Number Of Colluders Research Articles

Near-Optimal Detection for Binary Tardos Code by Estimating Collusion Strategy

A previously proposed optimal detector for bias-based fingerprinting codes such as Tardos and Nuida requires two kinds of important information: the number of colluders and the collusion strategy used to generate the pirated codeword. An estimator has now been derived for these two parameters. The bias in the pirated codeword is measured by observing the number of zeros and ones and compared with possible bias patterns calculated using information about the collusion strategy and number of colluders. Computer simulation demonstrated that the collusion strategy and number of colluders can be estimated with high probability and that the traceability of a detector using the proposed estimator is extremely close to being optimal.

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.

Simplified MAP Detector for Binary Fingerprinting Code Embedded by Spread Spectrum Watermarking Scheme

When a binary fingerprinting codeword is embedded into multimedia content by using a spread-spectrum (SS) watermarking scheme, it is difficult for colluders to perform a symbol-wise attack using their codewords. As discussed in regard to SS fingerprinting schemes, averaging their copies is a cost-effective attack from the signal processing point of view. If the number of colluders is known, an optimal detector can be used against an averaging attack with added white Gaussian noise. The detector first estimates the variance of additive noise, and then calculates correlation scores using a log-likelihood-based approach. However, the number of colluders is not usually known in a real situation. In this paper, we simplify the optimal detector by making statistical approximations and using the characteristics of the parameters for generating codewords. After that, we propose an orthogonal frequency division multiplexing-based SS watermarking scheme to embed the fingerprinting codeword into multimedia content. In a realistic situation, the signal embedded as a fingerprint is in principle attenuated by lossy compression. Because the signal amplitude in a pirated codeword is attenuated, we should adaptively estimate the parameters before calculating the scores. Different from the optimal detector, the simplified detector can easily accommodate changes in signal amplitude by examining the distorted codeword extracted from a pirated copy. We evaluate the performance of the simplified detector through simulation using digital images as well as codewords.

Binary and $$q$$ q -ary Tardos codes, revisited

The Tardos code is a much studied collusion-resistant fingerprinting code, with the special property that it has asymptotically optimal length $$m\propto c_0^2$$ m ? c 0 2 , where $$c_0$$ c 0 is the number of colluders. In this paper we give alternative security proofs for the Tardos code, working with the assumption that the strongest coalition strategy is position-independent. We employ the Bernstein inequality and Bennett inequality instead of the typically used Markov inequality. This proof technique requires fewer steps and slightly improves the tightness of the bound on the false negative error probability. We present new results on code length optimization, for both small and asymptotically large coalition sizes.

On the Collusion-Resistance of the Fingerprinting Algorithm Based on Additive Watermarking

Digital fingerprinting is the technique used for traitor tracing in multimedia content distribution. It embeds unique customer information (e.g., customer ID) into multimedia content and generates different copies for different customers. The unique customer information can be extracted from the illegal copy and used to tell the traitor who re-distributes his copy to other unauthorized customers. Till now, there exist some fingerprint embedding algorithms, such as additive embedding, multiplicative embedding, quantization embedding, etc. However, the ability to resist collusion attacks is not studied. Collusion attack is the biggest threat to fingerprinting algorithms, which combines several copies together to produce a new copy without customer information. In this paper, the fingerprinting algorithm based on additive embedding is considered, its robustness against various collusion attacks are analyzed. The theoretical results are given, together with some simulation results. As is shown, the considered fingerprinting algorithm is robust against some of the collusion attacks, while becomes fragile to some other collusion attacks with the rise of the number of colluders. It is expected to provide some advices to design or analyze fingerprinting algorithms.

On the Saddle-Point Solution and the Large-Coalition Asymptotics of Fingerprinting Games

We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1/(k^2 2ln2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotic behavior, numerical solutions to the game for small k are also presented.

Regular Simplex Fingerprints and Their Optimality Properties

This paper addresses the design of additive fingerprints that are maximally resilient against linear collusion attacks on a focused correlation detector, as defined below. Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> be the length of the host vector and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> les <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> + 1 the number of users. The focused detector performs a correlation test in order to decide whether a user of interest is among the colluders. Both the fingerprint embedder and the colluders are subject to squared-error distortion constraints. We show that simplex fingerprints maximize a geometric figure of merit for this detector. In that sense they outperform orthogonal fingerprints but the advantage vanishes as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> rarr infin. They are also optimal in terms of minimizing the probability of error of the focused detector when the attack is a uniform averaging of the marked copies followed by the addition of white Gaussian noise. Reliable detection is guaranteed provided that the number of colluders <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> Lt radic( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ). Moreover, we study the probability of error performance of simplex fingerprints for the focused correlation detector when the colluders use nonuniform averaging plus white Gaussian noise attacks.

Fingerprinting Codes for Multimedia Data against Averaging Attack

Code construction for digital fingerprinting, which is a copyright protection technique for multimedia, is considered. Digital fingerprinting should deter collusion attacks, where several fingerprinted copies of the same content are mixed to disturb their fingerprints. In this paper, we consider the averaging attack, which is known to be effective for multimedia fingerprinting with the spread spectrum technique. We propose new methods for constructing fingerprinting codes to increase the coding rate of conventional fingerprinting codes, while they guarantee to identify the same number of colluders. Due to the new fingerprinting codes, the system can deal with a larger number of users to supply digital contents.

Public Key Broadcast Encryption Schemes With Shorter Transmissions

Broadcast encryption allows a sender to securely distribute messages to a dynamically changing set of users over an insecure channel. In a public key broadcast encryption (PKBE) scheme, this encryption is performed in the public key setting, where the public key is stored in a user's device, or directly transmitted to the receivers along with ciphertexts. In this paper, we propose two PKBE schemes for stateless receivers which are transmission-efficient. A distinctive feature in our first construction is that, different than existing schemes in the literature, only a fraction of the public key related to the set of intended receivers is required in the decryption process. This feature results in the first PKBE scheme with O(r) transmission cost and O(1) user storage cost for r revoked users. Our second construction is a generalized version of the first one providing a tradeoff between ciphertext size and public key size. With appropriate parametrization, we obtain a PKBE scheme with (Oradicn) transmission cost and O(1) user storage cost for any large set of n users. The transmission cost of our second scheme is at least 30\% less than that of the recent result of Boneh et al.'s PKBE scheme, which is considered as being the current state-of-the-art. By combining the two proposed schemes, we suggest a PKBE scheme that achieves further shortened transmissions, while still maintaining O(1) user storage cost. The proposed schemes are secure against any number of colluders and do not require costly re-keying procedures followed by revocation of users.

Joint Coding and Embedding Techniques for Multimedia Fingerprinting

Digital fingerprinting protects multimedia content from illegal redistribution by uniquely marking every copy of the content distributed to each user. The collusion attack is a powerful attack where several different fingerprinted copies of the same content are combined together to attenuate or even remove the fingerprints. One major category of collusion-resistant fingerprinting employs an explicit step of coding. Most existing works on coded fingerprinting mainly focus on the code-level issues and treat the embedding issues through abstract assumptions without examining the overall performance. In this paper, we jointly consider the coding and embedding issues for coded fingerprinting systems and examine their performance in terms of collusion resistance, detection computational complexity, and distribution efficiency. Our studies show that coded fingerprinting has efficient detection but rather low collusion resistance. Taking advantage of joint coding and embedding, we propose a permuted subsegment embedding technique and a group-based joint coding and embedding technique to improve the collusion resistance of coded fingerprinting while maintaining its efficient detection. Experimental results show that the number of colluders that the proposed methods can resist is more than three times as many as that of the conventional coded fingerprinting approaches.

Anti-collusion fingerprinting for multimedia

Digital fingerprinting is a technique for identifying users who use multimedia content for unintended purposes, such as redistribution. These fingerprints are typically embedded into the content using watermarking techniques that are designed to be robust to a variety of attacks. A cost-effective attack against such digital fingerprints is collusion, where several differently marked copies of the same content are combined to disrupt the underlying fingerprints. We investigate the problem of designing fingerprints that can withstand collusion and allow for the identification of colluders. We begin by introducing the collusion problem for additive embedding. We then study the effect that averaging collusion has on orthogonal modulation. We introduce a tree-structured detection algorithm for identifying the fingerprints associated with K colluders that requires O(Klog(n/K)) correlations for a group of n users. We next develop a fingerprinting scheme based on code modulation that does not require as many basis signals as orthogonal modulation. We propose a new class of codes, called anti-collusion codes (ACCs), which have the property that the composition of any subset of K or fewer codevectors is unique. Using this property, we can therefore identify groups of K or fewer colluders. We present a construction of binary-valued ACC under the logical AND operation that uses the theory of combinatorial designs and is suitable for both the on-off keying and antipodal form of binary code modulation. In order to accommodate n users, our code construction requires only O(/spl radic/n) orthogonal signals for a given number of colluders. We introduce three different detection strategies that can be used with our ACC for identifying a suspect set of colluders. We demonstrate the performance of our ACC for fingerprinting multimedia and identifying colluders through experiments using Gaussian signals and real images.

A Collusion Problem and Its Solution

Consider a group of colluders, each with certain knowledge such as the identity of some other colluders, some cryptographic keys, and some data, possibly multiply encrypted. Two colluders can combine their knowledge if their current knowledge satisfies certain conditions. Their cryptographic keys can help decrypt each other's encrypted data, expanding their knowledge and revealing more collusion opportunities, and the process of collusion continues. The question we address is whether it is possible for the colluders to uncover a target set of unencrypted data. In this paper we formulate the collusion problem and provide an algorithm that determines whether a collusion problem has a solution and, if so, computes one. A solution is a specific way by which the colluders can uncover the hidden information. The solution generated by our algorithm is generally not one that involves the minimum number of colluders. We show, however, that to find such a solution is NP-complete. Complex communications protocols employing cryptographic building blocks are being developed to transfer information among some users and hide from others. The algorithm presented here can be applied to determine whether and how a subset of protocol users can discover during or after the protocol's execution the information that is to be hidden from them.