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'''Discrete logarithm records''' are the best results achieved to date in solving the [[discrete logarithm]] problem, which is the problem of finding solutions ''x'' to the equation ''g''<sup>''x''</sup>&nbsp;=&nbsp;''h'' given elements ''g'' and ''h'' of a finite cyclic group&nbsp;''G''.  The difficulty of this problem is the basis for the security of several [[cryptographic]] systems, including [[Diffie–Hellman]] key agreement, [[ElGamal encryption]], the [[ElGamal signature scheme]], the [[Digital Signature Algorithm]], and the [[elliptic curve cryptography]] analogs of these. Common choices for ''G'' used in these algorithms include the multiplicative group of integers modulo&nbsp;''p'', the multiplicative group of a [[finite field]], and the group of points on an [[elliptic curve]] over a [[finite field]].
<br><br>SOG is leading the perfect knife producer that has won the wishes of many people. SOG Flash II is a superb and affordable knife that is used for each-day-carry goal It seems to be just like Benchmade Griptilian in look. It is a durable and razor sharp knife. The Flash II begins spring assisted open technology feature that can make sure that smoother and faster opening of the knife. The SOG Flash II has received a total of a hundred and ten+ constructive customer critiques with a score of four.5 out of five on Amazon. It is vitally reasonably priced to purchase as a result of it costs less than $forty.<br><br>The two toned, [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=In+terms+of+using+a+knife+every+individual+might+have+distinct+provisions+and+practical+activity.+Whether+or+not+a+person+go+on+mountain+climbing+or+could+want+to+use+it+as+self-safety+if+ever+they%27re+misfortune+and+mugged+by+these+robbers%2C+they+may+need+the+right+gadget+for+every+circumstances.+Some+could+use+it+for+camping+and+others+on+household+chores.+Electricians+and+plumbers+might+also+use+it+as+a+component+for+his+or+her+job.+Hunters+may+additionally+want+to+have+this+for+excoriating+animals+on+the+sphere.%0D%0A%0D%0AThe+Previous+Timer+Knife+is+the+all+time+traditional+knife%2C+the+perfect+made+American+pocket+knife+at+this+time+with+a+historical+past+that+goes+again+almost+one+hundred+ten+years.+Not+all+pocket+knives+are+created+equal.+They+all+are+very+unique+and+have+their+very+personal+features+and+enchantment.++If+you+cherished+this+post+and+also+you+wish+to+be+given+guidance+with+regards+to+%5Bhttp%3A%2F%2Fwww.Thebestpocketknifereviews.com%2F+Pocket+Knife+reviews%5D+kindly+pay+a+visit+to+the+web-page.+%5Bhttp%3A%2F%2Fwww.google.de%2Fsearch%3Fq%3DSelecting+Selecting%5D+a+pocket+knife+can+be+a+bit+confusing+and+perhaps+even+overwhelming+with+all+the+different+options+accessible+to+choose+from.+There%27s+a+wide+variety+of+kinds%2C+handles%2C+blades%2C+and+more+to+choose+from.+To+study+out+extra+about+pocket+knives+and+extensive+selection+in+the+market%2C+please+visit++for+professional+ideas+and+beneficial+perception+earlier+than+buying.+A+knife+is+a+matter+of+preference%2C+but+these+are+five+of+our+favorite+pocket+knives+that+consistently+get+rave+reviews.+Tip+up+or+tip+down+pocket+clip%0D%0A%0D%0AAt+3.75+ounces%2C+the+Spyderco+Para-Navy+2+makes+an+awesome+daily+carry+knife+as+well+as+a+common+use+knife%2C+and+it+does+this+because+of+the+great+stability+between+weight%2C+robustness+and+performance.+The+tip+on+the+blade+is+somewhat+bit+thin+for+me+to+suggest+it+as+a+survival+knife%2C+but+it+would+easily+withstand+any+camping+trip+or+each+day+carry+perform+you+may+consider.+searching+knife+%2C+the+knife+has+a+greater+size+%2C+stiffness+%2C+good+supplies+and+A+slender%2C+straight+blade+with+a+central+level+is+best+for+peeling+spuds+and+whittling+kindling.+The+just+about+infinite+variety+in+brand+names+of+knives+is+confusing+at+greatest%2C+and+misleading+at+worst.+Opening+a+letter.+Sure%2C+you+can+use+your+finger%2C+however+using+a+knife+is+simply+more+manly.+Boker+Plus+Credit+score+Card+Knife%0D%0A%0D%0APocket+knives+are+excellent+companions+as+a+result+of+they%27re+transportable+and+make+themselves+helpful+in+lots+of+conditions%2C+whether+or+not+it%27s+tenting+exterior+or+let%27s+simply+say+you+are+lost+in+a+desert+island.+When+it%27s+a+must+to+select+one%2C+make+sure+that+the+knife+meets+your+wants+and+is+in+response+to+your+preferences.+There+are+a+zillion+selections+today.+There+are+traditional+knives%2C+single+knives%2C+Swiss+knives%2C+tactical+knives+%2C+multi-device+knives+and+and+so+forth.+Selecting+out+a+knife+is+a+private+thing.+It%27s+like+choosing+out+a+pockets.+All+of+it+comes+down+to+personal+taste.+Men+have+a+bond+with+their+instruments.+Choose+something+that+feels+good+in+your+hand%2C+and+that+you%27re+going+to+take+pleasure+in+and+need+to+carry+around.+Home+%3E%3E+Make+Your+Own+Emergency+Kit+%3E%3E+The+Greatest+Pocket+Knife+Fastened-Blade+Knives%0D%0A%0D%0ANow+this+has+been+a+huge+debate+for+nevertheless+lengthy+pocket+knives+have+been+round.+The+giants+Victorinox+are+always+competitors+for+being+the+best+however+then+once+more+there+are+a+whole+lot+of+various+corporations.+I+private+love+CRKT+and+Gerber%2C+however+then+once+more+everyone+has+their+own+private+choice.+I+will+leave+the+difficult+choice+up+to+you+on+what+one+of+the+best+pocket+knife+model+is%2C+but+then+again+most+manufacturers+have+no+less+than+just+a+few+wonderful+knives+so+with+any+alternative+you+may+be+nice%21%0D%0A%0D%0AThis+is+a+knife+that+can+take+some+abuse.+From+the+stable+RotoLock+mechanism+to+the+prime+quality+Elmax+PM+one+hundred+seventy+steel+on+the+4.5mm+thick+blade%2C+you+may+trust+this+knife+to+come+back+by+way+of+for+you.+The+handles+are+CNC+machined+perpendicular+to+the+blades+size+to+make+sure+a+pleasant+stable+grip+on+the+knife.+Timothy+Dahl+of+Charles+%26+Hudson+%2C+also+reviewing+the+unique+model%2C+%E2%80%9Cabsolutely+liked+this+knife.+The+appear+and+feel+is+nice+and+the+design+is+exclusive.%E2%80%9D+7.one+hundred+twenty+five%22+three%22+3.2+oz+CPM-S30V+%24%24%24%24+4.9+Buck+112FG+Ranger+with+Finger+Grooves%2C+Lockback+Folding+Knife+3.375%22+5+oz+154CM+%24%24%24+four.8+Kershaw+Black+Shallot+Knife+with+Tungsten+DLC+Black+coating+on+Blade+%26+Deal+with+4.625%22+1.875%22+1.5+oz+420HC+Stainless+Metal+%24+4.8+Kershaw+Ken+Onion+Tactical+Blur+Folding+Knife+with+Speed+Secure%0D%0A%0D%0AEarlier+than+you+let+your+youngster+run+off+into+the+woods+unsupervised+it%27s+essential+examine+him+to+that+kid+we+all+knew+growing+up.+Is+your+child+that+baby%3F+If+so%2C+then+it+would+be+best+to+wait+a+handful+of+extra+years+or+extra+before+permitting+them+to+have+a+pocket+knife.+It+boils+down+as+to+whether+or+not+you%27re+feeling+your+baby+is+mature%2C+respectful+and+intelligent+sufficient+to+care+for+a+potentially+dangerous+instrument.+Take+a+while+and+observe+your+child%2C+especially+how+he+or+she+reacts+round+other+youngsters.+Peer+stress+can+lead+to+unlucky+resolution+making.%0D%0A%0D%0AKummerfeldt+says+the+neatest%2C+safest+place+to+stash+your+knife+is+in+a+straightforward-entry+spot+in+your+backpack.+You+are+asking+for+trouble+by+sporting+a+fixed-blade+knife+in+your+belt.+If+you+fall%2C+the+knife+may+rotate+inward+and+you+could+land+right+on+the+blade.+had+quite+a+bit+SOG+fashion+knife+participate+in+tactical+battlefield+for+the+soldiers.+electrical+meeting+shall+choose+its+Victorinox+knife.+Victorinox+is+an+effective+If+you+happen+to+use+a+knife+as+ornament+or+souvenir%2C+you+possibly+can+choose+its+SOG+knife.+SOG+has+a+very+small+product+line%3B+you+may+fairly+put+on+in+hand+everybody+may+be+assured+of+the+most+effective+choices.+Choosing+knife+depends+on+each+individual%2C+each+type+and+finance.+Hope+you+may+choose+up+a+knife+best.+Use+a+Q-tip+or+toothbrush+to+scrub+the+knife+after+using+it.%0D%0A%0D%0AThe+Flash+II+has+a+army+background%2C+but+unlike+the+Civilian%2C+it+has+uses+that+go+above+and+past+the+rippin%27+and+the+tearin%27.+It%27s+meant+to+be+a+partial+utility+knife+that+can+employ+the+serrated+lower+end+for+easy+sawing+procedures+whereas+the+graceful+higher+portion+can+handle+scraping+or+slashing.+The+blade+is+3.5+inches+of+AUS-8+metal+that+carries+a+bolt+for+easy+one-handed+opening.+It+is+spring+loaded%2C+so+it+snaps+to+attention+with+a+minimal+of+effort.+The+ergonomic+deal+with+means+it%27s+arduous+to+drop+and+almost+unattainable+to+have+pulled+out+of+your+grip.+An+awesome+folding+knife+for+the+survivalist+that+wants+just+a+little+bit+of+every+thing.++Buy&create=Create aluminum] and G10 handle provides the knife a really shiny and clean look when closed or open. The knife weights 3.2 ounces and opens in a split second. The AXIS help has a safety lock that may lock the knife within the open or close place. Safeties are common on super harmful objects reminiscent of weapons however, given how the AXIS lock is, it's becoming. Chilly Steel , a California based knife maker , was shaped in 1980 and presents all kinds of fine knives, machetes , and martial arts equipment at cheap prices. Cold Steel additionally has bi-annual manufacturing unit second sales on their products.<br><br>Functionally, I've at all times really preferred the Spyderco thumb-gap. This blade's thumb gap is sufficiently big that you could flip the knife open one-handed with all however the thickest of gloves. The backbone of the blade is textured (referred to as jimping) so you'll be able to actually get an excellent grip on it along with your thumb when chopping and the sample within the [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=In+terms+of+using+a+knife+every+individual+might+have+distinct+provisions+and+practical+activity.+Whether+or+not+a+person+go+on+mountain+climbing+or+could+want+to+use+it+as+self-safety+if+ever+they%27re+misfortune+and+mugged+by+these+robbers%2C+they+may+need+the+right+gadget+for+every+circumstances.+Some+could+use+it+for+camping+and+others+on+household+chores.+Electricians+and+plumbers+might+also+use+it+as+a+component+for+his+or+her+job.+Hunters+may+additionally+want+to+have+this+for+excoriating+animals+on+the+sphere.%0D%0A%0D%0AThe+Previous+Timer+Knife+is+the+all+time+traditional+knife%2C+the+perfect+made+American+pocket+knife+at+this+time+with+a+historical+past+that+goes+again+almost+one+hundred+ten+years.+Not+all+pocket+knives+are+created+equal.+They+all+are+very+unique+and+have+their+very+personal+features+and+enchantment.++If+you+cherished+this+post+and+also+you+wish+to+be+given+guidance+with+regards+to+%5Bhttp%3A%2F%2Fwww.Thebestpocketknifereviews.com%2F+Pocket+Knife+reviews%5D+kindly+pay+a+visit+to+the+web-page.+%5Bhttp%3A%2F%2Fwww.google.de%2Fsearch%3Fq%3DSelecting+Selecting%5D+a+pocket+knife+can+be+a+bit+confusing+and+perhaps+even+overwhelming+with+all+the+different+options+accessible+to+choose+from.+There%27s+a+wide+variety+of+kinds%2C+handles%2C+blades%2C+and+more+to+choose+from.+To+study+out+extra+about+pocket+knives+and+extensive+selection+in+the+market%2C+please+visit++for+professional+ideas+and+beneficial+perception+earlier+than+buying.+A+knife+is+a+matter+of+preference%2C+but+these+are+five+of+our+favorite+pocket+knives+that+consistently+get+rave+reviews.+Tip+up+or+tip+down+pocket+clip%0D%0A%0D%0AAt+3.75+ounces%2C+the+Spyderco+Para-Navy+2+makes+an+awesome+daily+carry+knife+as+well+as+a+common+use+knife%2C+and+it+does+this+because+of+the+great+stability+between+weight%2C+robustness+and+performance.+The+tip+on+the+blade+is+somewhat+bit+thin+for+me+to+suggest+it+as+a+survival+knife%2C+but+it+would+easily+withstand+any+camping+trip+or+each+day+carry+perform+you+may+consider.+searching+knife+%2C+the+knife+has+a+greater+size+%2C+stiffness+%2C+good+supplies+and+A+slender%2C+straight+blade+with+a+central+level+is+best+for+peeling+spuds+and+whittling+kindling.+The+just+about+infinite+variety+in+brand+names+of+knives+is+confusing+at+greatest%2C+and+misleading+at+worst.+Opening+a+letter.+Sure%2C+you+can+use+your+finger%2C+however+using+a+knife+is+simply+more+manly.+Boker+Plus+Credit+score+Card+Knife%0D%0A%0D%0APocket+knives+are+excellent+companions+as+a+result+of+they%27re+transportable+and+make+themselves+helpful+in+lots+of+conditions%2C+whether+or+not+it%27s+tenting+exterior+or+let%27s+simply+say+you+are+lost+in+a+desert+island.+When+it%27s+a+must+to+select+one%2C+make+sure+that+the+knife+meets+your+wants+and+is+in+response+to+your+preferences.+There+are+a+zillion+selections+today.+There+are+traditional+knives%2C+single+knives%2C+Swiss+knives%2C+tactical+knives+%2C+multi-device+knives+and+and+so+forth.+Selecting+out+a+knife+is+a+private+thing.+It%27s+like+choosing+out+a+pockets.+All+of+it+comes+down+to+personal+taste.+Men+have+a+bond+with+their+instruments.+Choose+something+that+feels+good+in+your+hand%2C+and+that+you%27re+going+to+take+pleasure+in+and+need+to+carry+around.+Home+%3E%3E+Make+Your+Own+Emergency+Kit+%3E%3E+The+Greatest+Pocket+Knife+Fastened-Blade+Knives%0D%0A%0D%0ANow+this+has+been+a+huge+debate+for+nevertheless+lengthy+pocket+knives+have+been+round.+The+giants+Victorinox+are+always+competitors+for+being+the+best+however+then+once+more+there+are+a+whole+lot+of+various+corporations.+I+private+love+CRKT+and+Gerber%2C+however+then+once+more+everyone+has+their+own+private+choice.+I+will+leave+the+difficult+choice+up+to+you+on+what+one+of+the+best+pocket+knife+model+is%2C+but+then+again+most+manufacturers+have+no+less+than+just+a+few+wonderful+knives+so+with+any+alternative+you+may+be+nice%21%0D%0A%0D%0AThis+is+a+knife+that+can+take+some+abuse.+From+the+stable+RotoLock+mechanism+to+the+prime+quality+Elmax+PM+one+hundred+seventy+steel+on+the+4.5mm+thick+blade%2C+you+may+trust+this+knife+to+come+back+by+way+of+for+you.+The+handles+are+CNC+machined+perpendicular+to+the+blades+size+to+make+sure+a+pleasant+stable+grip+on+the+knife.+Timothy+Dahl+of+Charles+%26+Hudson+%2C+also+reviewing+the+unique+model%2C+%E2%80%9Cabsolutely+liked+this+knife.+The+appear+and+feel+is+nice+and+the+design+is+exclusive.%E2%80%9D+7.one+hundred+twenty+five%22+three%22+3.2+oz+CPM-S30V+%24%24%24%24+4.9+Buck+112FG+Ranger+with+Finger+Grooves%2C+Lockback+Folding+Knife+3.375%22+5+oz+154CM+%24%24%24+four.8+Kershaw+Black+Shallot+Knife+with+Tungsten+DLC+Black+coating+on+Blade+%26+Deal+with+4.625%22+1.875%22+1.5+oz+420HC+Stainless+Metal+%24+4.8+Kershaw+Ken+Onion+Tactical+Blur+Folding+Knife+with+Speed+Secure%0D%0A%0D%0AEarlier+than+you+let+your+youngster+run+off+into+the+woods+unsupervised+it%27s+essential+examine+him+to+that+kid+we+all+knew+growing+up.+Is+your+child+that+baby%3F+If+so%2C+then+it+would+be+best+to+wait+a+handful+of+extra+years+or+extra+before+permitting+them+to+have+a+pocket+knife.+It+boils+down+as+to+whether+or+not+you%27re+feeling+your+baby+is+mature%2C+respectful+and+intelligent+sufficient+to+care+for+a+potentially+dangerous+instrument.+Take+a+while+and+observe+your+child%2C+especially+how+he+or+she+reacts+round+other+youngsters.+Peer+stress+can+lead+to+unlucky+resolution+making.%0D%0A%0D%0AKummerfeldt+says+the+neatest%2C+safest+place+to+stash+your+knife+is+in+a+straightforward-entry+spot+in+your+backpack.+You+are+asking+for+trouble+by+sporting+a+fixed-blade+knife+in+your+belt.+If+you+fall%2C+the+knife+may+rotate+inward+and+you+could+land+right+on+the+blade.+had+quite+a+bit+SOG+fashion+knife+participate+in+tactical+battlefield+for+the+soldiers.+electrical+meeting+shall+choose+its+Victorinox+knife.+Victorinox+is+an+effective+If+you+happen+to+use+a+knife+as+ornament+or+souvenir%2C+you+possibly+can+choose+its+SOG+knife.+SOG+has+a+very+small+product+line%3B+you+may+fairly+put+on+in+hand+everybody+may+be+assured+of+the+most+effective+choices.+Choosing+knife+depends+on+each+individual%2C+each+type+and+finance.+Hope+you+may+choose+up+a+knife+best.+Use+a+Q-tip+or+toothbrush+to+scrub+the+knife+after+using+it.%0D%0A%0D%0AThe+Flash+II+has+a+army+background%2C+but+unlike+the+Civilian%2C+it+has+uses+that+go+above+and+past+the+rippin%27+and+the+tearin%27.+It%27s+meant+to+be+a+partial+utility+knife+that+can+employ+the+serrated+lower+end+for+easy+sawing+procedures+whereas+the+graceful+higher+portion+can+handle+scraping+or+slashing.+The+blade+is+3.5+inches+of+AUS-8+metal+that+carries+a+bolt+for+easy+one-handed+opening.+It+is+spring+loaded%2C+so+it+snaps+to+attention+with+a+minimal+of+effort.+The+ergonomic+deal+with+means+it%27s+arduous+to+drop+and+almost+unattainable+to+have+pulled+out+of+your+grip.+An+awesome+folding+knife+for+the+survivalist+that+wants+just+a+little+bit+of+every+thing.++Buy&create=Create FRN handle] supplies resists slipping from all angles. It additionally comes with a clip that be hooked up to the blade itself in all four doable positions. Either facet works, and it both be blade-up or blade-down in your [http://Www.Indonesia-Exporter.com/forum/index.php?a=member&m=438390 pocket relying] on how it's hooked up to the clip.<br><br>You possibly can take a look at our evaluation section to seek out out reviews of different particular person knives separately which will probably be useful for you. You can make your choice after evaluating the knives and pick the one out that you simply determine to purchase. You may assume what are the few elements that ought to be considered in case you are about to select the precise knife. This is one of the commonest doubts that most of you might need. Now we have listed a few of the major elements under When you have any feedback about our top knife picks or something referring to the site please don't hesitate to get in touch<br><br>Subject to your utilization, Ceramic Blade Pocket Knives can supply a lightweight, stronger knife that doesn't must be sharpened as typically, just isn't going to move on a metallic taste to meals and will not rust. However, you could take special care of ceramic blade pocket knives , since they'll chip or break plus they cost more than traditional solid metal pocket knives do. We will always use new seashore towels, so I was thrilled to search out these Marvel Microfiber Seaside Towels on Private Creations. My son Ryan is a fan of Ironman, so this beach towel was excellent for him.<br><br>The center section is a 4-burner gasoline grill with 540 square inches of main cooking floor and a one hundred fifty sq. inch removable warming rack above. The primary grates are forged iron and are held in tight by three screws so they will not bounce out in transit. The burners line up aspect to side, so you possibly [http://www.victorinox.com/ch small knives] can cook dinner indirectly or have a hot zone on one facet and a heat zone on the opposite. There are porcelain coated warmth diffuser plates above the chrome steel burners and a tray under the grill is angled to catch and drain grease into a small, easy-to-remove pan.<br><br>A good folding knife is an merchandise that has a place in every pocket. It should be lightweight, sturdy, straightforward to deploy, have good ergonomics, lock up tight, and maintain an edge well. And since I don’t like having to work any more than I have to, it would be good if it’s inexpensive too. With that record [http://abcnews.go.com/blogs/politics/2013/03/tsa-to-allow-pocket-knives-on-planes/ army knife] in thoughts, I’d say the Gerber Mini Covert is value a look. Are there knives with better steels, more durable construction, extra robust mechanisms? Completely. However for the $33 bucks and alter I paid for it, the Gerber Mini Covert has more than glad this peasant. Find it online right here<br><br>The knife's size and curvy tail finish make for an excellent pocket companion, even and not using a clip. The entire package deal is sufficiently small to suit into your coin pocket (geez, I at all times want to write " coin slot " but that's one thing totally totally different). The only ding I have against the knife comes within the type of the very exposed and pointy rear tang. Its a minor ding, its not as sharp as it is on other knives, and the pattern requires it to be exposed, however still. Superb, positively above common, however not excellent.<br><br>Twenty-pound paper is the standard weight for many enterprise needs. To start with, utilizing top quality linen or other specialty paper is not obligatory. Nonetheless, it is probably a good suggestion to maintain a small supply of nice paper readily available for instances when you need [http://www.thebestpocketknifereviews.com/ http://Www.thebestpocketknifereviews.Com/] to make written communications with high-level people, comparable to prospective purchasers, or consultants you would like to thank for assist given. Folks develop first impressions of you through your written communications In case you are protecting it simple, that's okay. A effectively-developed letterhead on 20-pound printer paper can look very professional.
 
==Integers modulo ''p''==
 
On 18 Jun 2005, [[Antoine Joux]] and Reynald Lercier announced the computation of a discrete logarithm modulo a 130-digit (431-bit) [[strong prime]] in  three weeks, using a 1.15&nbsp;GHz 16-processor HP AlphaServer GS1280 computer and a [[number field sieve]] algorithm.<ref>Antoine Joux, “Discrete logarithms in GF(''p'') &ndash; 130 digits,” June 18, 2005, http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0506&L=nmbrthry&T=0&P=20.</ref>
 
On 5 Feb 2007 this was superseded by the announcement by Thorsten Kleinjung of the computation of a discrete logarithm modulo a 160-digit (530-bit) [[safe prime]], again using the [[number field sieve]].  Most of the computation was done using idle time on various PCs and on a parallel computing cluster.<ref>Thorsten Kleinjung, “Discrete logarithms in GF(''p'') &ndash; 160 digits,” February 5, 2007, http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0702&L=NMBRTHRY&P=R45&D=0&I=-3&T=0.</ref>
 
==Finite fields==
 
The current record (as of January 2014) in a finite field of characteristic 2 was announced by Robert Granger, Thorsten Kleinjung, and Jens Zumbrägel on 31 January 2014. This team was able to compute discrete logarithms in GF(2<sup>9234</sup>) using about 400,000 core hours. New features of this computation include a modified method for obtaining the logarithms of degree two elements and a systematically optimized descent strategy.<ref>Jens Zumbrägel, "Discrete Logarithms in GF(2^9234)", 31 January 2014, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9aa2b043.1401.</ref>
 
Previous records in a finite field of characteristic 2 were announced by:
 
* Antoine Joux on 21 May 2013. His team was able to compute discrete logarithms in the field with 2<sup>6168</sup> = (2<sup>257</sup>)<sup>24</sup> elements using less than 550 CPU-hours. This computation was performed using the same index calculus algorithm as in the recent computation in the field with 2<sup>4080</sup> elements. <ref> Antoine Joux, "Discrete logarithms in GF(2<sup>6168</sup>) [=GF((2<sup>257</sup>)<sup>24</sup>)]", May 21, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1305&L=NMBRTHRY&F=&S=&P=3034.</ref>
 
* Robert Granger, Faruk Göloğlu, Gary McGuire, and Jens Zumbragel on 11 Apr 2013.  The new computation concerned the field with 2<sup>6120</sup> elements and took 749.5 core-hours.
 
* Antoine Joux on Mar 22nd, 2013. This used the same algorithm <ref>  Antoine Joux. A new index calculus algorithm with complexity $L(1/4+o(1))$ in very small characteristic, 2013, http://eprint.iacr.org/2013/095 </ref> for small characteristic fields as the previous computation in the field with 2<sup>1778</sup> elements. The new computation concerned the field with 2<sup>4080</sup> elements, represented as a degree 255 extension of the field with 2<sup>16</sup> elements. The computation took less than 14100 core hours. <ref> Antoine Joux, "Discrete logarithms in GF(2<sup>4080</sup>)", Mar 22, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1303&L=NMBRTHRY&F=&S=&P=13682.</ref>
 
* Robert Granger, Faruk Göloğlu, Gary McGuire, and Jens Zumbragel on 19 Feb 2013.  They used a new variant of the medium-sized base field [[function field sieve]], for binary fields, to compute a discrete logarithm in a field of 2<sup>1971</sup> elements. In order to use a medium-sized base field, they represented the field as a degree 73 extension of the field of 2<sup>27</sup> elements. The computation took 3132 core hours on an SGI Altix ICE 8200EX cluster using Intel (Westmere) Xeon E5650 hex-core processors.<ref>Faruk Gologlu et al.,  On the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in <math>\mathbb{F}_{2^{1971}}</math>, 2013, http://eprint.iacr.org/2013/074.</ref>
 
* Antoine Joux on 11 Feb 2013. This used a new algorithm for small characteristic fields. The computation concerned a field of 2<sup>1778</sup> elements, represented as a degree 127 extension of the field with 2<sup>14</sup> elements. The computation took less than 220 core hours. <ref> Antoine Joux, "Discrete logarithms in GF(2<sup>1778</sup>)", Feb. 11, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1302&L=NMBRTHRY&F=&S=&P=2317.</ref>
 
The current record (as of 2013) in a finite field of characteristic 2 of prime degree was announced by the CARAMEL group on April 6, 2013. They used the [[function field sieve]] to compute a discrete logarithm in a field of 2<sup>809</sup> elements. <ref>The CARAMEL group:  Razvan Barbulescu and Cyril Bouvier and Jérémie Detrey and Pierrick Gaudry and Hamza Jeljeli and Emmanuel Thomé and Marion Videau and Paul Zimmermann, “Discrete logarithm in GF(2<sup>809</sup>) with FFS”, April 6, 2013, http://eprint.iacr.org/2013/197.</ref>  The previous record in a finite field of characteristic 2 of prime degree was announced by Antoine Joux and Reynald Lercier on 23 Sep 2005. They used the [[function field sieve]] to compute a discrete logarithm in a field of 2<sup>613</sup> elements. The computation took 17 days on four 16-processor (1.3&nbsp;GHz) nodes of the Itanium 2-based Bull computer Teranova.<ref>Antoine Joux, “Discrete logarithms in GF(2<sup>607</sup>) and GF(2<sup>613</sup>),” September 23, 2005, http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0509&L=NMBRTHRY&P=R1490&D=0&I=-3&T=0.</ref>
 
The current record (as of 2014) for a field of characteristic 3 was announced by Francisco Rodríguez-Henríquez on 27 January 2014. The computation used Joux's algorithm to compute discrete logarithms in the 1303-bit field GF(3<sup>6&nbsp;&middot;&nbsp;137</sup>), taking 918 CPU hours. This work did not exploit any "special" aspects of the field such as Kummer or twisted-Kummer properties.<ref>Francisco Rodríguez-Henríquez, “Announcement,” 27 January 2014, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;763a9e76.1401.</ref><ref> Gora Adj and Alfred Menezes and Thomaz Oliveira and Francisco Rodríguez-Henríquez, "Computing Discrete Logarithms in F_{3^{6*137}} using Magma", 26 Jan 2014, http://eprint.iacr.org/2014/057.</ref>  The previous record was announced in 2012 by a joint Fujitsu, NICT, and Kyushu University team, that computed a discrete logarithm in the field of 3<sup>6&nbsp;&middot;&nbsp;97</sup> elements and a size of 923 bits,<ref>Kyushu University, NICT and Fujitsu Laboratories Achieve
World Record Cryptanalysis of Next-Generation Cryptography, 2012, http://www.nict.go.jp/en/press/2012/06/PDF-att/20120618en.pdf.</ref> using a variation on the [[function field sieve]] and beating the previous record in a field of 3<sup>6&nbsp;&middot;&nbsp;71</sup> elements and size of 676 bits
<ref>Takuya Hayashi et al., Solving a 676-bit Discrete Logarithm Problem in GF(3<sup>6''n''</sup>), 2010, http://eprint.iacr.org/2010/090.</ref> by a wide margin.
 
Over fields of "moderate"-sized characteristic, notable computations as of 2005 included those a field of 65537<sup>25</sup> elements (401 bits) announced on 24 Oct 2005, and in a field of 370801<sup>30</sup> elements (556 bits) announced on 9 Nov 2005.<ref>A. Durand, “New records in computations over large numbers,” The Security Newsletter, January 2005, http://eric-diehl.com/letter/Newsletter1_Final.pdf.</ref> The current record (as of 2013) for a finite field of "moderate" characteristic was announced on 6 January 2013. The team used a new variation of the [[function field sieve]] for the medium prime case to compute a discrete logarithm in a field of 33341353<sup>57</sup> elements (a 1425-bit finite field).<ref>Antoine Joux, “Discrete Logarithms in a 1425-bit Finite Field,” January 6, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1301&L=NMBRTHRY&F=&S=&P=2214.</ref><ref name="Joux-ePrint"> Faster index calculus for the medium prime case. Application to 1175-bit and 1425-bit finite fields,  Eprint Archive, http://eprint.iacr.org/2012/720</ref>  The same technique had been used a few weeks earlier to compute a discrete logarithm in a field of 33553771<sup>47</sup> elements (an 1175-bit finite field).<ref>Antoine Joux, “Discrete Logarithms in a 1175-bit Finite Field,” December 24, 2012, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1212&L=NMBRTHRY&F=&S=&P=13902.</ref><ref  name="Joux-ePrint"/>
 
==Elliptic curves==
 
[[Certicom]] Corp. has issued a series of Elliptic Curve Cryptography challenges. Level I involves fields of 109-bit and 131-bit sizes. Level II includes 163, 191, 239, 359-bit sizes. All Level II challenges are currently believed to be computationally infeasible.<ref>Certicom Corp., “The Certicom ECC Challenge,” http://www.certicom.com/index.php/the-certicom-ecc-challenge.</ref>
 
The Level I challenges which have been met are:<ref>Certicom Research, Certicom ECC Challenge (Certicom Research, November 10, 2009), http://www.certicom.com/images/pdfs/challenge-2009.pdf.</ref>
 
* ECC2K-108, involving taking a discrete logarithm on a [[Koblitz curve]] over a field of 2<sup>108</sup> elements. The prize was awarded on 4 April 2000 to a group of about 1300 people represented by Robert Harley.  They used a parallelized [[Pollard's rho algorithm for logarithms|Pollard rho method]] with speedup.
 
* ECC2-109, involving taking a discrete logarithm on a curve over a field of 2<sup>109</sup> elements.  The prize was awarded on 8 April 2004 to a group of about 2600 people represented by Chris Monico.  They also used a version of a parallelized [[Pollard's rho algorithm for logarithms|Pollard rho method]], taking 17 months of calendar time.
 
* ECCp-109, involving taking a discrete logarithm on a curve modulo a 109-bit prime.  The prize was awarded on 15 Apr 2002 to a group of about 10308 people represented by Chris Monico. Once again, they used a version of a parallelized [[Pollard's rho algorithm for logarithms|Pollard rho method]] method, taking 549 days of calendar time.
 
None of the 131-bit (or larger) challenges have been met as of 2010.
 
In July 2009, Joppe W. Bos, Marcelo E. Kaihara, Thorsten Kleinjung, Arjen K. Lenstra and Peter L. Montgomery announced that they had carried out a discrete logarithm computation on an elliptic curve modulo a 112-bit prime. The computation was done on a cluster of over 200 PlayStation 3 game consoles over about 6 months.  They used the common parallelized version of [[Pollard's rho algorithm for logarithms|Pollard rho method]].<ref>1. Joppe W. Bos and Marcelo E. Kaihara, “PlayStation 3 computing breaks 2^60 barrier:  112-bit prime ECDLP solved,” EPFL Laboratory for cryptologic algorithms - LACAL, http://lacal.epfl.ch/112bit_prime</ref>
 
==References==
 
{{reflist}}
 
[[Category:Asymmetric-key algorithms]]
[[Category:Modular arithmetic]]
[[Category:Logarithms]]
[[Category:Computational hardness assumptions]]
[[Category:World records]]

Latest revision as of 15:35, 8 January 2015



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