Max Alekseyev on Wed, 27 Mar 2013 16:08:53 +0100

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 efficiency of modular exponentiation

• To: pari-users@pari.math.u-bordeaux.fr
• Subject: efficiency of modular exponentiation
• From: Max Alekseyev <maxale@gmail.com>
• Date: Wed, 27 Mar 2013 11:08:41 -0400
• Delivery-date: Wed, 27 Mar 2013 16:08:53 +0100
• Dkim-signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=20120113; h=mime-version:x-received:date:message-id:subject:from:to :content-type:content-transfer-encoding; bh=KDVsARALwsQfA80jWnRb0eGB2z+yvOeD5pXxe1LiVX0=; b=p1fBsOYNYi0Fy9yfIj3PU7jgc0/l58zrIo+0lx4ypzvDjaiuzk19mBPo0rhZaoOCcl tE80tzJL+RACXqiHXYm70dpMlkLK8NDh/nYgeBvRpejhxZd9tnytRa/WqfIJS7lOjgpY TGiz5Sq+fxLf1TOYnVT7YkMogq3uJx2RjtK8O957sfliyxzbAoI1VnbEp07eOYV1KgUZ k/2EU0FO4/5yhLCfzx2P6JPMKpl7zvZxPaxq9zyntIIg+ezcd7h/tapVMP32QKlEMvUD rcUwvDzkU23CDoxpcx8dC8f/n+/rraV/IWQ1hjGJecHNXDSrTgAMWKYAAUom03nPTvTL 8yvw==

```Just out of curiosity: why the magma script is much faster the gp script?
(both scripts are quoted below)

Regards,
Max

---------- Forwarded message ----------
From: Antoine Joux <Antoine.Joux@cryptoexperts.com>
Date: Sat, Mar 23, 2013 at 10:30 AM
Subject: Re: Discrete Logarithms in GF(2^4080)
To: NMBRTHRY@listserv.nodak.edu

Dear All,

In order to ease verification, here is a magma script which is much
faster than the initial gp script.

R:=RealField(1500);
pival:=Pi(R);
gf2:=GF(2);
gf2pol<x>:=PolynomialRing(gf2);
gf<a>:=ext<gf2|x^16+x^5+x^3+x+1>;
gfpol<X>:=PolynomialRing(gf);
A:=a+a^256;
gfext<u>:=ext<gf|X^255+A>;
g:=u+a;
Z:=&+[(Floor(pival*2^(val+1)) mod 2)*u^(val div 16)*a^(15-(val mod
16)):  val in [0..4079]];
lg:=5935377918714230432230999933717750972257258069856089974977949660023882324802076896698410404982590206958649631628772672466127669634274818596358582683302117352838165909188471579953420256387758687914285280177954658284572333669860436891005920917402903089607764477430547370770112475381244907965544496884807875672058922056500650363713396354721008645927686282457785486271699937105302489522475021983391024140847168793050589732859677058978247175646259738344232835001918988149268862458058654691394256198576710650030125544077411432323340939433051485194567571240185673981732045983714973267283534300647601226252568098892440462401965111622976003259591077704702584200763046171986480349330806899873312846204834058399352574005416231688261510545134741182277970358473883943958563579015179820120979292270637497907072612180871069400619450857723011268017454116823535827228473296516703273009238893345386444533871542383504242463001681961734268277378540067885920080290584936097716155329313777328195435585629703275367750105825453097378643622824901407930221204813818805961136841682239404338275246672278987523193876833029445938199819122011285813404240449718569721922907241151390900428524224234201221755939491010573105885453826465599986918927823875647571538;
if Z eq (g^lg) then
print "Verification OK";
else
print "Verification Failed";
end if;

All the best,
Antoine Joux

Le 23 mars 2013 Ã 01:01, Antoine Joux <Antoine.Joux@m4x.org> a Ãcrit :

> Dear Number Theorists,
>
> We are pleased to announce a new record for the computation of
> discrete logarithms in finite fields. We were able to compute discrete
> logarithms in GF(2^4080) using about 14100 CPU.hours. This
> computation was performed using the same index calculus algorithm as
> in our recent computation [Jo13]. A draft describing the algorithm is
> available as [Jo13a].
>
> As far as we know, the previous discrete logarithm record in
> characteristic 2 is GF(2^1971), using a L(1/3) algorithm
> (see [Go+13a,Go+13b]).
>
> The main features of our new index calculus algorithm are:
>
>    - An asymptotic complexity L(1/4+o(1))
>
>    - A small smoothness basis of size q^4 for discrete logs in a
>    field GF(q^(2k)) with k close to q. Indeed, this smoothness basis
>    contains polynomials of degree 1 and 2 with coefficients in
>    GF(q^2). As a consequence, the computation of the logarithms of
>    smoothness basis elements takes polynomial time.
>
>    - A new descent algorithm that together with classical descent
>    techniques allows to express arbitrary elements in the finite
>    field in terms of smoothness basis elements. This new descent step
>    is essential to reach the announced complexity.
>
>
> We first defined GF(2^16), from the irreducible polynomial
> x^16+x^5+x^3+x+1.  We denote by 'a' a root of this polynomial and use the
> polynomial basis 1, a, ..., a^15 to represent elements in GF(2^16).
>
> We then defined GF(2^4080) using the following Kummer extension
>
>      GF((2^16)^255) = GF(2^16)[u]/(u^255+A),
>
> where A is the Trace of a [to GF(2^8)], i.e:
> A=a^256+a=a^14 + a^12 + a^7 + a^6 + a^5 + a^4 + a.
>
> We choose as basis for the discrete logarithms, the value : g = u+a.
>
> As usual, we set to ourselves the challenge of computing the logarithm of:
>
>  Z= sum(i=0,254,u^i*Pol(binary(floor(Pi*Q^(i+1))%Q),a))   [in Pari-gp syntax, with Q=2^16]
>
> The cardinality of the multiplicative group of GF(2^4080) is:
>
>     2^4080-1=
>     3^2*5^2*7*11*13*17^2*31*41*61*97*103*137*151*241*257*307*331*409*673*953*1021*1321*
>     1361*2143*2857*3061*4421*6529*8161*11119*12241*13669*26317*43691*51001*61681*
>     106591*131071*354689*383521*550801*949111*12717361*15571321*23650061*40932193*
>     394783681*1326700741*2949879781*4278255361*4562284561*46908728641*
>     611787251461*1392971637361*1467129352609*2368179743873*2879347902817*15455023589221*
>     33910825580641 * 116772720677761 * 418562986357561 * 737539985835313 *
>     171664686650370481 * 4967178060528306401 * 7226904352843746841 *
>     9520972806333758431 * 26831423036065352611 * 51366149455494753931 *
>     373200722470799764577 * 1230412270786066204321 * 8088220746627020943841 *
>     10146032011084172688350401 *
>     5702451577639775545838643151 * 4251553088834471719044481725601 *
>     630894905395143528221826310327361  * 18741457027056199460701768016571521 *
>     420245688628846194691190674873072272865640768049748318922486401 *
>     P78 * P116 * C295
>
> where:
>    P78=116244395157193581337282640791798084114394917399572436767868837818708235649281
>
>    P116=59759045572704532151734514229676903701763064698110618010245423428627221235639899045664816790870237783305610352947361
>
>    C295=1533490284616846723598415409259297911652711222168775547792306046025942905067111993457984065168194835553976875731082408417656840935335955173202390542791678493470066047287643716215585615327618503502688057052265147924415358240017501698179552850131324250482005642180027027923716821703262527243660161
>
> Since P116 has 385 bits, computing discrete logarithms in GF(2^4080) is
> clearly out of reach of generic algorithms.
>
>
> As usual, the computation was done in three steps:
>  - the generation of multiplicative relations,
>  - the linear algebra,
>  - the final computation of individual logarithms.
>
> As mentioned above, the factor basis that has been used contains all
> irreducible monic polynomials of degree 1 and 2 in u (with arbitrary
> coefficients in GF(2^16)).  Thanks to the action of the 8-th power of
> Frobenius, this basis can be reduced to approximately 2^22 elements.
>
> Note that performing linear algebra on 2^22 elements would be quite
> costly. However, as explained in [Jo13a], in the case of Kummer
> extension, we are in fact able to split the computation into several
> much smaller ones. In the present case, we have to solve 130 linear
> systems, the smallest one contains 130 linear polynomials (up to
> Frobenius), the next system contains 2^14 elements (corresponding to
> polynomials of the form u^2+u+alpha). Finally, we also have 128
> systems containing 2^15 elements. Each of these 128 systems was solved
> in 9 hours (including the generation of the corresponding equations)
> on 8-cores. They were run independently in parallel and the total
> CPU time is less than 9300 CPU.hours.
>
>
> Individual Logarithms:
>
> We followed a descent approach similar to [JoLe06]. As in our previous
> computation [Jo13], this descent includes three separate parts.
> First, we used continued fractions to find an expression of a value
> related to Z as a product of relatively low degree polynomials. Here,
> the highest degree polynomial has degree 29. Then using classical
> descent (rercursively), we expressed these polynomials using
> polyomials of degree 12 or less. Finally, the new descent phase allows
> us to continue the descent down to degree 2.
>
> [In [Jo13], the maximal degree after the continued fraction step was
> 18 and the new descent was only used for polynomials of degree 5 or less.]
>
> The continued fraction steps took a few hours on 8 cores. Hitting
> degree 29 so quickly was quite lucky. The classical descent took 12
> hours on one core for the slowest of the polynomials appearing after
> the continued fraction step. The total cost for classical descent was
> less than 50 CPU.hours.
>
> After the classical descent, we had 149 polynomials of degree 12, 128
> polynomials of degree 11 and 125 polynomials of degree 10 (we neglect
> polynomials of degree 9 or less, which belongs to a lower level of the
> descent tree and whose contribution to the total runtime is
> negliglible). Among those, the most costly where the polynomials of
> degree 12. On average, for each of those, it took 4 hours on a Intel
> Core i7 processor to find a decomposition into polynomials of degree 9
> or less. Once this was done, the remaining time to get the value of
> the logarithm was about 13 hours per degree 12 polynomial (using the
> same machine as for the linear algebra step). The polynomials of
> degree 11 and 10 were less expensive, respectively requiring
> a total time of 13 and 4 hours each.
>
> Once this logarithms were collected, concluding the computation took
> about 20 CPU.hours.
>
> The total cost of the descent step was less than 4800 CPU.hours.
>
> Finally, we find:
> Z = g ^ 593537791871423043223099993371775097225725806985608997497794966002388232480207689669841040498259020695864963162877267246612766963427481859635858268330211735283816590918847157995342025638775868791428528017795465828457233366986043689100592091740290308960776447743054737077011247538124490796554449688480787567205892205650065036371339635472100864592768628245778548627169993710530248952247502198339102414084716879305058973285967705897824717564625973834423283500191898814926886245805865469139425619857671065003012554407741143232334093943305148519456757124018567398173204598371497326728353430064760122625256809889244046240196511162297600325959107770470258420076304617198648034933080689987331284620483405839935257400541623168826151054513474118227797035847388394395856357901517982012097929227063749790707261218087106940061945085772301126801745411682353582722847329651670327300923889334538644453387154238350424246300168196173426827737854006788592008029058493609771615532931377732819543558562!
> 9703275367750105825453097378643622824901407930221204813818805961136841682239404338275246672278987523193876833029445938199819122011285813404240449718569721922907241151390900428524224234201221755939491010573105885453826465599986918927823875647571538
>
>
> Antoine Joux  (CryptoExperts and UVSQ, France, Antoine.Joux@m4x.org),
>
>
> References:
> ===========
>
> [Go+13a] Discrete logarithms in a GF(2^1971).
>          Faruk Gologlu, Robert Granger, Gary McGuire and Jens Zumbragel
>          NMBRTHRY list, Feb. 20th, 2013.
>          https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1302&L=NMBRTHRY&F=&S=&P=4793
>
> [Go+13a]  On the Function Field Sieve and the Impact of Higher Splitting Probabilities:
>               Application to Discrete Logarithms in GF(2^1971)
>                Faruk Gologlu, Robert Granger, Gary McGuire and Jens Zumbragel
>                Eprint Archive. http://eprint.iacr.org/2013/074
>
> [JoLe06] The Function Field Sieve in the Medium Prime Case. Antoine Joux and
>             Reynald Lercier. EUROCRYPT'2006
>
> [Jo13]  Discrete logarithms in GF(2^1778). Antoine Joux.
>          NMBRTHRY list, Feb. 11th, 2013.
>           https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1302&L=NMBRTHRY&F=&S=&P=2317
>
> [Jo13a]  A new index calculus algorithm with complexity L(1/4+o(1)) in very small characteristic.
>           Antoine Joux.
>            Eprint Archive. http://eprint.iacr.org/2013/095
>
>
> Appendix: Pari/GP verification script
> =======================
> Warning: This verification takes several minutes
>
> \p 2000
> allocatemem(100000000)
> Q=2^16
> Z= sum(i=0,254,u^i*Pol(binary(floor(Pi*Q^(i+1))%Q),a))
> pola=(a^16+a^5+a^3+a+1)*Mod(1,2)
> polu=u^255+ Mod(a^14 + a^12 + a^7 + a^6 + a^5 + a^4 + a,pola)
> g=(u+a)*Mod(1,2)*Mod(1,pola)*Mod(1,polu)
> lg=59353779187142304322309999337177509722572580698560899749779496600238823248020768966984104049825902069586496316287726724661276696342748185963585826833021173528381659091884715799534202563877586879142852801779546582845723336698604368910059209174029030896077644774305473707701124753812449079655444968848078756720589220565006503637133963547210086459276862824577854862716999371053024895224750219833910241408471687930505897328596770589782471756462597383442328350019189881492688624580586546913942561985767106500301255440774114323233409394330514851945675712401856739817320459837149732672835343006476012262525680988924404624019651116229760032595910777047025842007630461719864803493308068998733128462048340583993525740054162316882615105451347411822779703584738839439585635790151798201209792922706374979070726121808710694006194508577230112680174541168235358272284732965167032730092388933453864445338715423835042424630016819617342682773785400678859200802905849360977161553293137773281954355856297032!
> 75367750105825453097378643622824901407930221204813818805961136841682239404338275246672278987523193876833029445938199819122011285813404240449718569721922907241151390900428524224234201221755939491010573105885453826465599986918927823875647571538
> if (g^lg == Z, print("Verification OK"), print("Verification FAILED"))
>

```
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