Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1312.3200

Constrained Colluding Eavesdroppers: An Information-Theoretic Model

Mahtab Mirmohseni, Panos Papadimitratos

Published Dec 11, 2013• 2 citations
arXivHTMLPDF

Abstract

We study the secrecy capacity in the vicinity of colluding eavesdroppers. Contrary to the perfect collusion assumption in previous works, our new information-theoretic model considers constraints in collusion. We derive the achievable secure rates (lower bounds on the perfect secrecy capacity), both for the discrete memoryless and Gaussian channels. We also compare the proposed rates to the non-colluding and perfect colluding cases.

DOI
10.3929/ethz-a-010089041

Related papers

On the effect of colluding eavesdroppers on secrecy capacity scaling

O. Ozan Koyluoglu, C. Emre Koksal, Hesham El Gamal

In a powerful secrecy attack, eavesdroppers can collude, i.e., they can share their observations. Securing information in such a scenario will be an even more challenging task compared to non-colluding case. We here analyze the effect of eavesdropper collusion on the achievable performance in both the path loss and ergodic multi-path fading models. We provide two results: 1) For the Poisson point process model in a random extended network, if the legitimate nodes have unit intensity (λ = 1) and the colluding eavesdroppers have an intensity of λe= O((log n)-(2+p)) for any p > 0, almost all of the nodes can achieve a secure rate of Ω(1/(√n)); I and 2) In the K-user Gaussian interference channel with E external colluding eavesdroppers, a secure degrees of freedom of η =[1/2-E/K]+per frequency-time slot is achievable for each user in the ergodic setting (in the absence of the eavesdropper channel state information).

European Wireless Conference•Jan 1, 2010•12 citations

Impact of secrecy on capacity in large-scale wireless networks

Jinbei Zhang, Luoyi Fu, Xinbing Wang

Since wireless channel is vulnerable to eavesdroppers, the secrecy during message delivery is a major concern in many applications such as commercial, governmental and military networks. This paper investigates information-theoretic secrecy in large-scale networks and studies how capacity is affected by the secrecy constraint where the locations and channel state information (CSI) of eavesdroppers are both unknown. We consider two scenarios: 1) non-colluding case where eavesdroppers can only decode messages individually; and 2) colluding case where eavesdroppers can collude to decode a message. For the non-colluding case, we show that the network secrecy capacity is not affected in order-sense by the presence of eavesdroppers. For the colluding case, the per-node secrecy capacity of Θ(1/√n) can be achieved when the eavesdropper density ψe(n) is O(n-β), for any constant β >; 0 and decreases monotonously as the density of eavesdroppers increases. The upper bounds on network secrecy capacity are derived for both cases and shown to be achievable by our scheme when ψe(n) = O(n-β) or ψe(n) = Ω(log α-2/α n), where α is the path loss gain. We show that there is a clear tradeoff between the security constraints and the achievable capacity.

IEEE Conference on Computer Communications (INFOCOM)•Mar 1, 2012•13 citations

Asymptotic Secrecy Analysis of Random Networks With Colluding Eavesdroppers

Satyanarayana Vuppala, Giuseppe Thadeu Freitas de Abreu

We perform an analysis of the secrecy outage and transmission capacity of random networks under Nakagami-m fading with colluding eavesdroppers, in the asymptotic case of large reference signal-to-noise ratios. In literature, secrecy analysis from the perspective of collusion pool has still not been well characterized. Considering a network model that accounts for uncertainties both in node locations and channel coefficients, we approximate the distribution of the aggregate path gain received by a collusion of eavesdroppers. Using this result, the asymptotic secrecy rate distribution, the secrecy outage probability, and the secrecy transmission capacity of random networks with colluding eavesdroppers are obtained, with basic factors such as density of nodes, the guarding distance, and the fading figure all accounted for by explicit parameters. Since secrecy in low-outage random networks are found to be most significant when the density of active nodes is small, the expressions derived can be seen as upper-bounds on the three aforementioned secrecy performance metrics in random networks, offering particular insight on the impact of fading, density, and guarding distances, while taking into account cooperation among eavesdroppers. Furthermore, we give a bound on colluding pool size, which is pragmatic in creating a nonprotected communication zone for the typical destination.

IEEE Systems Journal•Apr 29, 2016•5 citations

On the individual secrecy for Gaussian broadcast channels with receiver side information

Yanling Chen, O. Ozan Koyluoglu, Aydin Sezgin

This paper studies the individual secrecy capacity region of the broadcast channel with receiver side information. First, an achievable rate region is established for the discrete memoryless case by employing superposition coding. Further, it is extended to the corresponding Gaussian case, where the individual secrecy capacity region is characterized in case of a weak or strong eavesdropper (compared to two legitimate receivers). For the case left, inner and outer bounds are established and the individual secrecy capacity region is characterized for the low and high SNR regimes. Note that the last case is distinctive due to the individual secrecy constraint, in the sense that positive rate pair is still possible although the eavesdropper may have the advantage against at least one of the legitimate receivers over the channel, unlike the situation if the joint secrecy constraint is imposed.

•Jun 1, 2015•5 citations

Information-Theoretic Key Agreement of Multiple Terminals—Part II: Channel Model

Amin Gohari, Venkat Anantharam

This is the second part of a two-part paper on information-theoretically secure secret key agreement. This part covers the secret key capacity under the channel model. In this model, multiple terminals wish to create a shared secret key that is secure from an eavesdropper with unlimited computational resources. The terminals are all connected to a noiseless and authenticated but insecure channel, called the “public channel.” Furthermore, the terminals have access to a secure but noisy discrete memoryless broadcast channel (DMBC). The first terminal can choose a sequence of inputs to the DMBC, which has outputs at the other terminals and at the eavesdropper. After each channel use, the terminals can engage in arbitrarily many rounds of interactive authenticated communication over the public channel. At the end, each legitimate terminal should be able to generate the secret key. In this paper, we derive new lower and upper bounds on the secrecy capacity. In each case, an example is provided to show that the new bound represents a strict improvement over the previously best known bound. This part of the paper is not standalone, and is written under the assumption that the reader has access to Part I, which is published in the same issue.

IEEE Transactions on Information Theory•Jul 21, 2010•74 citations

Information-theoretic Secrecy in Multi-user Channels

Ersen Ekrem

Inherent openness of the wireless medium imposes stronger challenges on the security of wireless communications. Information-theoretic security addresses these challenges at the physical layer by using tools from wireless communication theory, signal processing and information theory. In information-theoretic security, physical layer communication is intelligently designed to exploit the characteristics of the wireless medium, such as fading, interference, cooperation, and multi-dimensional signaling, in order to provide or improve security. In this dissertation, we study the security of several fundamental wireless network configurations from an information-theoretic perspective. First, we study the Gaussian multiple-input multiple-output (MIMO) wiretap channel. In this channel, the transmitter sends a common message to both the legitimate user and the eavesdropper. In addition to the common message, a private message is sent only to the legitimate user, which needs to be kept hidden as much as possible from the eavesdropper. We obtain the entire capacity-equivocation region for this channel model. In particular, we show the sufficiency of jointly Gaussian auxiliary random variables and channel input to evaluate the existing single-letter description of the capacity-equivocation region due to Csiszar-Korner. Next, we study the secure broadcasting problem, where a transmitter wants to have secure communication with multiple legitimate users in the presence of an external eavesdropper. We study several special cases of the secure broadcasting problem. First, we consider the degraded multi-receiver wiretap channel, and establish its secrecy capacity region. Second, we consider the parallel less noisy multi-receiver wiretap channel, and obtain its common message secrecy capacity and sum secrecy capacity. Third, we consider the parallel degraded multi-receiver wiretap channel for the two-user and two-sub-channel case, and obtain its entire secrecy capacity region. Finally, we consider a parallel channel model with two sub-channels, where the transmitter can use only one of the subchannels at any time, and characterize its secrecy capacity region. Then, we study the two-user Gaussian MIMO broadcast channel with common and confidential messages. In this channel model, the transmitter sends a common message to both users, and a confidential message to each user which needs to be kept perfectly secret from the other user. We obtain the entire capacity region of this channel. We also explore the connections between this channel model and its non-confidential counterpart, i.e., the Gaussian MIMO broadcast channel with common and private message. Next, we consider the Gaussian MIMO multi-receiver wiretap channel and obtain its secrecy capacity region for the most general case. We first show that even for the single-input single-output (SISO) case, existing converse techniques fall short of proving the secrecy capacity region, to emphasize the need for a new proof technique, which we develop by using the relationships between the Fisher information and the differential entropy. Using this new proof technique, we obtain the secrecy capacity region of the degraded MIMO channel. We then establish the secrecy capacity region of the general MIMO channel by using the channel enhancement technique in conjunction with the capacity result we obtained for the degraded MIMO channel. For the general MIMO channel, we show that dirty-paper coding (DPC) combined with stochastic encoding attains the entire secrecy capacity region. Then, we study the multi-receiver wiretap channel for a more general scenario, where, in addition to confidential messages, the transmitter sends public messages to the legitimate users, on which there are no secrecy constraints. First, we consider the degraded discrete memoryless channel, and obtain inner and outer bounds for the capacity region. These inner and outer bounds match for certain cases, providing the capacity region. Second, we obtain an inner bound for the general discrete memoryless channel by using Marton's inner bound. Third, we consider the degraded Gaussian MIMO channel, and show that jointly Gaussian auxiliary random variables and channel input are sufficient to exhaust the inner and outer bounds. Finally, we provide an inner bound for the capacity region of the general Gaussian MIMO channel. Next, we focus on the multiple access wiretap (MAC-WT) channel whose capacity region is unknown. We consider a special class of MAC-WT channels which we call the weak eavesdropper class, where each user's link to the legitimate receiver is stronger than its link to the eavesdropper. For this class of channels, we develop an outer bound for the secrecy capacity region, which partially matches the achievable region in an n-letter form. We evaluate a looser version of our outer bound for the Gaussian case, and show that our outer bound is within 0.5 bits/channel use of the achievable rates along the individual secrecy rates for all weak eavesdropper Gaussian MAC-WT. Then, we investigate the effects of user cooperation on the secrecy of broadcast channels by considering the cooperative relay broadcast channel (CRBC). We propose an achievable scheme that combines Marton's coding scheme for broadcast channels and Cover and El Gamal's compress-and-forward (CAF) scheme for relay channels. For the Gaussian CRBC, we show that both users can have positive secrecy rates, which is not possible for scalar Gaussian broadcast channels without cooperation. We further investigate the effects of user cooperation on secrecy by considering the multiple access channel with generalized feedback (MAC-GF), which can be viewed as the MAC-dual of the CRBC. We propose a CAF-based achievable secrecy rate region for the MAC-GF. Specializing our results to a Gaussian MAC-GF, we present numerical results which demonstrate that cooperation can improve secrecy for the MAC-GF. Next, we study the two-user one-eavesdropper discrete memoryless compound wiretap channel, and provide the best known lower bound for the secrecy capacity of this compound channel. We evaluate this achievable secrecy rate for the Gaussian MIMO case by using DPC. We show that this achievable secrecy rate achieves at least half of the secrecy capacity of this Gaussian MIMO compound wiretap channel, and also attains the secrecy capacity when the eavesdropper is degraded with respect to one of the two users. Then, we study the degraded compound multi-receiver wiretap channel (DCMRWC), which, in addition to a group of eavesdroppers, has two groups of users, namely the stronger group and the weaker group. We study two different communication scenarios for this channel. In the first scenario, there is only one eavesdropper, and the transmitter sends a confidential message to each group of legitimate users while keeping both messages secret from the eavesdropper. In the second scenario, we study the DCMRWC with layered messages without any restriction on the number of eavesdroppers. For both scenarios, we obtain the secrecy capacity region for the discrete memoryless channel, the parallel channel, and the Gaussian parallel channel. For the Gaussian MIMO channel, we obtain the secrecy capacity region when there is only one user in the second group. Next, we study the two-user fading broadcast channel and obtain its ergodic secrecy capacity region. We show that, thanks to fading, both users can have simultaneous secure communication with the transmitter, although this is not possible in the scalar non-fading Gaussian broadcast channel where only one user can have secure communication. This simultaneous secrecy of both users is achieved by an opportunistic communication scheme, in which, at each time instant, the transmitter communicates with the user having a better channel gain. Then, we study the secure lossy transmission of a vector Gaussian source to a legitimate user in the presence of an eavesdropper, where both the legitimate user and the eavesdropper have vector Gaussian side information. We obtain an outer bound for the rate, equivocation and distortion region. Moreover, we obtain the maximum equivocation at the eavesdropper when there is no constraint on the transmission rate. By using this maximum equivocation result, we show two facts. First, for this problem, in general, Wyner-Ziv scheme is suboptimal, although, it is optimal in the absence of an eavesdropper. And, second, even when there is no transmission rate constraint, an uncoded transmission scheme is suboptimal; the presence of an eavesdropper necessitates the use of a coded scheme to attain the maximum equivocation. Finally, we revisit the secure lossy source coding problem. In all works on this problem, either the equivocation of the source at the eavesdropper or the equivocation of the legitimate user's reconstruction of the source at the eavesdropper is used to measure secrecy. We first propose the relative equivocation of the source at the eavesdropper with respect to the legitimate user as a new secrecy measure. We argue that this new secrecy measure is the one that corresponds to the natural generalization of the equivocation in a wiretap channel to the context of secure lossy source coding. Under this new secrecy measure, we provide a single-letter description of the rate, relative equivocation and distortion region, as well as its specializations to degraded and reversely degraded cases. We investigate the relationships between the optimal scheme that attains this region and the Wyner-Ziv scheme.

•Jan 1, 2012•0 citations
We study the secrecy capacity in the vicinity of colluding eavesdroppers. Contrary to the perfect collusion assumption in previous works, our new information-theoretic model considers constraints in collusion. We derive the achievable secure rates (lower bounds on the perfect secrecy capacity), both for the discrete memoryless and Gaussian channels. We also compare the proposed rates to the non-colluding and perfect colluding cases.
In a powerful secrecy attack, eavesdroppers can collude, i.e., they can share their observations. Securing information in such a scenario will be an even more challenging task compared to non-colluding case. We here analyze the effect of eavesdropper collusion on the achievable performance in both the path loss and ergodic multi-path fading models. We provide two results: 1) For the Poisson point process model in a random extended network, if the legitimate nodes have unit intensity (λ = 1) and the colluding eavesdroppers have an intensity of λe= O((log n)-(2+p)) for any p > 0, almost all of the nodes can achieve a secure rate of Ω(1/(√n)); I and 2) In the K-user Gaussian interference channel with E external colluding eavesdroppers, a secure degrees of freedom of η =[1/2-E/K]+per frequency-time slot is achievable for each user in the ergodic setting (in the absence of the eavesdropper channel state information).
Since wireless channel is vulnerable to eavesdroppers, the secrecy during message delivery is a major concern in many applications such as commercial, governmental and military networks. This paper investigates information-theoretic secrecy in large-scale networks and studies how capacity is affected by the secrecy constraint where the locations and channel state information (CSI) of eavesdroppers are both unknown. We consider two scenarios: 1) non-colluding case where eavesdroppers can only decode messages individually; and 2) colluding case where eavesdroppers can collude to decode a message. For the non-colluding case, we show that the network secrecy capacity is not affected in order-sense by the presence of eavesdroppers. For the colluding case, the per-node secrecy capacity of Θ(1/√n) can be achieved when the eavesdropper density ψe(n) is O(n-β), for any constant β >; 0 and decreases monotonously as the density of eavesdroppers increases. The upper bounds on network secrecy capacity are derived for both cases and shown to be achievable by our scheme when ψe(n) = O(n-β) or ψe(n) = Ω(log α-2/α n), where α is the path loss gain. We show that there is a clear tradeoff between the security constraints and the achievable capacity.
We perform an analysis of the secrecy outage and transmission capacity of random networks under Nakagami-m fading with colluding eavesdroppers, in the asymptotic case of large reference signal-to-noise ratios. In literature, secrecy analysis from the perspective of collusion pool has still not been well characterized. Considering a network model that accounts for uncertainties both in node locations and channel coefficients, we approximate the distribution of the aggregate path gain received by a collusion of eavesdroppers. Using this result, the asymptotic secrecy rate distribution, the secrecy outage probability, and the secrecy transmission capacity of random networks with colluding eavesdroppers are obtained, with basic factors such as density of nodes, the guarding distance, and the fading figure all accounted for by explicit parameters. Since secrecy in low-outage random networks are found to be most significant when the density of active nodes is small, the expressions derived can be seen as upper-bounds on the three aforementioned secrecy performance metrics in random networks, offering particular insight on the impact of fading, density, and guarding distances, while taking into account cooperation among eavesdroppers. Furthermore, we give a bound on colluding pool size, which is pragmatic in creating a nonprotected communication zone for the typical destination.
This paper studies the individual secrecy capacity region of the broadcast channel with receiver side information. First, an achievable rate region is established for the discrete memoryless case by employing superposition coding. Further, it is extended to the corresponding Gaussian case, where the individual secrecy capacity region is characterized in case of a weak or strong eavesdropper (compared to two legitimate receivers). For the case left, inner and outer bounds are established and the individual secrecy capacity region is characterized for the low and high SNR regimes. Note that the last case is distinctive due to the individual secrecy constraint, in the sense that positive rate pair is still possible although the eavesdropper may have the advantage against at least one of the legitimate receivers over the channel, unlike the situation if the joint secrecy constraint is imposed.
This is the second part of a two-part paper on information-theoretically secure secret key agreement. This part covers the secret key capacity under the channel model. In this model, multiple terminals wish to create a shared secret key that is secure from an eavesdropper with unlimited computational resources. The terminals are all connected to a noiseless and authenticated but insecure channel, called the “public channel.” Furthermore, the terminals have access to a secure but noisy discrete memoryless broadcast channel (DMBC). The first terminal can choose a sequence of inputs to the DMBC, which has outputs at the other terminals and at the eavesdropper. After each channel use, the terminals can engage in arbitrarily many rounds of interactive authenticated communication over the public channel. At the end, each legitimate terminal should be able to generate the secret key. In this paper, we derive new lower and upper bounds on the secrecy capacity. In each case, an example is provided to show that the new bound represents a strict improvement over the previously best known bound. This part of the paper is not standalone, and is written under the assumption that the reader has access to Part I, which is published in the same issue.
Inherent openness of the wireless medium imposes stronger challenges on the security of wireless communications. Information-theoretic security addresses these challenges at the physical layer by using tools from wireless communication theory, signal processing and information theory. In information-theoretic security, physical layer communication is intelligently designed to exploit the characteristics of the wireless medium, such as fading, interference, cooperation, and multi-dimensional signaling, in order to provide or improve security. In this dissertation, we study the security of several fundamental wireless network configurations from an information-theoretic perspective. First, we study the Gaussian multiple-input multiple-output (MIMO) wiretap channel. In this channel, the transmitter sends a common message to both the legitimate user and the eavesdropper. In addition to the common message, a private message is sent only to the legitimate user, which needs to be kept hidden as much as possible from the eavesdropper. We obtain the entire capacity-equivocation region for this channel model. In particular, we show the sufficiency of jointly Gaussian auxiliary random variables and channel input to evaluate the existing single-letter description of the capacity-equivocation region due to Csiszar-Korner. Next, we study the secure broadcasting problem, where a transmitter wants to have secure communication with multiple legitimate users in the presence of an external eavesdropper. We study several special cases of the secure broadcasting problem. First, we consider the degraded multi-receiver wiretap channel, and establish its secrecy capacity region. Second, we consider the parallel less noisy multi-receiver wiretap channel, and obtain its common message secrecy capacity and sum secrecy capacity. Third, we consider the parallel degraded multi-receiver wiretap channel for the two-user and two-sub-channel case, and obtain its entire secrecy capacity region. Finally, we consider a parallel channel model with two sub-channels, where the transmitter can use only one of the subchannels at any time, and characterize its secrecy capacity region. Then, we study the two-user Gaussian MIMO broadcast channel with common and confidential messages. In this channel model, the transmitter sends a common message to both users, and a confidential message to each user which needs to be kept perfectly secret from the other user. We obtain the entire capacity region of this channel. We also explore the connections between this channel model and its non-confidential counterpart, i.e., the Gaussian MIMO broadcast channel with common and private message. Next, we consider the Gaussian MIMO multi-receiver wiretap channel and obtain its secrecy capacity region for the most general case. We first show that even for the single-input single-output (SISO) case, existing converse techniques fall short of proving the secrecy capacity region, to emphasize the need for a new proof technique, which we develop by using the relationships between the Fisher information and the differential entropy. Using this new proof technique, we obtain the secrecy capacity region of the degraded MIMO channel. We then establish the secrecy capacity region of the general MIMO channel by using the channel enhancement technique in conjunction with the capacity result we obtained for the degraded MIMO channel. For the general MIMO channel, we show that dirty-paper coding (DPC) combined with stochastic encoding attains the entire secrecy capacity region. Then, we study the multi-receiver wiretap channel for a more general scenario, where, in addition to confidential messages, the transmitter sends public messages to the legitimate users, on which there are no secrecy constraints. First, we consider the degraded discrete memoryless channel, and obtain inner and outer bounds for the capacity region. These inner and outer bounds match for certain cases, providing the capacity region. Second, we obtain an inner bound for the general discrete memoryless channel by using Marton's inner bound. Third, we consider the degraded Gaussian MIMO channel, and show that jointly Gaussian auxiliary random variables and channel input are sufficient to exhaust the inner and outer bounds. Finally, we provide an inner bound for the capacity region of the general Gaussian MIMO channel. Next, we focus on the multiple access wiretap (MAC-WT) channel whose capacity region is unknown. We consider a special class of MAC-WT channels which we call the weak eavesdropper class, where each user's link to the legitimate receiver is stronger than its link to the eavesdropper. For this class of channels, we develop an outer bound for the secrecy capacity region, which partially matches the achievable region in an n-letter form. We evaluate a looser version of our outer bound for the Gaussian case, and show that our outer bound is within 0.5 bits/channel use of the achievable rates along the individual secrecy rates for all weak eavesdropper Gaussian MAC-WT. Then, we investigate the effects of user cooperation on the secrecy of broadcast channels by considering the cooperative relay broadcast channel (CRBC). We propose an achievable scheme that combines Marton's coding scheme for broadcast channels and Cover and El Gamal's compress-and-forward (CAF) scheme for relay channels. For the Gaussian CRBC, we show that both users can have positive secrecy rates, which is not possible for scalar Gaussian broadcast channels without cooperation. We further investigate the effects of user cooperation on secrecy by considering the multiple access channel with generalized feedback (MAC-GF), which can be viewed as the MAC-dual of the CRBC. We propose a CAF-based achievable secrecy rate region for the MAC-GF. Specializing our results to a Gaussian MAC-GF, we present numerical results which demonstrate that cooperation can improve secrecy for the MAC-GF. Next, we study the two-user one-eavesdropper discrete memoryless compound wiretap channel, and provide the best known lower bound for the secrecy capacity of this compound channel. We evaluate this achievable secrecy rate for the Gaussian MIMO case by using DPC. We show that this achievable secrecy rate achieves at least half of the secrecy capacity of this Gaussian MIMO compound wiretap channel, and also attains the secrecy capacity when the eavesdropper is degraded with respect to one of the two users. Then, we study the degraded compound multi-receiver wiretap channel (DCMRWC), which, in addition to a group of eavesdroppers, has two groups of users, namely the stronger group and the weaker group. We study two different communication scenarios for this channel. In the first scenario, there is only one eavesdropper, and the transmitter sends a confidential message to each group of legitimate users while keeping both messages secret from the eavesdropper. In the second scenario, we study the DCMRWC with layered messages without any restriction on the number of eavesdroppers. For both scenarios, we obtain the secrecy capacity region for the discrete memoryless channel, the parallel channel, and the Gaussian parallel channel. For the Gaussian MIMO channel, we obtain the secrecy capacity region when there is only one user in the second group. Next, we study the two-user fading broadcast channel and obtain its ergodic secrecy capacity region. We show that, thanks to fading, both users can have simultaneous secure communication with the transmitter, although this is not possible in the scalar non-fading Gaussian broadcast channel where only one user can have secure communication. This simultaneous secrecy of both users is achieved by an opportunistic communication scheme, in which, at each time instant, the transmitter communicates with the user having a better channel gain. Then, we study the secure lossy transmission of a vector Gaussian source to a legitimate user in the presence of an eavesdropper, where both the legitimate user and the eavesdropper have vector Gaussian side information. We obtain an outer bound for the rate, equivocation and distortion region. Moreover, we obtain the maximum equivocation at the eavesdropper when there is no constraint on the transmission rate. By using this maximum equivocation result, we show two facts. First, for this problem, in general, Wyner-Ziv scheme is suboptimal, although, it is optimal in the absence of an eavesdropper. And, second, even when there is no transmission rate constraint, an uncoded transmission scheme is suboptimal; the presence of an eavesdropper necessitates the use of a coded scheme to attain the maximum equivocation. Finally, we revisit the secure lossy source coding problem. In all works on this problem, either the equivocation of the source at the eavesdropper or the equivocation of the legitimate user's reconstruction of the source at the eavesdropper is used to measure secrecy. We first propose the relative equivocation of the source at the eavesdropper with respect to the legitimate user as a new secrecy measure. We argue that this new secrecy measure is the one that corresponds to the natural generalization of the equivocation in a wiretap channel to the context of secure lossy source coding. Under this new secrecy measure, we provide a single-letter description of the rate, relative equivocation and distortion region, as well as its specializations to degraded and reversely degraded cases. We investigate the relationships between the optimal scheme that attains this region and the Wyner-Ziv scheme.