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-   -   conversion to GF(2) (https://www.mersenneforum.org/showthread.php?t=4000)

bigbud 2005-04-14 12:15

conversion to GF(2)
 
would it be possible to "convert"

GF(211921F4A73A3761B88323D6A34DF9E984D08FF25C491DF5DB3251310FAB9C36ACE903E01D41B9BF1EA5CBEAA79FD1D7036835E45933E34825B87C9AB45C2C4F^1)

to GF(2), for utilizing Coppersmith's index-calculus ?

eg, convert from one GF representation to another, with all numbers converted to GF2, espeically the polynomial etc.

Best Regards, Bud

R.D. Silverman 2005-04-14 13:45

[QUOTE=bigbud]would it be possible to "convert"

GF(211921F4A73A3761B88323D6A34DF9E984D08FF25C491DF5DB3251310FAB9C36ACE903E01D41B9BF1EA5CBEAA79FD1D7036835E45933E34825B87C9AB45C2C4F^1)

to GF(2), for utilizing Coppersmith's index-calculus ?

eg, convert from one GF representation to another, with all numbers converted to GF2, espeically the polynomial etc.

Best Regards, Bud[/QUOTE]


I will assume that the large hex number is prime. At least it is odd :smile:

What do you mean by "convert"?? You are not changing representations
within a fixed field; you are asking about some (hypothetical) relation
between GF(p) and GF(2)???

You can conduct an index calculus attack on GF(p) by the number field
sieve. The field given above would require a massive effort. Good luck.

Can you clarify?

bigbud 2005-04-14 17:58

Dear Dr Silverman,

Perhaps this is a stupid question, but the question is:

Is it possible to convert modular arithmetices (espcially) discrete logs in prime fields GF(p) to GF(2) representation ?

Like if we have an diskrete log like a^x mod p = b (which is to solve for x) in a prime field GF(p) - all parameters a,b,x,p elements of this field.

Is there any possibility to convert those number to GF(2) like a,b elements in GF(2), p probably the irreducible polynomial, with the goal to solve the log in this field ?

dave_dm 2005-04-14 18:18

Is your question the following?

Does there exist a polynomial-time reduction from the problem "Discrete log in GF(p)" to the problem "Discrete log in GF(2^n)" ?

I put the 2^n in there because I know of an *extremely* efficient algorithm to compute discrete logs in GF(2) :showoff:

Dave

bigbud 2005-04-14 18:35

Hello,

And thanks for the reply, the answer would indeed be yes,
can we somehow translate our problem to GF(2) for usage of the powerful dlp solving algorithms which exists ?

Are you talking about coppersmith's index-calculus ?

Best Regards

R.D. Silverman 2005-04-14 20:12

[QUOTE=bigbud]Dear Dr Silverman,

Perhaps this is a stupid question, but the question is:

Is it possible to convert modular arithmetices (espcially) discrete logs in prime fields GF(p) to GF(2) representation ?

Like if we have an diskrete log like a^x mod p = b (which is to solve for x) in a prime field GF(p) - all parameters a,b,x,p elements of this field.

Is there any possibility to convert those number to GF(2) like a,b elements in GF(2), p probably the irreducible polynomial, with the goal to solve the log in this field ?[/QUOTE]

The question isn't "stupid" per se, but I don't understand what you mean
by "convert modular arithmetic in prime fields to GF(2) representation".

What do you mean by "convert"? by "representation".

All finite fields of a given order are isomorphic. Thus, any map from
GF(a) to GF(b) will either be surjective or injective, depending on the
size of the fields. There will never exist a 1-1 map unless the sizes are the same.

The only time solving a discrete log in one field helps to solve a DL in another is if the first field is a sub-field of the second AND the target
log in the second field is in the orbit of the generator used in the first field.

May I suggest that you do some reading about the structure of Finite Fields?
Lidl & Neiderreiter's book is superb.

NFS *IS* an index calculus algorithm. One can solve logs in GF(p) with it.

JHansen 2005-04-14 21:17

[QUOTE=R.D. Silverman]NFS *IS* an index calculus algorithm. One can solve logs in GF(p) with it.[/QUOTE]

I've never seen an implementation of NFS for discrete logs. Do you know anybody who has made an efficient implementation? If so, how large can p be, if you want to be able to solve a discrete log in GF(p) in 1GHz month, say?

--
Cheers,
Jes

maxal 2005-04-14 21:26

[QUOTE=JHansen]I've never seen an implementation of NFS for discrete logs. Do you know anybody who has made an efficient implementation?[/QUOTE]Look at [url]http://www.cs.toronto.edu/~cvs/dlog/[/url]
[QUOTE=JHansen]If so, how large can p be, if you want to be able to solve a discrete log in GF(p) in 1GHz month, say?[/QUOTE]If you measured that I'd be interested in knowing the answer.

R.D. Silverman 2005-04-15 12:23

[QUOTE=JHansen]I've never seen an implementation of NFS for discrete logs. Do you know anybody who has made an efficient implementation? If so, how large can p be, if you want to be able to solve a discrete log in GF(p) in 1GHz month, say?

--
Cheers,
Jes[/QUOTE]

O. Schirakauer has an implementation, AFAIK. The big headache is the LA.
One must solve the matrix mod p-1, not just mod p.

ColdFury 2005-04-16 01:13

[QUOTE=R.D. Silverman]O. Schirakauer has an implementation, AFAIK. The big headache is the LA.
One must solve the matrix mod p-1, not just mod p.[/QUOTE]

Indeed, this is where my toy implementation failed.


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