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First, the majority of all multiplication errors can be classified as operand-related', i.e. they are multiples of one of the operands (e.g. 8 × 4 = 28) [ 10, 11, 13].
Integer bugs span a wide range of vulnerabilities, including type mismatching and multiplication errors.
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The multiplication error, is not considered in this paper.
where is the magnitude of the per-channel multiplication error projected onto ; see [5].
The multiplication error and the addition error are bounded by the MRRE (assuming rounding mode = truncation) as given by (11) and (12): (11).
Since the bound is always verified for every, the upper bound can be computed by substituting in (16) by the standard deviation of the per-channel multiplication error.
Hence, from (4) and (5) it can be seen that the per-channel multiplication error is strictly dependent on both the filter coefficients and the host signal.
The rationale for this behavior can be found in the inner working of DFT-RDM, which is essentially an RDM-like scheme for every DFT channel, and in the influence of a nonflat psd on the per-channel multiplication error.
Due to the effects of the circular convolution, the random variable representing the th received DFT coefficient can be written as, where models the deviation from a pure multiplication (which would correspond to full-length DFTs) and so it will be referred to as per-channel multiplication error.
On the other hand, according to [5], if the host signal is white, the per-channel multiplication error is approximately independent on both the host and the watermark signal, so the correlation between neighboring channels, which usually leads to higher per-channel error probabilities, becomes small.
From the analysis of DFT-RDM for colored hosts developed in Section 4, any colored host will have unavoidably different watermark signal powers for different DFT channels; consequently, there will be some DFT channels more exposed than others to the per-channel multiplication error, as it has been explained above.
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