3.4 Finding the Keyspace for the SVD Method
3.4.3 Experiment 1 - Range of Perturbations
The range of perturbations which can be applied to the matrices ˜G while fulfilling require-ments a) and b) above are examined.
10−20 10−15 10−10 10−5 100
Figure 3.3: Err over the astronomical image set against the value of ǫ in equation (3.12), applied at the folding stage. The hidden coefficients are securely hidden by adding the perturbation ǫ to the first element of ˜G in (3.12) and then recovered without applying a perturbation. A single standard deviation from this mean is shown above and below the Err by the error bars. The dashes ⊳ indicate where perturbations smaller than ǫmin occur.
The Experiments described in this Section were performed with a single perturbation ǫ added to element ˜G(1, 1), when folding each image in the test set.
To examine the effect of perturbations smaller than ǫmin ≈ 4.44 × 10−16, the first Experiment will apply a minimum perturbation of 10−20. The maximum perturbation size for this Experiment will be 10 (chosen as it is far in excess of what could be considered to be a perturbation of the ˜G). The set E of perturbations chosen for Experiment 1 is therefore
E = {10i}1i=−20.
In addition to the two standard simulations designed to investigate incorrect and cor-rect recovery from the folding procedure, an additional simulation is performed. This is to simulate a third party trying to guess the correct perturbation, by examining how close their guess has to be.
Simulation 1
Simulation of a third party attempting to recover the hidden portion of an image, without knowledge of the private key. For each ǫ ∈ E, added with equation (3.11), the images were folded. The perturbation, applied with equation (3.11) at the recovery stage, is denoted by ǫR, and for this Experiment is ǫR= 0.
Figure 3.3 shows the Err in the recovery of the hidden coefficients, for each ǫ ∈ E applied at the folding stage. For ǫ > ǫmin, displayed on Figure 3.3 to the right of the ⊳ symbol, the Err > 120%.
An Err greater than 100% for all ǫ > ǫmin indicates that, as expected the instability in the calculation of the vectors spanning the space V⊥, is sensitive to all perturbations greater than ǫmin. Thus under these test conditions, any perturbation greater than ǫmin
can prevent unauthorized access to the hidden coefficients e.
Figure 3.3 shows that for ǫ < 10−18 the Err < 10−12%. Thus ǫ < 10−18 applied at the folding stage, does not prevent a third party, without knowledge of the correct perturbation, from recovering the hidden coefficients.
This simulation indicates that, all ǫ > ǫmin in equation (3.11), fulfil requirement a) for the astronomical image set. Because of this and because ǫ ≤ 10−18 did not prevent recovery of the hidden coefficients ˜e, the set of test perturbations was restricted further to
E1 = {10i}1i=−16. (3.20)
The main focus now being on perturbations greater than 10−15.
Remark 4. Notice that in Figure 3.3 there are three data points to the left of the ⊳ symbol, where the Err > 120%. This indicates that correct recovery of the hidden coefficients ˜e, is also prevented by perturbations less than ǫmin. This is because the maximum value of any element of the matrices G can be smaller than 3, reducing the size of the minimum perturbation ǫmin, which is guaranteed to perturb the matrix.
Simulation 2
To examine which perturbation sizes in E fulfil requirement b), all the images were folded and recovered with the same perturbation ǫ ∈ E1, applied to ˜G(1, 1) in equation (3.12).
The results plotted in Figure 3.4 show that for perturbations ǫ = 10i, i = −16, . . . , −12, the recovered images ˜IK are identical to the approximated images IK. In other words the approximated image IK is not numerically altered by folding and recovery procedure.
All perturbations ǫ > 10−11 introduce error to ˜IK, which increases with the perturba-tion size ǫ. This is shown in Figure 3.4 by the δPSNR. More importantly for all ǫ ≤ 10−3 the δPSNR < 2 × 10−4%, less than the maximum of x = 1 × 10−2%. Additionally at this level their is no visual difference between the approximated image IK and the recovered image ˜IK. Therefore the results of this simulation indicate that ǫ ≤ 10−3 in equation (3.12) is sufficient to fulfil requirement b) for the astronomical image set.
Simulation 3
Simulation 1 demonstrated that any perturbation ǫ ≥ ǫmin applied at the folding stage, can prevent a third party accessing the vectors of hidden coefficients e, if ǫ = 0 at the
10−10 10−8 10−6 10−4 10−2 100 10−10
10−5 100
ε PSNRE%
PSNRE%
Figure 3.4: δPSNR over the astronomical image set against the value of ǫ in equation (3.12), applied at the folding and recovery stage. The images are folded and recovered by adding ǫ to element ˜G(1, 1) in (3.12). ǫ < 10−11 is not shown on the Figure because this resulted in a δPSNR = 0. A single standard deviation is shown above and below the δPSNR by the error bars, where the end of the error bar is not shown, if this single standard deviation below the mean is less than or equal to zero.
recovery stage. The aim of this simulation is to determine if a difference of ǫmin, between the perturbation applied at the folding and recovery stages, is sufficient to prevent recovery of the hidden coefficients.
To examine this for each ǫ ∈ E1 from (3.20), the images were folded with ǫ, and then recovered with ǫR= ǫ + ǫmin.
For each ǫ ∈ E1, the resulting Err over the image set was greater than 123.42% with a standard deviation less than 17.62%. This indicates that for all ǫ ∈ E1, a perturbation differing by as little as ǫmin from the correct one, can prevent correct recovery of the hidden coefficients.
Discussion
The results of simulation 1 in Figure 3.3 show, requirement a) is satisfied by all discrete perturbations
E1 = {10i}1i=−16. (3.21)
That is, any ǫ ∈ E1 applied in equation (3.12) when the images were folded, prevented recovery of the hidden coefficients ˜eq, q = 1, . . . , QN, when ǫR= 0.
Further to this, simulation 3 showed this result to hold, even when the recovery pertur-bation was only ǫmin different, to the perturbation applied at the folding stage. To verify this result, in the remaining Experiments all simulations designed to verify requirement a), will apply a perturbation of ǫR= ǫ + ǫmin, when performing the recovery.
The results of simulation 2 in Figure 3.4 show requirement b) is satisfied by ǫ =
10i, i = −16, . . . , −3. Therefore a set of perturbations E2, fulfilling requirements a) and b) can be proposed. For the astronomical images processed in blocks with N = 8, initially approximated to a PSNRa= 45 ± 4.5 × 10−3dB, this set is,
E2 = {10i}−3i=−16. (3.22)