### Table 8. Large Integer Op erati ons

2007

"... In PAGE 17: ...6 8, ... P 9 76 8. Table8 summ ar ize s the op erations tes ted and the corres pon din g grou ps of op eran ds. Wit h resp ec t to a parti cular lib rary un der a pa rticular op eratin g sys tem, eac h op eration usin g grou p A of op erand s is tes ted using either RD T SC or Timing m etho d.... ..."

### Table 1: Bit complexity of the standard algorithm with large integer arithmetic.

### Table 9. Large Integer Op erati on Rank ings Pen tium IV, Wi ndo ws XP

2007

"... In PAGE 20: ... Th e op erations ran kin gs are, MUL: M ulti pli ca- tion ran king, E 3: Mo du lar Ex pon entiation Ranki ng with exp onen t = 3, E 65 53 7: Mo du lar Exp on entiation Rank ing with exp onen t = 65537, E : Mo dular Exp o- nen tiation Ran kin g w ith exp onen t of th e same siz e as th e mo dulu s, G CD: Great- es t C ommon Divisor ran kin g, an d xGCD: Extend ed Gr eatest Comm on Div isor ran kin g. Table9 lists the perf ormance res ults und er Pen tiu m IV, Win do ws XP . In terms of the overal l ran k LI B R W inX P , GM P has th e best ran k and PIOLO GIE th e wors t.... ..."

### Table 11. Large Integer Op erati ons Ranking s Ult raSP AR C, Sola ris

2007

"... In PAGE 21: ... PIO LOGIE rank is sli gh tly bette r than Cryp toPP du e to M odu lar Exp on entiation s rank with E = 3 and GCDs rank. Table11 lists th e perfor m an ce result s und er Ultra-S PAR C, Solari s. In terms of th e overall ran k (LIB- R)S PAR C, GMP has th e best ran k an d Cry ptoP P the wor st.... ..."

### Table 10. Large Integer Op erati on Rank ings Pen tium IV, RedHat 9.0

2007

"... In PAGE 20: ... Cryp toPP GCDs rank is hi gher than xG C D s ran k unli ke all other librar ies whi le it is compl etely th e op pos ite for PIO LOGIE ; in both cas es the re ason behind th at is the choice of algori thms (se e Tab le 6). Table10 lists the perfor m ance res ults under Pen tiu m IV, Re dHat 9. 0.... ..."

### Table 2 compares the performance of our MutableLargeInteger class with the built in JDK 1.2 BigInteger in the workstation environment. The benchmarked version of MutableLargeInteger uses only card com- patible datatypes. No variables are de ned in the local scope and the number of temporary objects has been minimized using the register al- location approach. The gures in the Table 3 tell us that the performance of the current MutableLargeInteger prototype on the card leaves room for improve- ment. As multiplication and inversion are the basic steps of the ECDSA algorithm, we can readily tell something about the performance of the

2000

"... In PAGE 15: ... For example, 192-bit ECDSA needs about 30000 inversions during the signature operation and about half of that for the key generation or checking of the signature. Looking at gures in the Table2 , we can deduce that inversion opera- tion of our current prototype is roughly 50 times as slow as with the JDK 1.2 implementation of BigInteger.... ..."

Cited by 3

### Table 3. Simulation of DBD with integer con- figurations

1999

"... In PAGE 7: ... Both capture the branch penalty seen by the pro- cessor. Comparing the simulations of DBD processor with the integer-benchmark configuration in Table3 and the base processor in Table 5, the three integer benchmarks demon- strate large gains in speedup, led by li benchmark with 1.... In PAGE 8: ... These bot- tlenecks at execution make the idle cycles in PU in waiting for BU less critical to overall performance of floating point benchmarks. For the simulations of DBD processor with the config- uration for floating point benchmarks shown in Table 4, performance of integer benchmarks is almost identical to that in Table3 . As the configuration is intended for float- ing point benchmarks, a significant performance gain is ob- served on the floating point benchmarks.... ..."

Cited by 5

### Table 4: Run times for the three multi-pair programs operations involved in the multiprecision array updates, and these matrix multiplica- tions can be performed concurrently at the outermost loop level. The parallel techniques mentioned in the previous paragraph can still be applied to the double precision and in- termediate precision iterations. It turns out, though, that the double precision iterations run so rapidly that parallel processing of these iterations is often not worth the overhead. Nonetheless, we have achieved modest acceleration on very large problems by using par- allel processing on some steps of double precision iterations. Some parallel performance results will be given in the next section. 8. Large Applications and Parallel Performance Four recent applications will be described here, each of which involves very large integer relation problems. Thus they are excellent test cases for the new multi-pair programs. Reduction of Euler sums: In section 3, we mentioned recent research on multiple zeta values, which play a key role in quantum eld theory [13]. More generally, one may de ne Euler sums by [9]

2000

"... In PAGE 12: ... But it turns out that these programs also run faster on a single processor system, compared with the standard PSLQ equivalents. Some one- processor timings are shown in Table4 for the suite of test problems used in Table 3. Note for example that the one-level multi-pair program is up to twice as fast as the one- level PSLQ program, and the three-level multi-pair program is up to 22% faster than the three-level PSLQ program.... ..."

Cited by 23

### Table 2 Performance comparison of the di erent big integer implementations with the number length of 192 bits.

2000

"... In PAGE 14: ...1. PERFORMANCE DATA Table2 compares the performance of our MutableLargeInteger class with the built in JDK 1.2 BigInteger in the workstation environment.... In PAGE 15: ... For example, 192-bit ECDSA needs about 30000 inversions during the signature operation and about half of that for the key generation or checking of the signature. Looking at gures in the Table2 , we can deduce that inversion opera- tion of our current prototype is roughly 50 times as slow as with the JDK 1.2 implementation of BigInteger.... ..."

Cited by 3