### Table 8: Distance determination of the Large Magellanic Cloud, obtained from Hipparcos data and many other ground-based observations. Distance modulus in mag (by type of intermediate objects, and by increasing distances)

"... In PAGE 6: ... 7 Large Magellanic Cloud distance The distance of the Large Magellanic Cloud is derived by some of the above authors, using various objects as intermediate calibrators. Table8 presents a summary of these determinations. Table 8: Distance determination of the Large Magellanic Cloud, obtained from Hipparcos data and many other ground-based observations.... ..."

### Table 6. eFFS signing time (ms) with 512-bit modulus.

"... In PAGE 8: ...cheme and the eFFS scheme (i.e., with the improvements and extensions mentioned above) using the large integer arithmetic routines from CryptoLib [8]. Table6 and Ta- ble 7 show the times for signing and verifying (with 512-bit modulus) 128-bit message digests, using different speedup techniques and different eFFS/FFS parameters (k; t).9 The results were obtained on a Pentium II 300 MHz machine running Linux.... In PAGE 8: ... For t = 1, the signature size is minimized, but the signing/verification key size is maximized. Moreover, for a fixed kt product, the signing/verification time is smaller when t is smaller (see Table6 and Table 7). Therefore, we recommend to use t = 1 except when adjustable verifica- tion is needed.... In PAGE 9: ... An improvement idea suggested in [12] is to use small prime numbers11 as the verification key components fvig and compute the sign- ing key components fsig by s2 i = v?1 i mod n. This im- provement (labeled as small v-key in Table6 and Table 7) has two advantages. First, the verification time is an order of magnitude smaller than without this improvement (and the signing time is not affected).... In PAGE 9: ...3. Chinese remainder theorem speedup We propose to use the following improvement (labeled as crt in Table6 ), which is based on the Chinese Re- mainder Theorem, to speed up signing operation. In FFS, the signing operation involves the computing of yi = ri (sbi1 1 : : : sbik k ) mod n where fsig do not change and only frig and fbijg change from message to message.... In PAGE 9: ... However, since not every prime number psatis- fies the condition that there exists an integer s such that s2 = p?1 mod n, we use the first k prime numbers that satisfy the condition as the verifica- tion key components. 12% to 20% (see Table6 ). The amount of additional mem- ory needed is only a few hundred bytes for storing a few large integers (with 512-bit modulus).... In PAGE 9: ... Similarly, if each smaller set contains eight si, then it is an 8-bit precomputa- tion. Compared to the basic FFS (with small v-key), 4-bit pre- computation plus crt speedup reduces the signing time by 45% to 55%, and 8-bit precomputation plus crt speedup re- duces the signing time by 60% to 70% (see Table6 ). For 4-bit precomputation with k = 128 and 512-bit modulus, a signer needs to store 128=4 (24 ? 1) = 480 products.... ..."

### Table 6. eFFS signing time (ms) with 512-bit modulus.

"... In PAGE 8: ...cheme and the eFFS scheme (i.e., with the improvements and extensions mentioned above) using the large integer arithmetic routines from CryptoLib [8]. Table6 and Ta- ble 7 show the times for signing and verifying (with 512-bit modulus) 128-bit message digests, using different speedup techniques and different eFFS/FFS parameters #28k;t#29.9 The results were obtained on a Pentium II 300 MHz machine running Linux.... In PAGE 8: ... For t =1, the signature size is minimized, but the signing/verification key size is maximized. Moreover, for afixedkt product, the signing/verification time is smaller when t is smaller (see Table6 and Table 7). Therefore, we recommend to use t = 1 except when adjustable verifica- tion is needed.... In PAGE 9: ... An improvement idea suggested in [12] is to use small prime numbers11 as the verification key components fv i g and compute the sign- ing key components fs i g by s 2 i = v ,1 i mod n.Thisim- provement (labeled as small v-key in Table6 and Table 7) has two advantages. First, the verification time is an order of magnitude smaller than without this improvement (and the signing time is not affected).... In PAGE 9: ...3. Chinese remainder theorem speedup We propose to use the following improvement (labeled as crt in Table6 ), which is based on the Chinese Re- mainder Theorem, to speed up signing operation. In FFS, the signing operation involves the computing of y i = r i #02 #28s b i1 1 #02 :::#02 s b ik k #29 mod n where fs i g do not change and only fr i g and fb ij g change from message to message.... In PAGE 9: ... However, since not every prime number psatis- fies the condition that there exists an integer s such that s 2 = p ,1 mod n, we use the first k prime numbers that satisfy the condition as the verifica- tion key components. 12% to 20% (see Table6 ). The amount of additional mem- ory needed is only a few hundred bytes for storing a few large integers (with 512-bit modulus).... In PAGE 9: ... Similarly, if each smaller set contains eight s i , then it is an 8-bit precomputa- tion. Compared to the basic FFS (with small v-key), 4-bit pre- computation plus crt speedup reduces the signing time by 45% to 55%, and 8-bit precomputation plus crt speedup re- duces the signing time by 60% to 70% (see Table6 ). For 4-bit precomputation with k = 128 and 512-bit modulus, a signer needs to store 128=4 #02 #282 4 , 1#29 = 480 products.... ..."

### Table 2: eFFS veri cation time (ms) with 512-bit modulus.

"... In PAGE 15: ...e implemented the basic Feige-Fiat-Shamir (FFS) scheme and the eFFS scheme (i.e., with the improvements and extensions mentioned above) using the large integer arithmetic routines from CryptoLib [8]. Table 1 and Table2 show the times for signing and verifying (with 512-bit modulus) 128-bit message digests, using di erent speedup techniques and di erent values for the eFFS/FFS parameter (k; t).11 The results were obtained on a Pentium II 300 MHz machine running Linux.... In PAGE 16: ... For t = 1, the signature size is minimized, but the signing/veri cation key size is maximized. Moreover, for a xed kt product, the signing/veri cation time is smaller when t is smaller (see Table 1 and Table2 ). Therefore, we recommend to use t = 1 except when adjustable veri cation is needed.... In PAGE 16: ... An improvement idea suggested in [12] is to use small prime numbers13 as the veri cation key components fvig and compute the signing key components fsig by s2 i = v?1 i mod n. This improvement (labeled as \small v-key quot; in Table 1 and Table2 ) has two advantages. First, the veri cation time is an order of magnitude smaller than without this improvement (and the signing time is not a ected).... In PAGE 18: ... This is because with the small v-key extension, small primes are used as public key components, and their products can be computed very e ciently. For example, with the small v-key extension, 8-bit precomputation in veri cation operations reduces the veri cation time by less than 10% (see Table2 ). In the remaining experiments, we use veri cation with small v-key and no precomputation.... ..."

### Table 1: eFFS signing time (ms) with 512-bit modulus.

"... In PAGE 15: ...e implemented the basic Feige-Fiat-Shamir (FFS) scheme and the eFFS scheme (i.e., with the improvements and extensions mentioned above) using the large integer arithmetic routines from CryptoLib [8]. Table1 and Table 2 show the times for signing and verifying (with 512-bit modulus) 128-bit message digests, using di erent speedup techniques and di erent values for the eFFS/FFS parameter (k; t).11 The results were obtained on a Pentium II 300 MHz machine running Linux.... In PAGE 16: ... For t = 1, the signature size is minimized, but the signing/veri cation key size is maximized. Moreover, for a xed kt product, the signing/veri cation time is smaller when t is smaller (see Table1 and Table 2). Therefore, we recommend to use t = 1 except when adjustable veri cation is needed.... In PAGE 16: ... An improvement idea suggested in [12] is to use small prime numbers13 as the veri cation key components fvig and compute the signing key components fsig by s2 i = v?1 i mod n. This improvement (labeled as \small v-key quot; in Table1 and Table 2) has two advantages. First, the veri cation time is an order of magnitude smaller than without this improvement (and the signing time is not a ected).... In PAGE 17: ...Chinese remainder theorem speedup We propose to use the following improvement (labeled as \crt quot; in Table1 ), which is based on the Chinese Remainder Theorem, to speed up the signing operation. In FFS, the signing operation involves the computing of yi = ri (sbi1 1 : : : sbik k ) mod n where fsig do not change and only frig and fbijg change from message to message.... In PAGE 17: ...4 Precomputation: memory-time tradeo One important feature of FFS is that a signer/veri er can trade memory for signing/veri cation time. We propose to use the following improvement (labeled \precomp quot; in Table1 and Table 2) to speed up signing/veri cation operation by using more memory at signer/veri er. To illustrate the basic idea of this improvement, consider the signing operation with k = 4.... In PAGE 17: ...f them. If each smaller set contains four si, then it is a 4-bit precomputation. Similarly, if each smaller set contains eight si, then it is an 8-bit precomputation. Compared to the basic FFS (with small v-key), 4-bit precomputation plus crt speedup reduces the signing time by 45% to 55%, and 8-bit precomputation plus crt speedup reduces the signing time by 60% to 70% (see Table1 ). For 4-bit precomputation with k = 128 and 512-bit modulus, a signer needs to store 128=4 (24 ? 1) = 480 products.... ..."

### Table 2. Average Storage Modulus (MPa)

2003

"... In PAGE 6: ... For every sample, the storage modulus for each individual test run at a given temperature was tabulated, and the average and standard deviation calculated. Table2... ..."

### Table 6 Lumped sensitivities with respect to rate coe cients (at the end of the fth day); displayed are the values of modulus greater than 1E-03.

1997

"... In PAGE 18: ... One drawback is that, when a parameter has a large in uence over a speci c or small number of species, the lumped coe cient may be large, although, from a chemical standpoint, the \global quot; in uence is negligible. As an example, look at Table6 . The large lumped sensitivity associated with DMS can be explained by a strong in uence of DMS initial concentration over itself, although the e ect of this parameter on important species is negligible.... ..."

Cited by 8

### Table 6.Lumped sensitivities with respect to rate coe cients (at the end of the fth day); displayed are the values of modulus greater than 1E-03.

1997

"... In PAGE 18: ... One drawback is that, when a parameter has a large in uence over a speci c or small number of species, the lumped coe cient may be large, although, from a chemical standpoint, the \global quot; in uence is negligible. As an example, look at Table6 . The large lumped sensitivity associated with DMS can be explained by a strong in uence of DMS initial concentration over itself, although the e ect of this parameter on important species is negligible.... ..."

Cited by 8

### Table 14 Parameters used in shear modulus models for 4340 steel. MTS shear modulus model

"... In PAGE 36: ...3. The parameters used in the shear modulus models are shown in Table14 . The parameters for the MTS model have been obtained from a least square fit to the data at a compression of 1.... ..."

### Table 17. The normally distributed data adjusted to a common modulus.

1998

"... In PAGE 10: ...able 16. The log of the data estimates showing the row mean (common modulus) and row meanminus the grand mean (constant offset). ............................................................80 Table17 .... ..."

Cited by 1