### TABLE II asymptotic multivariate merit factor for various quadratic constructions

2005

Cited by 4

### TABLE I GALOIS FIELD, GF(8)

### Table 1: Some constrains of quasigroups.

1994

"... In PAGE 1: ... In this paper, we are interested in the prob- lems in quasigroups given by Fujita, Slaney and Bennett in their award-winning IJCAI paper [3]. The constraints in Table1 are taken from [3]: Among the Latins squares satisfying these con- straints, we are also interested in those squares with a hole, i.e.... In PAGE 3: ... 3 Cyclic Group Construction The propositional reasoning program we used to attack quasigroup problems is called SATO (SAt- is ability Testing Optimized) which is an e cient implementation of the Davis-Putnam algorithm written by Zhang [8]. For a quasigroup of order v, the number of propositional clauses obtained from clauses like QG1 and QG2 in Table1 is O(v6) because there are six distinct variables in QG1 and QG2. For a large v, in addition to the large number of clauses, the search space involved in these problems is also huge.... ..."

Cited by 11

### Table 6-7: Raw data of performance in composite Galois field

in Contents

"... In PAGE 13: ...able 5-1: The best composition of each n.......................................................................... 41 Table6 -1: Timing of decryption in different optimization on PC.... In PAGE 13: ...able 6-1: Timing of decryption in different optimization on PC....................................... 43 Table6 -2: Raw Data of Performance in basic implementation on PC.... In PAGE 13: ...able 6-2: Raw Data of Performance in basic implementation on PC................................ 45 Table6 -3: Performance of each operator on PC.... In PAGE 13: ...able 6-3: Performance of each operator on PC.................................................................. 45 Table6 -4: Percentage of each operator on PC .... In PAGE 13: ...able 6-4: Percentage of each operator on PC .................................................................... 46 Table6 -5: Final Performance in single finite field on PC.... In PAGE 13: ...able 6-5: Final Performance in single finite field on PC................................................... 46 Table6 -6: Timing of each operator under different composition.... In PAGE 13: ...able 6-6: Timing of each operator under different composition........................................ 47 Table6 -7: Raw data of performance in composite Galois field.... In PAGE 13: ...able 6-7: Raw data of performance in composite Galois field.......................................... 48 Table6 -8: Timing of each operator in composite Galois field.... In PAGE 13: ...able 6-8: Timing of each operator in composite Galois field............................................ 48 Table6 -9: Compare PMI+ with RSA on PC .... In PAGE 13: ...able 6-9: Compare PMI+ with RSA on PC ....................................................................... 49 Table6 -10: The enhanced performance after group level multiplication.... In PAGE 13: ...able 6-10: The enhanced performance after group level multiplication............................ 50 Table6 -11: The raw data of performance in single Galois field on smartcard .... In PAGE 13: ...able 6-11: The raw data of performance in single Galois field on smartcard ................... 50 Table6 -12: The timing of each operator in single Galois field on smartcard .... In PAGE 13: ...able 6-12: The timing of each operator in single Galois field on smartcard ..................... 51 Table6 -13: The final performance in single Galois field on smartcard.... In PAGE 13: ...able 6-13: The final performance in single Galois field on smartcard.............................. 51 Table6 -14: The raw data of performance in composite Galois field on smartcard .... In PAGE 13: ...able 6-14: The raw data of performance in composite Galois field on smartcard ............ 52 Table6 -15: The timing of each operator in composite Galois field on smartcard .... In PAGE 13: ...able 6-15: The timing of each operator in composite Galois field on smartcard .............. 52 Table6 -16: The final performance in composite Galois field on smartcard .... In PAGE 57: ...6 16.6 Lookup table size 800B 968B 1152B 1352B 1568B 1800B Table6 -1: Timing of decryption in different optimization on PC ... In PAGE 59: ...78 / 4715446 13.36 / 288925038 0 0 0 81s Table6 -2: Raw Data of Performance in basic implementation on PC We calculated the time per operator and make addition as the standard to normalize the time. We implemented multiplication by the bit-level multiplication .... In PAGE 59: ...12.7 1 65.3 58.6 785.5 Table6 -3: Performance of each operator on PC ... In PAGE 60: ...6 46.2% 0.2% 19.3% 11% 21.1% 2.2% Table6 -4: Percentage of each operator on PC Here is the timing of each operator in each size of block. Block Size 80bit 88 bit 96 bit 104 bit 112 bit Complexity Key Generation (ms) 42 54 81 111 150 O(n4) Encryption (ms) 0.... In PAGE 60: ...3 4.8 O(n2) Public key size (KB) 33 44 57 40 45 O(n3) Table6 -5: Final Performance in single finite field on PC ... In PAGE 61: ...874 9.773 Table6 -6: Timing of each operator under different composition The experimental result is close to our inference before. Obviously, the former factoring of 96 is better than the latter.... In PAGE 62: ...85 10.75 Table6 -8: Timing of each operator in composite Galois field 6.... In PAGE 62: ....1.5. Comparison We see first three columns in Table6 -9 are the cryptosystems with the same security strength. Though the first column and second column are PMI+ cryptosystems, the second one implemented in composite field is much faster than traditional ... In PAGE 63: ....8 2.0 20.4 11.6 2.2 86.99 Security ECC-160 ECC-160 ECC-160 ECC-224 ECC-224 ECC-224 Table6 -9: Compare PMI+ with RSA on PC 6.2.... In PAGE 63: ... Here we set g=8, the look-up table size is n bytes. In Table6 -10, we see that Group level multiplication reduce 20% time of Bit level multiplication . ... In PAGE 64: ...0.19% 22.5% 20.98% data (bytes) 79 81 84 86 89 91 xdata (bytes) 84 164 85 173 86 182 code (bytes) 2902 3173 2921 3182 2912 3183 Table6 -10: The enhanced performance after group level multiplication 6.... In PAGE 64: ...71s / 8960 5.4s / 3840 6s / 3840 Total time 1020s 1230s 1378 342s 438s 491s 54s 70s 80s Table6 -11: The raw data of performance in single Galois field on smartcard ... In PAGE 65: ...48.5 1 51.7 36.4 Table6 -12: The timing of each operator in single Galois field on smartcard Chip Intel8052AH 3.57MHZ Intel8052AH 10MHZ Dollars-DS80C32 0 33MHZ block size 80 88 96 80 88 96 80 88 96 data size (bytes) 89 99 101 89 99 101 93 99 108 xdata size (bytes) 164 173 182 164 173 182 161 178 180 code size (bytes) 3.... In PAGE 65: ...4 43 49 5.4 7.0 8.0 Table6 -13: The final performance in single Galois field on smartcard ... In PAGE 66: ...38s 7.49s Table6 -14: The raw data of performance in composite Galois field on smartcard We calculate the time per operator and make addition as the standard to normalize the time. We find that a multiplication equals 8 additions which equals to at least 100 additions in traditional implementation.... In PAGE 66: ...08 5.73 Table6... In PAGE 67: ...34s 0.75s Table6 -16: The final performance in composite Galois field on smartcard ... ..."

### Table 6-8: Timing of each operator in composite Galois field

in Contents

"... In PAGE 13: ...able 5-1: The best composition of each n.......................................................................... 41 Table6 -1: Timing of decryption in different optimization on PC.... In PAGE 13: ...able 6-1: Timing of decryption in different optimization on PC....................................... 43 Table6 -2: Raw Data of Performance in basic implementation on PC.... In PAGE 13: ...able 6-2: Raw Data of Performance in basic implementation on PC................................ 45 Table6 -3: Performance of each operator on PC.... In PAGE 13: ...able 6-3: Performance of each operator on PC.................................................................. 45 Table6 -4: Percentage of each operator on PC .... In PAGE 13: ...able 6-4: Percentage of each operator on PC .................................................................... 46 Table6 -5: Final Performance in single finite field on PC.... In PAGE 13: ...able 6-5: Final Performance in single finite field on PC................................................... 46 Table6 -6: Timing of each operator under different composition.... In PAGE 13: ...able 6-6: Timing of each operator under different composition........................................ 47 Table6 -7: Raw data of performance in composite Galois field.... In PAGE 13: ...able 6-7: Raw data of performance in composite Galois field.......................................... 48 Table6 -8: Timing of each operator in composite Galois field.... In PAGE 13: ...able 6-8: Timing of each operator in composite Galois field............................................ 48 Table6 -9: Compare PMI+ with RSA on PC .... In PAGE 13: ...able 6-9: Compare PMI+ with RSA on PC ....................................................................... 49 Table6 -10: The enhanced performance after group level multiplication.... In PAGE 13: ...able 6-10: The enhanced performance after group level multiplication............................ 50 Table6 -11: The raw data of performance in single Galois field on smartcard .... In PAGE 13: ...able 6-11: The raw data of performance in single Galois field on smartcard ................... 50 Table6 -12: The timing of each operator in single Galois field on smartcard .... In PAGE 13: ...able 6-12: The timing of each operator in single Galois field on smartcard ..................... 51 Table6 -13: The final performance in single Galois field on smartcard.... In PAGE 13: ...able 6-13: The final performance in single Galois field on smartcard.............................. 51 Table6 -14: The raw data of performance in composite Galois field on smartcard .... In PAGE 13: ...able 6-14: The raw data of performance in composite Galois field on smartcard ............ 52 Table6 -15: The timing of each operator in composite Galois field on smartcard .... In PAGE 13: ...able 6-15: The timing of each operator in composite Galois field on smartcard .............. 52 Table6 -16: The final performance in composite Galois field on smartcard .... In PAGE 57: ...6 16.6 Lookup table size 800B 968B 1152B 1352B 1568B 1800B Table6 -1: Timing of decryption in different optimization on PC ... In PAGE 59: ...78 / 4715446 13.36 / 288925038 0 0 0 81s Table6 -2: Raw Data of Performance in basic implementation on PC We calculated the time per operator and make addition as the standard to normalize the time. We implemented multiplication by the bit-level multiplication .... In PAGE 59: ...12.7 1 65.3 58.6 785.5 Table6 -3: Performance of each operator on PC ... In PAGE 60: ...6 46.2% 0.2% 19.3% 11% 21.1% 2.2% Table6 -4: Percentage of each operator on PC Here is the timing of each operator in each size of block. Block Size 80bit 88 bit 96 bit 104 bit 112 bit Complexity Key Generation (ms) 42 54 81 111 150 O(n4) Encryption (ms) 0.... In PAGE 60: ...3 4.8 O(n2) Public key size (KB) 33 44 57 40 45 O(n3) Table6 -5: Final Performance in single finite field on PC ... In PAGE 61: ...874 9.773 Table6 -6: Timing of each operator under different composition The experimental result is close to our inference before. Obviously, the former factoring of 96 is better than the latter.... In PAGE 62: ...3 / 7040000 1.98 / 3840000 22s Table6 -7: Raw data of performance in composite Galois field Comparing Table 6.4 and Table 6.... In PAGE 62: ....1.5. Comparison We see first three columns in Table6 -9 are the cryptosystems with the same security strength. Though the first column and second column are PMI+ cryptosystems, the second one implemented in composite field is much faster than traditional ... In PAGE 63: ....8 2.0 20.4 11.6 2.2 86.99 Security ECC-160 ECC-160 ECC-160 ECC-224 ECC-224 ECC-224 Table6 -9: Compare PMI+ with RSA on PC 6.2.... In PAGE 63: ... Here we set g=8, the look-up table size is n bytes. In Table6 -10, we see that Group level multiplication reduce 20% time of Bit level multiplication . ... In PAGE 64: ...0.19% 22.5% 20.98% data (bytes) 79 81 84 86 89 91 xdata (bytes) 84 164 85 173 86 182 code (bytes) 2902 3173 2921 3182 2912 3183 Table6 -10: The enhanced performance after group level multiplication 6.... In PAGE 64: ...71s / 8960 5.4s / 3840 6s / 3840 Total time 1020s 1230s 1378 342s 438s 491s 54s 70s 80s Table6 -11: The raw data of performance in single Galois field on smartcard ... In PAGE 65: ...48.5 1 51.7 36.4 Table6 -12: The timing of each operator in single Galois field on smartcard Chip Intel8052AH 3.57MHZ Intel8052AH 10MHZ Dollars-DS80C32 0 33MHZ block size 80 88 96 80 88 96 80 88 96 data size (bytes) 89 99 101 89 99 101 93 99 108 xdata size (bytes) 164 173 182 164 173 182 161 178 180 code size (bytes) 3.... In PAGE 65: ...4 43 49 5.4 7.0 8.0 Table6 -13: The final performance in single Galois field on smartcard ... In PAGE 66: ...38s 7.49s Table6 -14: The raw data of performance in composite Galois field on smartcard We calculate the time per operator and make addition as the standard to normalize the time. We find that a multiplication equals 8 additions which equals to at least 100 additions in traditional implementation.... In PAGE 66: ...08 5.73 Table6... In PAGE 67: ...34s 0.75s Table6 -16: The final performance in composite Galois field on smartcard ... ..."

### Table 6-16: The final performance in composite Galois field on smartcard

in Contents

"... In PAGE 13: ...able 5-1: The best composition of each n.......................................................................... 41 Table6 -1: Timing of decryption in different optimization on PC.... In PAGE 13: ...able 6-1: Timing of decryption in different optimization on PC....................................... 43 Table6 -2: Raw Data of Performance in basic implementation on PC.... In PAGE 13: ...able 6-2: Raw Data of Performance in basic implementation on PC................................ 45 Table6 -3: Performance of each operator on PC.... In PAGE 13: ...able 6-3: Performance of each operator on PC.................................................................. 45 Table6 -4: Percentage of each operator on PC .... In PAGE 13: ...able 6-4: Percentage of each operator on PC .................................................................... 46 Table6 -5: Final Performance in single finite field on PC.... In PAGE 13: ...able 6-5: Final Performance in single finite field on PC................................................... 46 Table6 -6: Timing of each operator under different composition.... In PAGE 13: ...able 6-6: Timing of each operator under different composition........................................ 47 Table6 -7: Raw data of performance in composite Galois field.... In PAGE 13: ...able 6-7: Raw data of performance in composite Galois field.......................................... 48 Table6 -8: Timing of each operator in composite Galois field.... In PAGE 13: ...able 6-8: Timing of each operator in composite Galois field............................................ 48 Table6 -9: Compare PMI+ with RSA on PC .... In PAGE 13: ...able 6-9: Compare PMI+ with RSA on PC ....................................................................... 49 Table6 -10: The enhanced performance after group level multiplication.... In PAGE 13: ...able 6-10: The enhanced performance after group level multiplication............................ 50 Table6 -11: The raw data of performance in single Galois field on smartcard .... In PAGE 13: ...able 6-11: The raw data of performance in single Galois field on smartcard ................... 50 Table6 -12: The timing of each operator in single Galois field on smartcard .... In PAGE 13: ...able 6-12: The timing of each operator in single Galois field on smartcard ..................... 51 Table6 -13: The final performance in single Galois field on smartcard.... In PAGE 13: ...able 6-13: The final performance in single Galois field on smartcard.............................. 51 Table6 -14: The raw data of performance in composite Galois field on smartcard .... In PAGE 13: ...able 6-14: The raw data of performance in composite Galois field on smartcard ............ 52 Table6 -15: The timing of each operator in composite Galois field on smartcard .... In PAGE 13: ...able 6-15: The timing of each operator in composite Galois field on smartcard .............. 52 Table6 -16: The final performance in composite Galois field on smartcard .... In PAGE 57: ...6 16.6 Lookup table size 800B 968B 1152B 1352B 1568B 1800B Table6 -1: Timing of decryption in different optimization on PC ... In PAGE 59: ...78 / 4715446 13.36 / 288925038 0 0 0 81s Table6 -2: Raw Data of Performance in basic implementation on PC We calculated the time per operator and make addition as the standard to normalize the time. We implemented multiplication by the bit-level multiplication .... In PAGE 59: ...12.7 1 65.3 58.6 785.5 Table6 -3: Performance of each operator on PC ... In PAGE 60: ...6 46.2% 0.2% 19.3% 11% 21.1% 2.2% Table6 -4: Percentage of each operator on PC Here is the timing of each operator in each size of block. Block Size 80bit 88 bit 96 bit 104 bit 112 bit Complexity Key Generation (ms) 42 54 81 111 150 O(n4) Encryption (ms) 0.... In PAGE 60: ...3 4.8 O(n2) Public key size (KB) 33 44 57 40 45 O(n3) Table6 -5: Final Performance in single finite field on PC ... In PAGE 61: ...874 9.773 Table6 -6: Timing of each operator under different composition The experimental result is close to our inference before. Obviously, the former factoring of 96 is better than the latter.... In PAGE 62: ...3 / 7040000 1.98 / 3840000 22s Table6 -7: Raw data of performance in composite Galois field Comparing Table 6.4 and Table 6.... In PAGE 62: ...85 10.75 Table6 -8: Timing of each operator in composite Galois field 6.... In PAGE 62: ....1.5. Comparison We see first three columns in Table6 -9 are the cryptosystems with the same security strength. Though the first column and second column are PMI+ cryptosystems, the second one implemented in composite field is much faster than traditional ... In PAGE 63: ....8 2.0 20.4 11.6 2.2 86.99 Security ECC-160 ECC-160 ECC-160 ECC-224 ECC-224 ECC-224 Table6 -9: Compare PMI+ with RSA on PC 6.2.... In PAGE 63: ... Here we set g=8, the look-up table size is n bytes. In Table6 -10, we see that Group level multiplication reduce 20% time of Bit level multiplication . ... In PAGE 64: ...0.19% 22.5% 20.98% data (bytes) 79 81 84 86 89 91 xdata (bytes) 84 164 85 173 86 182 code (bytes) 2902 3173 2921 3182 2912 3183 Table6 -10: The enhanced performance after group level multiplication 6.... In PAGE 64: ...71s / 8960 5.4s / 3840 6s / 3840 Total time 1020s 1230s 1378 342s 438s 491s 54s 70s 80s Table6 -11: The raw data of performance in single Galois field on smartcard ... In PAGE 65: ...48.5 1 51.7 36.4 Table6 -12: The timing of each operator in single Galois field on smartcard Chip Intel8052AH 3.57MHZ Intel8052AH 10MHZ Dollars-DS80C32 0 33MHZ block size 80 88 96 80 88 96 80 88 96 data size (bytes) 89 99 101 89 99 101 93 99 108 xdata size (bytes) 164 173 182 164 173 182 161 178 180 code size (bytes) 3.... In PAGE 65: ...4 43 49 5.4 7.0 8.0 Table6 -13: The final performance in single Galois field on smartcard ... In PAGE 66: ...38s 7.49s Table6 -14: The raw data of performance in composite Galois field on smartcard We calculate the time per operator and make addition as the standard to normalize the time. We find that a multiplication equals 8 additions which equals to at least 100 additions in traditional implementation.... In PAGE 66: ...08 5.73 Table6... ..."

### Table 3: Newly solved Quasigroup problems.

1994

"... In PAGE 5: ... Our program is also able to reproduce all the results reported in [11, 3, 7, 6]. We present in Table3 our results which an- swered for the rst time some open problems listed in [1]. When a case is solved by the cyclic group construction, instead of presenting the newly found Latin squares, we list the vectors e, f and g, which can be used to construct the entire square by the cyclic group construction.... ..."

Cited by 11

### Table 1. Extension fields

2007

"... In PAGE 4: ... 4 Finite Field Arithmetic We construct the finite extension field Fp12 as a tower of finite extensions: Quadratic on top of a cubic on top of a quadratic. The quadratic/cubic non- residues and reduction polynomials are detailed in Table1 . The multiplication and squaring algorithms chosen to implement field arithmetic are listed in Ta- ble 2.... ..."

Cited by 2

### Table 1. Representation of multivariate quadratic public polynomials in memory

"... In PAGE 12: ... 1 1 0 1 0 1 0 To find the result matrix in Table1 and vector in Table 2 is multiplied and resulting vector will be Although consumes a lot of memory (around 200kB with overheads), this approach creates results very fast and efficiently. 4.... ..."

### Table 2. Representation of multivariate quadratic public polynomials in memory

"... In PAGE 12: ...To find the result matrix in Table 1 and vector in Table2 is multiplied and resulting vector will be Although consumes a lot of memory (around 200kB with overheads), this approach creates results very fast and efficiently. 4.... ..."