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It is somewhat ironic that the very technique that leads t o a devastating attack on the knapsack can, in a slightly modified setting, become the basis for constructing a public key system It is worth noting that there is one significant difference between the NTRU lattice problem in (611) and the knapsack lattice problem considered in Section 62 The successful knapsack attack breaks a single message, but it does not enable the attackcr to recover the private key However, in (611) we are trying to recover the private key from the public key, and, intuitively, this should be a much harder problem So it might seem to be unfair to compare this NTRU lattice problcni to the knapsack lattice problem It is possible to give a lattice reduction attack on a single NTRU message, which is more analogous to the knapsack setting.



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Similarly, the path of waves reflected from the object and traveling back to the GPR receiver consists of two parts, w3 through the soil and w4 through the air, with corresponding angles 3 and 4 A detailed analysis of this situation and corresponding complex calculations can be found in [11] In the following, we introduce some simplifications of the geometry of GPR data acquisition, leading to simplified calculations that can be found in, eg [32] There are several arguments in favor of these simplifications: the main interest in GPR is for subsurface imaging; it is easy to determine the air/ground interface on A- or B-scans, due to the strong reflection at such interfaces; GPR antennas often operate very close to the ground; the distance between T and R is often negligible in comparison with other distances.

var localSorter:Collator = new Collator(LocaleID.DEFAULT, CollatorMode.SORTING);





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Sorting mode is selected by default if the argument is omitted. The initial mode does nothing more than initially set properties of the Collator that affect comparison. These may all be set after construction time as well, and include

However, the NTRU lattice reduction attack is intractable (as far as is known), even in this seemingly simpler.

ignoreCase When true, a lowercase letter and its capital are considered equivalent. ignoreCharacterWidth When true, half-width versions of Japanese and Chinese characters are treated identically to their full-width versions. ignoreDiacritics When true, accents and diacritics are ignored.

ignoreSymbols When true, symbolic characters including whitespace, punctuation,

The primary claim to fame for NTRU is its eficiency~-the encryption, decrypt,ion. and the key generation process are all very fast by public key

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Accordingly, at a first approximation, it can be said that H = 0 and Y = 0, and consequently, Yr = Yt = Ya , where Ya is the distance of the central point between T and R from the starting position in a B-scan line This further means that w1 = w4 = 0 and w2 = w3 = w, with w being the (one-way) wavepath between the antennas and the object: w= D2 + Ya Yo.

numericComparison When true, numbers in strings are treated numerically, rather than character-by-character. For example, using string sorting, 100 < 2 ( 1 comes before 2 ), but using numeric sorting, 2 < 100.

As in other globalization classes, these usually take on the right properties depending on their locale and the CollatorMode used (as you saw in Example 41-5, because the sort worked perfectly without modifying the Collator). Tweaking them individually can be useful, though particularly numericComparison.

standards. The inventors of NTRU claim that when comparing a moderate NTRU security level to RSA with a 512-bit modulus, NTRU is approximately 5.9 times faster for encryption, 14.4 times faster for decryption and 5.0 times faster during key creation [68]. In addition, when comparing the highest NTRU security level to RSA with a 1024 bit RSA modulus, it is claimed that NTRU is the same speed for encryption, 3.2 times faster for decryption, and 5 . 3 times faster for key creation. For this reason alone, NTRU might be an attractive cryptosystem to use in resource constrained environments such as embedded systems. NTRU is somewhat unique since there have been several published attacks, but yet the cipher is not considered broken. In the next sections, we briefly outline a few attacks against NTRU. Some of these attacks have led to modifications in the suggested parameters for NTRU.

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