Fingerprinting by Design: Embedding and Authentication

Abstract

In this chapter we consider the design of fingerprints for the purpose of authenticating a message. We begin with a background discussion of fingerprinting and related ideas, progressing to a communications point of view. Fingerprint embedding for message authentication is motivated by the desire to make an authentication tag less accessible to an eavesdropper. We consider metrics for good fingerprint design, and apply these to develop an embedding scheme for wireless communications. Wireless software defined radio experiments validate the theory and demonstrate the applicability of our approach.

Appendix: Precoding and Power-Allocation with CSI

Alice can improve the performance of the system by shaping her transmissions based on her available CSI. Generally, the frame can be decomposed as

$$\begin{aligned} \mathbf {X} = {{\mathbf {F}_\text {S}}}{{\mathbf {P}_\text {S}}}^{\frac{1}{2}} \mathbf {S} \end{aligned}$$

(30)

where ${{\mathbf {F}_\text {S}}}$ is an $M \times M$ unitary matrix, ${{\mathbf {P}_\text {S}}}$ is an $M \times M$ diagonal matrix that allocates power between the columns of ${{\mathbf {F}_\text {S}}}$, and $\mathbf {S}$ is the modulated and possibly coded $M \times L$ data matrix. In general, to achieve optimality as described below, Alice allocates energy among the eigenvectors of either the channel covariance or its expectation. That is, the columns of ${{\mathbf {F}_\text {S}}}$ are the channel eigenvectors, and the entries of ${{\mathbf {P}_\text {S}}}$ allocate the transmission energy between them. The total power budget is constrained by

$$\begin{aligned} \text {Tr}({{\mathbf {P}_\text {S}}}) = M \; . \end{aligned}$$

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In the following we consider three cases where Alice has (1) no CSI, (2) perfect CSI, or (3) knowledge of the statistics of the channel. We briefly review the capacity-optimal precoding and power allocation strategies for each case.

1.1 No CSI

When the transmitter has no CSI, e.g., in the absence of feedback from the receiver, then there are no preferred transmission modes and transmission is isotropic, so that

$$\begin{aligned} {{\mathbf {F}_\text {S}}}= & {} \mathbf {I}\end{aligned}$$

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$$\begin{aligned} {{\mathbf {P}_\text {S}}}= & {} \mathbf {I} \end{aligned}$$

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resulting in $\mathbf {\Phi } = \mathbf {I}$.

1.2 Perfect CSI

In this case the transmitter has knowledge of the realization of $\mathbf H$, and the capacity-achieving channel input covariance $\mathbf {\Phi }$ has eigenvectors equal to those of $\mathbf {H}^\dagger \mathbf {H}$. Because the eigenvectors are orthogonal, the optimal power allocation is given by the water-filling solution [23]. That is, the transmissions are shaped using

$$\begin{aligned} {{\mathbf {F}_\text {S}}}= & {} \mathbf {V} \end{aligned}$$

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$$\begin{aligned} P_\text {S}(i)= & {} (\nu - n(i))^+ \; ,\end{aligned}$$

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$$\begin{aligned} \text {where } \mathbf {H}^\dagger \mathbf {H}= & {} \mathbf {V D V^\dagger } \; . \end{aligned}$$

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Here $P_\text {S}(i)$ (resp., D(i)) is the ith element on the diagonal of ${{\mathbf {P}_\text {S}}}$ (resp., $\mathbf {D}$), $n(i) = \sigma _w^2/D(i)$ is the ith channel noise component, and $\nu$ is chosen to satisfy the power constraint

$$\begin{aligned} \sum _{i=1}^{M} P_\text {S}(i) = M \; . \end{aligned}$$

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In Rayleigh fading ($K=0 \Rightarrow {\bar{\mathbf {H}} = 0}$), we have ${{\mathbf {F}_\text {S}}}= \mathbf {U}_T$.

1.3 Statistical CSI

Although not as good as precise knowledge of the realization of $\mathbf {H}$, when the transmitter has knowledge of the Gaussian channel statistics (mean and covariance), she is still able to improve beyond isotropic transmissions. Conditioned on the knowledge of the channel statistics, the capacity-achieving channel input has eigenvectors equal to those of $E[\mathbf {H}^\dagger \mathbf {H}]$ [25]. That is, the transmissions are shaped using

$$\begin{aligned} {{\mathbf {F}_\text {S}}}= & {} \mathbf {V}\end{aligned}$$

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$$\begin{aligned} \text {where } E[{\mathbf {H}}^\dagger \mathbf {H}]= & {} {\mathbf {VDV}}^\dagger \; . \end{aligned}$$

(39)

In Rayleigh fading ($K=0$), we have ${{\mathbf {F}_\text {S}}}= \mathbf {U}_T$.

We note that this power allocation does not correspond to a water-filling solution. When the transmitter does not know $\mathbf {H}$, precoding the input with ${{\mathbf {F}_\text {S}}}$ does not yield orthogonal channels because energy spills across eigenmodes. Thus in the case of statistical CSI application of water-filling does not yield an optimal solution. An efficient iterative algorithm to determine the optimal ${{\mathbf {P}_\text {S}}}$ is given in [16].