua del contenido entregando copias personalizadas, es decir, marcadas, a cada uno de los usuarios. El conjunto de todas las marcas de usuario se conoce como c ódigo de fingerprinting. No obstante, existe otra amenaza m as grave. Si diversos traidores confabulan para crear una copia que es una combinación de sus copias del contenido, entonces la copia pirata generada contendr a una marca corrompida que di ficultar a el proceso de identificaci on de traidores. Esta tesis versa sobre el estudio y an alisis de c odigos para su uso en sistemas de trazabilidad o de fi ngerprinting bajo la presencia de ataques de confabulaci on. Otra de las cuestiones importantes que se tratan es el diseño de algoritmos de identi caci on e ficientes, es decir, algoritmos que se ejecuten en tiempo polin omico en la longitud del c odigo. En los Cap tulos 1 y 2 presentamos el tema e introducimos la notaci on que utilizaremos. Tambi en presentaremos algunas propiedades que caracterizan los c odigos de fi ngerprinting, conocidas bajo los nombres de propiedad de separaci on, propiedad identi cadora de padres (IPP) y propiedad de trazabilidad (TA), que est an sujetas a estudio en este trabajo. El Cap tulo 3 est a dedicado al estudio del algoritmo de decodi caci on de lista con informaci on de canal de Kötter-Vardy en la resoluci on de determinados problemas que aparecen en sistemas de fingerprinting. El ambito de estudio del cap ítulo son sistemas basados en c odigos de Reed-Solomon. Empleando el algoritmo de Kötter-Vardy como parte central de los algoritmos de identifi caci on, se analizan tres propuestas en el cap ítulo: identi caci on en c odigos TA, identifi caci on en c odigos IPP e identifi caci on en c odigos de fingerprinting binarios concatenados. Tambi en se analiza c omo mediante un cuidadoso ajuste de una matriz de abilidad, es decir, de la informaci on del canal, se pueden encontrar a todos los traidores que es posible identi car e ficientemente. En el Capí tulo 4 presentamos una versi on relajada de los c odigos separables. Relajando la propiedad de separaci on nos llevar a a obtener dos nociones diferentes: c odigos cuasi separables y c odigos cuasi seguros contra incriminaciones. De los resultados principales se puede observar que las cotas inferiores de las tasas asint oticas para c odigos cuasi separables y cuasi seguros contra incriminaciones son mayores que las cotas inferiores actualmente conocidas para c odigos separables ordinarios. Adem as, tambi en estudiamos como estas nuevas familias de c odigos pueden utilizarse para demostrar la existencia de familias de c odigos de ngerprinting de baja probabilidad de error y dotados de un algoritmo de identi caci on en tiempo polin omico. En el Capí tulo 5 presentamos construcciones expl citas de c odigos cuasi seguros contra incriminaciones, basadas en matrices de bajo sesgo. Mostramos como tales matrices nos proporcionan una herramienta para construir dichos c odigos. Poniendo en com un los resultados de este cap tulo con los del Capí tulo 4, podemos ver que, bas andonos en c odigos cuasi seguros contra incriminaciones, existen construcciones expl ícitas de c odigos de fi ngerprinting de tasa positiva, baja probabilidad de error y con un proceso de identi caci on en tiempo polin omico. Demostrar que existen dichas construcciones expl citas era uno de los principales objetivos de este trabajo. Finalmente, en el Capí tulo 6, estudiamos la relaci on existente entre las propiedades de separaci on y trazabilidad de los c odigos de Reed-Solomon. Es un resultado bien conocido el hecho que un c odigo TA es un c odigo IPP, y que un c odigo IPP es un c odigo separable. Las implicaciones en el sentido opuesto son falsas en general. No obstante, existe una conjetura acerca de la equivalencia de estas tres propiedades en el caso de cóodigos de Reed-Solomon. Obtener una respuesta a esta conjetura es de una importancia relevante en el campo del fi ngerprinting, puesto que la caracterización de estas propiedades est a directamente relacionada con una cota superior en la tasa del c odigo, es decir, con el n umero de usuarios que puede gestionar un sistema de fi ngerprinting. En este cap ítulo investigamos esta equivalencia y proporcionamos una respuesta afirmativa para un gran n umero de familias de c odigos de Reed-Solomon. Los resultados obtenidos parecen sugerir que la conjetura es cierta

A study of the separating property in Reed-Solomon codes by bounding the minimum distance

The version of record is available online at: http://dx.doi.org/10.1007/s10623-021-00988-zAccording to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.This work has been supported by the Spanish Government Grant TCO-RISEBLOCK (PID2019-110224RB-I00) MINECO .Peer ReviewedPostprint (published version

Random Codes and Graphs for Secure Communication

This dissertation considers two groups of problems related to secure communication. The first line of research is devoted to theoretical problems of copyright protection of digital content. Embedding identification data in the content is a well-developed technique of content protection known under the name of fingerprinting. Schemes that provide such protection are known as fingerprinting codes in the literature. We study limits of the number of users of a fingerprinting system as well as constructions of low-complexity fingerprinting codes that support a large number of users. The second problem that is addressed in the dissertation relates to connectivity analysis of ad hoc wireless networks. One of the basic requirements in such environments is to ensure that none of the nodes are completely isolated from the network. We address the problem of characterizing threshold parameters for node isolation that enable the system designer to choose the power needed for network operation based on the outage probability of links in the network. The methods of this research draw from coding theory, information theory and random graphs. An idea that permeates most results in this dissertation is the application of randomization both in the analysis of fingerprinting and node isolation. The main contributions of this dissertation belong in the area of fingerprinting and are described as follows. We derive new lower and upper bounds on the optimal trade-off between the number of users and the length of the fingerprints required to ensure reliability of the system, which we call fingerprinting capacity. Information-theoretic techniques employed in our proofs of bounds on capacity originate in coding theorems for channels with multiple inputs. Constructions of fingerprinting codes draw on methods of coding theory related to list decoding and code concatenation. We also analyze random graph models for ad hoc networks with link failures and secure sensor networks that employ randomized key distribution. We establish a precise zero-one law for node isolation in the model with link failures for nodes placed on the circle. We further generalize this result to obtain a one-law for secure sensor networks on some surfaces

A Security Analysis of Some Physical Content Distribution Systems

Content distribution systems are essentially content protection systems that protect premium multimedia content from being illegally distributed. Physical content distribution systems form a subset of content distribution systems with which the content is distributed via physical media such as CDs, Blu-ray discs, etc. This thesis studies physical content distribution systems. Specifically, we concentrate our study on the design and analysis of three key components of the system: broadcast encryption for stateless receivers, mutual authentication with key agreement, and traitor tracing. The context in which we study these components is the Advanced Access Content System (AACS). We identify weaknesses present in AACS, and we also propose improvements to make the original system more secure, flexible and efficient

Robust parent-identifying codes and combinatorial arrays

An $n$-word $y$ over a finite alphabet of cardinality $q$ is called a descendant of a set of $t$ words $x^1,\dots,x^t$ if $y_i\in\{x^1_i,\dots,x^t_i\}$ for all $i=1,\dots,n.$ A code \cC=\{x^1,\dots,x^M\} is said to have the $t$-IPP property if for any $n$-word $y$ that is a descendant of at most $t$ parents belonging to the code it is possible to identify at least one of them. From earlier works it is known that $t$-IPP codes of positive rate exist if and only if $t\le q-1$. We introduce a robust version of IPP codes which allows {unconditional} identification of parents even if some of the coordinates in $y$ can break away from the descent rule, i.e., can take arbitrary values from the alphabet, or become completely unreadable. We show existence of robust $t$-IPP codes for all $t\le q-1$ and some positive proportion of such coordinates. The proofs involve relations between IPP codes and combinatorial arrays with separating properties such as perfect hash functions and hash codes, partially hashing families and separating codes. For $t=2$ we find the exact proportion of mutant coordinates (for several error scenarios) that permits unconditional identification of parents

Asymptotics of Fingerprinting and Group Testing: Tight Bounds from Channel Capacities

In this work we consider the large-coalition asymptotics of various fingerprinting and group testing games, and derive explicit expressions for the capacities for each of these models. We do this both for simple decoders (fast but suboptimal) and for joint decoders (slow but optimal). For fingerprinting, we show that if the pirate strategy is known, the capacity often decreases linearly with the number of colluders, instead of quadratically as in the uninformed fingerprinting game. For many attacks the joint capacity is further shown to be strictly higher than the simple capacity. For group testing, we improve upon known results about the joint capacities, and derive new explicit asymptotics for the simple capacities. These show that existing simple group testing algorithms are suboptimal, and that simple decoders cannot asymptotically be as efficient as joint decoders. For the traditional group testing model, we show that the gap between the simple and joint capacities is a factor 1.44 for large numbers of defectives.Comment: 14 pages, 6 figure

Almost separating and almost secure frameproof codes over q-ary alphabets

The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-015-0060-zIn this paper we discuss some variations of the notion of separating code for alphabets of arbitrary size. We show how the original definition can be relaxed in two different ways, namely almost separating and almost secure frameproof codes, yielding two different concepts. The new definitions enable us to obtain codes of higher rate, at the expense of satisfying the separating property partially. These new definitions become useful when complete separation is only required with high probability, rather than unconditionally. We also show how the codes proposed can be used to improve the rate of existing constructions of families of fingerprinting codes.Peer Reviewe