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Old 2016-09-01, 14:16   #1
Joe O
 
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"Improved Pollard rho method for computing discrete logarithms over finite extension fields"
by Ping Wang, , Fangguo Zhang,
Has anyone read this? I can only find it behind a pay wall.
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Old 2016-09-13, 21:11   #2
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Plug in the title to scholar.google.com and follow the link to the pdf.
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Old 2016-09-14, 13:40   #3
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Thank You!
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Old 2016-09-14, 14:03   #4
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I'd be interested in hearing what you or others think. I've had "Experiment with Wang/Zhang 2012 Rho cycle finding" on my todo list for a couple years but haven't tried any implementation. My current implementation is very traditional, and I have it race with a BSGS. Bai/Brent 2008 is also worth looking at.
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Old 2016-09-14, 20:50   #5
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Quote:
Originally Posted by Joe O View Post
"Improved Pollard rho method for computing discrete logarithms over finite extension fields"
by Ping Wang, , Fangguo Zhang,
Has anyone read this? I can only find it behind a pay wall.
Please also see over here by.
Please too refer up to here by.

t = gl (mod p).

Quote:
Originally Posted by Raman View Post
The Pollard’s Rho approach can also be used to solve the Discrete Logarithm problem. Let xi = ta[SUB]i[/SUB]gb[SUB]i [/SUB](mod p) and a0 = 0, b0 = 0, x0 = 1. Here, we define a pseudo random function such as:
(ai+1, bi+1) = (ai + 1 (mod φ(p)), bi), if 0 < xi < p/3
(ai+1, bi+1) = (2ai (mod φ(p)), 2bi (mod φ(p))), if p/3 < xi < 2p/3
(ai+1, bi+1) = (ai, bi + 1 (mod φ(p))), if 2p/3 < xi < p
And so, we have that
xi+1 = tx[SUB]i[/SUB] (mod p), if 0 < xi < p/3
xi+1 = xi2 (mod p), if p/3 < xi < 2p/3
xi+1 = gx[SUB]i[/SUB] (mod p), if 2p/3 < xi < p

We compare the different values of xi to find out a match. Similar to the tortoise-hare algorithm (Floyd’s cycle finding algorithm), we compare x1 with x2, x2 with x4, x3 with x6 and so on. We compare xm with x2m each time, to look for a match. We have two variables for x, a and b, each of which hold the iterations for m and 2m respectively.
If a match is found out, say xj = xk, then we have ta[SUB]j[/SUB]gb[SUB]j[/SUB] = ta[SUB]k[/SUB]gb[SUB]k[/SUB] (mod p). Since the period is φ(p), or a divisor of φ(p), comparing the powers of t and g, and writing t = gl, we will have (aj – ak).l ≡ bk – bj (mod φ(p)), from which we could calculate the value of l as to be l = (bk – bj)(aj – ak)-1 (mod φ(p)).
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Old 2016-09-15, 05:11   #6
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re: "originally posted by Raman"

So, you are now trying to write a book, or what? (the correct term should be "plagiarize" a book).
A little bit too empty that book, for out tastes here...
Still a lot of things to add to it...
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Old 2016-09-15, 08:29   #7
Raman
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Quote:
Originally Posted by LaurV View Post
re: "originally posted by Raman"

So, you are now trying to write a book, or what? (the correct term should be "plagiarize" a book).
A little bit too empty that book, for out tastes here...
Still a lot of things to add to it...
Hi,

So that how did you get that link?

such that

What concepts do you think that are being newly appropriate / fancy enough / well suited up to add over to that highlighted book that you had mentioned that above?

That was something that I had submitted for my B.Tech degree thesis.
Who had uploaded it into archive.org?
Apparently it cannot be my university from which I had studied and graduated because there were lots of additions that I had made after I had submitted my final thesis for my B.Tech degree, and the latest changes are being present.

Some of them were originally prepared for Google Groups Notes, although this feature is being discontinued right now.
To compile everything into a website in the future. It should be safe there. I shouldn't trust third party sources for web site hosting but Google sites should be safe at least for centuries. I had previously lost data from Yahoo Briefcase, Yahoo Geocities, Orkut, etc. are all being now closed. Orkut was predicted by me that it would close in 2013, yes that it had been closed up on over Tuesday 30 September 2014. Yahoo Games is also now being deactivated too.

Right now that without a personal website things will be metastable in untrustworthy third party web site hosting, i.e. existing for long time - but destroyed at any time at happening of unfortunate things.
Hosting website will cost money.
Where as, let me at least stay with in a job for a while - the an.
Where as, for a while - the an - up - an? - before itself, by earlier, a while ago - for the an.
By using be being - right now - does do - does not - do not - not not !?


The latest changes that I had made to the material was in the year 2010.
At that time, I knew nothing about Cornacchia's algorithm, Ideonal Numbers, Class Numbers, Values of k such that there are no numbers N such that if N cannot be written as a²+kb² form, then N³ can be written as a²+kb² form (or apparently any odd power of N), etc. (end of thinking capacity!) and then so on.
although there is article up on over sum of two squares representation, which I first studied in second half of 2009. The final B.Tech degree thesis was being submitted up on over Friday 8 May 2009.

See that the newest / latest Mersenne Primes 57885161 and 74207281 are not being there included, Mersenne Prime 42643801 was only being known on Thursday 4 June 2009, although it was first being reported on Sunday 12 April 2009 went unnoticed. Similarly to as like newest / latest Mersenne Prime 74207281 was only being known on Thursday 7 January 2016, although it was first being reported on Thursday 17 September 2015 went unnoticed - for the - an a.
See that the newest / latest Wagstaff Primes 13347311 and 13372531 are not being there included also too.

Linux / Unix Operating System does do save a ~ back up file, where as while Windows Operating System does not - do not - not not !?

Quote:
Originally Posted by Raman
MERSENNE Prime numbers p for which (2p-1) is being a prime number.
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217,
4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917,
20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609,
57885161, 74207281
WAGSTAFF Prime numbers p for which (2p+1)/3 is being a prime number.
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347,
701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369,
117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, 13347311,
13372531
REPUNIT Prime numbers p for which (10p-1)/9 is being a prime number.
(Any more new terms are there being missing out?)
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343
Prime numbers p for which (10p+1)/11 is being a prime number.
(Are there being any specific name for this sequence?)
5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011

To consider with in other bases as like for the - an a
b = 2, 3, 5, 6, 7, 10, 11, 12
cases also too.
Man, these material are being well suited up / appropriate / fancy enough for the round neck / polo collar T-shirt / shirt! Or a scrolling marquee screen saver (Guess what operating system! - returning back again against - for the - an
a) - Cf. After Dark Screen Saver that I do enjoy up with in - What does Cf. mean up - an? - before itself, by earlier, a while ago - for the an.
By using be being - right now - does do - does not - do not - not not !?


Round neck / polo collar T-shirt / shirt
of 7 different colours
to wear up on over 7 different days of the week
Matching data to be printed out up on over it
Logic puzzle / games / sports - matches - data / matter / material.

Well, let me know very well about this thing around - variably.
think stuff style item step idea


SUNDAY - RED
MONDAY - WHITE
TUESDAY - BLUE
WEDNESDAY - PINK
THURSDAY - GREEN
FRIDAY - YELLOW
SATURDAY - BLACK

RED - (front side) Ideonal Numbers, (back side) 2-Ideonal numbers (just as like 10-adic integers only certainly), i.e. values of k such that all the prime numbers representable in the (a/2)²+k(b/2)² form can be given by a set of residue classes (mod 4k), with every prime number of these residue classes being uniquely writable in the (a/2)²+k(b/2)² form, (right sleeve) formula for product of two quadratic forms into a²+kb² quadratic form, (left sleeve) and then formula for product of a quadratic form with a a²+kb² quadratic form into same quadratic form.
WHITE - (front side) Mersenne Prime numbers, (back side) Wagstaff Prime numbers, (right sleeve) Repunit Prime numbers, (left sleeve) Prime numbers p for which (10p+1)/11 is being a prime number.
BLUE - (front side) Amicable pairs, (back side) Aliquot cycles of length ≥ 4. Does no one do know an Aliquot cycle of length 3 or 7 or 10 to 27 or ≥ 29, or an Amicable pair of opposite parity?
PINK - Open end Aliquot sequences (below 10000?) - not a valid sequence at all because computations in progress might reduce these numbers up on over time. To consider with in only lowest distinct members certainly, etc. (end of thinking capacity!) and then so on. Space is being available - not not.
GREEN - Values of k such that there are no numbers N such that if N cannot be written as a²+kb² form, then N³ can be written as a²+kb² form (or apparently any odd power of N).
YELLOW - (front side) Mersenne Prime / Composite Number Factors (2 to 149?), (back side) Fermat Prime / Composite Number Factors (1 to 32?).
BLACK - (front side) π digits, (back side) e digits. Other transcendental / algebraic / irrational constants are being interesting to consider up also too - Euler-Mascheroni constant, Catalan's constant, Apéry's constant, √2, √3, etc. (end of thinking capacity!) and then so on. Space is being available - not not.

i.e. = e.g. ??

Well, let me know very well about this thing around - variably.
think stuff style item step idea


Mathematical constants can be calculated to infinitely many digits of precision up on over a given fixed computer, but physical constants, e.g. gravitational constant, speed of light in vacuum, charge of electron, mass of electron, mass of proton, mass of neutron, permittivity of vacuum, permeability of vacuum, Avagadro's constant, Boltzmann's constant, Planck's constant, Stefan-Boltzmann's constant, molar gas constant, Hubble's constant, Dirac's constant, Wien's displacement constant, acceleration due to gravity on earth, atmospheric pressure on earth, mass of earth, radius of earth, cosmological constant, etc. (end of thinking capacity!) and then so on ??

The average atmospheric pressure on the surface of Mars at 610 Pascals is being very close to the triple point of water at 611 Pascals. So that on the surface of Mars, it is possible that three states of matter - solid, liquid, gas can co-exist. Although that the atmospheric pressure on the surface of Mars can vary from 30 Pascals to 1155 Pascals.

How many states of matter are there? Eighteen and counting! Solid, liquid, gas, plasma, degenerate matter, photonic matter, quantum, bose-einstein condensate, fermionic condensate, superconductivity, superfluid, supersolid, quantum spin liquid, string-net liquid, supercritical fluid, dropleton, Jahn-Teller metal, quark-gluon plasma.

I do like up with in piling / accumulating of things - during at all @ - time period frame duration - does - not not as - stamp collection - a - for the an !?

There is much further more difference between white colour font text and then yellow colour font text.

@ ? - How old are you at all ??

To consider with in only Mersenne Forum's round neck / polo collar T-shirts / shirts rather than instead of Mersenne Forum's guinness world records prime number badges certainly, etc. (end of thinking capacity!) and then so on ?! Space is being available - not not also too !?

Trying out to adding text until / till repeatedly getting from 16384 Mersenne Forum's post character limit to Mersenne Forum's post character limit is being reached up - could be able to - consider it up - up over!

To consider with in only returning back again against from 16384 Mersenne Forum's post character limit to Mersenne Forum's post character limit certainly, etc. (end of thinking capacity!) and then so on ?? Space is being available - not not also too !!

To consider with in only one single lengthy / long post rather than instead of many / multiple posts certainly, etc. (end of thinking capacity!) and then so on ? Space is being available - not not also too !

Last fiddled with by Raman on 2016-09-15 at 08:37
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Old 2016-09-15, 12:49   #8
LaurV
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Quote:
Originally Posted by Raman View Post
Hi,
So that how did you get that link?
By searching for relevant terms in the fragment you quoted.
That internet is a bitch... you know?
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Old 2016-09-18, 14:42   #9
Raman
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Quote:
Originally Posted by LaurV View Post
By searching for relevant terms in the fragment you quoted.
That internet is a bitch... you know?
Hi,

But what is the point in searching for something that you don't seek an answer to? Do you use Google for some custom data?
What do you exactly mean by 'Internet is a bitch'? A spy that reports a person's own wrongdoings to someone else?
You searched for text inside the quote tag? Why didn't you do so for text outside the quote tag?
I wrote a full prose. You replied to only the first paragraph?
To consider using emoticons.
Emoticon = smiley? What is the difference exactly between emoticons and emotions?

Man, not only some number theory data is being archived, but also some group theory data. I am not sure if I did it or someone else made it into the archive way back machine.
For some reason, in mersenne forum, font text size tag is not being working out at all.
Editing one's own post time limit is being one hour exactly.
Yellow yellow dirty fellow, white white orbit white.
Yellow colour font text is being relevant data being obscured to avoid reader's distractions.

Giving away points to convey is being more important than using different kinds of words at all.
Short posts without some type of junk is being better than long posts with some type of junk. Trying out to control it up uncontrollably. Good thing to take a few days' break between two successive posts.
This is likely being caused by addiction and also by spending too much time with it. They all come to me by instinct. Good thing to divert my own attention, deviate from it and then go out away somewhere outside, enjoy places, every day different places to go out. Going route ≠ coming route.
Should have been spending out more time playing out rather than instead of working out every day. All work and no play makes Jack a dull boy. I am not Jack any way and I cannot be any way. Rather than = instead of?
My mind is being time variant, mercurial, changing out from between calm state condition to wild state condition frequently often. Frequently = often? Lol = Laughing out loud?

At the poles, one year is equal to one day at the equator with sun very low in elevation.
Quote tag can be used out for as a separator, rather than instead of quoting text from other people or from myself.
Wondering out why I do list out things pointwise!
And then numbering things pointwise with alphabets rather than by using numbers?

Aliquot sequence 314718 is being approaching the iteration number 17000! When ever things are being compared to it, aliquot sequence 4788 is being: 10540 to be equivalent when ever things are being compared to it?
Aliquot sequence 4788: iteration 10400, iteration 10490 nice split up results!
Aliquot sequence 4788: iteration 10473, iteration 10489 quick ecm curves results?
Aliquot sequence 11040: iteration 9812 nearly nice split up results!
689 divides Aliquot sequence 4788: iteration 10497. How many groups of order 689 are being there? Why? Just asking. Not seeking out an answer to it. I know it by myself. Two groups of order 689 are being there. Cyclic group and then Frobenius group. Why?

To predict out to what given fixed family does an open end aliquot sequence does do terminate up.
And then to consider by measuring or by estimating the probability with in which an open end aliquot sequence does do terminate up to a given fixed family. By using be being.

To respect out (give take out reputation) people who win out games, sports, puzzles by skill rather than instead of by chance. Rather than ≠ instead of?
FactorDB has got bandwidth limits and query limits.
Attachment tag form format has been getting limits. Why? I have been observing that it is being set up to at a some arbitrary limit levels. Why?
Limited mersenne forum attachment tag formats are being to be available up out.
I observed that Linux / Unix operating system text editor case file always seeks opening saved stored file pointer to the last restore point score. Why?
To consider out with in recharging at 13% battery life limit levels. Why?
To consider out with in alerting at 10% battery life limit levels. Why?
Mersenne Forum right now that many multiple attachments are being supported out even for the common man up! Why?
Confidentially that image tags it is being case they have been disabled out also too up? Why?

Quote:
Originally Posted by schickel View Post
PS. The guide is pretty benign: 2^4 * 7
That is not being a driver at all. But only, let alone a stable guide. Not a downdriver at all - not - not obviously. Cannot be it. That but only = let alone?
What do you think the maximum and minimum aliquot sequence growth rate of the next subsequent iteration can be? Why?

Quote:
Originally Posted by Raman View Post
How about posting all of my opinions over here, rather than in their appropriate threads?
And then or also too with in ECM curves with in:

Microsoft Golf 1.0 Lies:
Deep Rough, Rough, Fairway, Green, Sand, Water, Dirt, Asphalt, Concrete, Pine Needles, Tall Grass, Mud, Dormant Grass, Lava, Stream, Ocean, Hazard, Tee, In The Hole, Out Of Bounds, Deep Grass.
Hazard = Dirt = Lava = Mud?
Water = Stream = Ocean?
Tall Grass = Deep Grass?

Memorize and then ≠ or recall?
How much do you think out that you can be recollecting - is being case?

Hi, Arrange with in order, man! - Sport / game / puzzle.
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Old 2016-09-18, 15:42   #10
danaj
 
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Can we keep the posts relevant to the subject? We're also not being paid by the word.
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