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

A construction of traceability set systems with polynomial tracing algorithm

© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.A family F of w-subsets of a finite set X is called a set system with the identifiable parent property if for any w-subset contained in the union of some t sets, called traitors, of F at least one of these sets can be uniquely determined, i.e. traced. A set system with traceability property (TSS, for short) allows to trace at least one traitor by minimal distance decoding of the corresponding binary code, and hence the complexity of tracing procedure is of order O(M), where M is the number of users or the code's cardinality. We propose a new construction of TSS which is based on the old Kautz-Singleton concatenated construction with algebraic-geometry codes as the outer code and Guruswami-Sudan decoding algorithm. The resulting codes (set systems) have exponentially many users (codevectors) M and polylog(M) complexity of code construction and decoding, i.e. tracing traitors. This is the first construction of traceability set systems with such properties.Peer ReviewedPostprint (author's final draft

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

Grigory Kabatiansky interview November 16-17, 1999

NOTE: to view these items please visit http://dynkincollection.library.cornell.eduInterview by Eugene Dynkin conducted with Grigory Kabatiansky on November 16-17, 1999

Códigos con propiedades de localización basados en matrices de bajo sesgo

En este artículo presentamos una construcción explicita de un código con propiedades de identificación de traidores, aplicable a entornos de fingerprinting. Nuestro trabajo parte del estudio de una familia de códigos conocidos como códigos separables, que en el campo del fingerprinting también se conocen como códigos seguros contra incriminaciones. A partir de estos códigos, nos centramos en una versión menos estricta de ellos, en los que no se requiere que la propiedad de separación se cumpla en todos los casos, sino con alta probabilidad. Este tipo de códigos se conocen como códigos cuasi seguros contra incriminaciones. En este trabajo mostramos como construir explícitamente estos códigos, basando nuestras construcciones en estructuras conocidas como matrices de bajo sesgo. Además, mostramos cómo es posible utilizar dichos códigos para construir de forma explícita una familia de códigos binarios con propiedades de identificación, baja tasa de error y decodificación eficiente.J. Moreira y M. Fernández han sido financiados por el Gobierno de España mediante los proyectos CONSOLIDER INGENIO 2010 CSD2007-00004 “ARES” y TEC2011-26491 “COPPI”, y por la Generalitat de Catalunya mediante la ayuda 2009 SGR-1362. G. Kabatiansky ha sido financiado por la Russian Foundation for Basic Research mediante las ayudas RFBR 13-07-00978 y RFBR 13-01-12458

On non-binary traceability set systems

We introduce non-binary IPP set systems with traceability properties that have IPP codes and binary IPP set systems with traceability capabilities as particular cases. We prove an analogue of the Gilbert–Varshamov bound for such systems.Marcel Fernandez: The work of M. Fernández has been supported by the Spanish Government Grant TEC2015- 68734-R and Catalan Government Grant SGR 782.Peer ReviewedPostprint (author's final draft