Date:         Fri, 15 Nov 2002 12:53:48 -0500
Reply-To:     Reynald LERCIER <lercier@club-internet.fr>
Sender:       Number Theory List <NMBRTHRY@LISTSERV.NODAK.EDU>
From:         Reynald LERCIER <lercier@club-internet.fr>
Subject:      Elliptic Curve Point Counting, 65538 bits

Dear Number Theorists,

In [Harley02b], Robert Harley explains how he adapted SST's algorithm
to count points on elliptic curves defined over GF(2^d) in nearly
quadratic time when Gaussian Normal Basis as described in [KPC+02] are
available in GF(2^d).

In our side, we would like to report that in this case we achieved the
same complexity with a completely different method. Furthermore, it
seems thanks to a recent remark due to F. Vercauteren [Vercauteren02]
that these ideas yield in full generality, i.e for finite fields
without Gaussian Normal Basis, an algorithm with a time complexity
close to O(n^2.5) which do not need a cumbersome O(n^3) time
complexity precomputation step (more exactly, O(n^(2.688+eps)) time
complexity at the expense of a O(n^2.5) space). In practice, this
method is probably easier to implement.

In fact, we developped it at first for jacobians of hyperelliptic
curves of higher genus while looking for an efficient way to implement
Mestre's ideas [Mestre02, LeLu02a]. But, since the genus one case is
slightly easier, we have done some further experiments for elliptic
curves (higher genus implementations are underway [LeLu02b]). As a
consequence, we are pleased to announce a successful computation for
an arbitrary elliptic curve defined over the finite field
GF(2^65538). The total run time was about 29 hours on a 731 MHz Alpha
EV6 CPU. The lifting phase was done in 16 hours and the norm
computation, in 13 hours. Long but accurate details are given below.

Even if we improved in this way the AGM method [Mestre01], it can be
adapted in a straightforward way to Satoh's algorithm for elliptic
curves over finite fields with small odd characteristics.

Details are available in a preprint which is submitted for publication
and which is available on request.

     R. Lercier (lercier@celar.fr)
     D. Lubicz (lubicz@celar.fr)

[Harley02a] R. Harley. Elliptic Curve Point Counting: 32003 bits, email
to the NMBRTHRY list

[Harley02b] R. Harley. Re: Elliptic Curve Point Counting: 32003 bits, email
to the NMBRTHRY list

[KPC+02] Hae Young Kim, Jung Youl Park, Jung Hee Cheon, Je Hong Park,
Jae Heon Kim, and Sang Geun Hahn. Fast elliptic curve point counting
using gaussian normal basis. In Claus Fieker and David R. Kohel,
editors, Algorithmic Number Theory, 5th International Symposium,
ANTS-V, pages 292-307, Berlin, July 2002. Springer Verlag.

[LeLu02a] R. Lercier and D. Lubicz, Cardinality of a genus 2
hyperelliptic curve over GF(2^4001), email to the NMBRTHRY list.

[LeLu02b] R. Lercier and D. Lubicz, Mestre's Extension of AGM
Algorithm to Hyperelliptic Curves in Charateristic 2, preprint.

[Mestre01] Jean-Francois Mestre. Lettre a Gaudry et
Harley. http://www.math.jussieu/~mestre, 2001.

[Mestre02] Jean-Francois Mestre. Notes of a talk given at the seminar
of cryptography of Rennes. http://www.maths.univ-rennes1.fr/crypto/
2001-02/mestre.ps, 2002.

[Satoh02] Takakazu Satoh. On p-adic point counting algorithms for
elliptic curves over finite fields. In Claus Fieker and David R. Kohel,
editors, Algorithmic Number Theory, 5th International Symposium,
ANTS-V, pages 43-66, Berlin, July 2002. Springer Verlag.

[Vercauteren02] Frederik Vercauteren. On AGM, personnal communication.

PS: Computational data
----------------------

Let GF(2^65538) be given as GF(2)[t]/(P(t)) with

     P(t) = (t^65539+1)/(t+1),

let E be the elliptic curve given by

 y^2 + xy = x^3 + a6,

with

 a6 = t^65536 + t^65535 + ... + t^5 + t^4 + 1,

given in a compressed hexadecimal form as

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dca91cdcc368023f6e25d35b95324bac50a0a612916fb19323bdecca32e270a8f6931\
dcee9fbcccc5966da8fd1d4243ab13a124f7c318fbe077a26f51b28584c499515ccd6\
8120b6532bc5e7d89ca4a23d796cadcd1c634d1363d8d47ebff06520d51cc813454eb\
474422451a03e6c4eb2f6bfbd09fc36fd38bdb2e3ffabd6d0f0bcc41e3062b4220452\
0f4d783293b54f58cd9cc7bf4b7015b231989c6dfb1000b61c795f2e7721651cdd0b9\
a761b522bc868b0517e756b609a6ddf6b980d67272e614f5e5bf1bc9b816dfd2edf6a\
8472b34a6774960023ac999642b3eadfd72a34efd117c55c7344010eafa67a40b59d7\
1aaa6a6883c27e90f26e69dd81d3f4a1d55df4b199ba4281c0627faead1c1229d98aa\
9c51fdcde2c30b3f95dafe3d4bc4b6c1edb7d5e7c3e4901a0055a614dbbbbedb64aad\
b0f67762b59c4685c820c3c4946324f1c55627a6db0a24d772bc8acd8878d2f65cd87\
4e4bc7979dec271cf1c626aec012f04556271a38767540cb0f7b2fc2ca6430e43c82b\
c6dc83c8a57da12b45efcd3301c8a71ba3004c4aa0e2ef442cb617ae45e8c6afdc3b2\
0f502629feb0d7e8fd6f242caa98b64116272e07af2b86e6246695d5cd3b835d87255\
4da620844a42d6d75b6b703df446a166f1ffeb728f56a46ca65faff3f468fca3efe93\
a4aa745e2897c426a7cbf373ca6e6edce2265daaac9773b38d1d75e3cf5de4b5b222d\
89de350e0344986899719141a0b0e59ad20533c1da3f144ca41cfd83ddc7281765fad\
38f91347a804f397801ee098b6231e64465e2cc20452cc97e8cb3f220ba9c69094e24\
781997738b1c2e9d00bd17419929ba86f78175711616da7ddabd6b9473605387487b7\
6aa04015f7fc337ba2377413270077ad21b45f23e587a0df38f32fd3caa8e28798a38\
9f76ee11683704e04663716b117e3d9cd8a0b82b1e664776c97c1291f8c17baa6e877\
25a0f08e4f669416f45f8cce134bb4aa0dba3859042b372e00ca060a7f51138387ba1\
9727f0e40bffb73bbb4f444ad295bba63bb8a2a37ce39628fafe0537791acdb9df09b\
803abdb1b0b6acf9a20db5f510327945c34531cddb34310999be5516c56ed2d0de15c\
58cd0e04aac7e11e2795a5e734b211eba0378f5a83d1116beb30837a831561423d702\
c3a9cd02d0a4df9fa9ed19f051d43b7f46e1a3b4576e646fda8d014bef808774b007c\
f4e5bffda2ebd874188673ccbaded01f4761462e2edde81414846fdaca361933ba11e\
1a2c3a7d06472b81441d637d9645fc56a81d19412e005e5fc2a771a39e04c873f632a\
c289f91f5c0b1fed7cfc6fb0962f17f4990e12961816b10b1a5be124c1d4ca5feb8fb\
ee71ef17e24dd5bba6f31e7b9bb25609ecf1158fcf045cc522466cd0ecb0e15fff050\
1e4a756ea68b70937957378ac4ed9651c103a875cac09f7c39ae93667dbfff2f16167\
f3200a85a03d955b218db85cf2e4d54d47f9b3b4f5cdf039f1b6c46e545ae40b7264a\
8a684a4f4b963138585037e6cdf9982a6a3d3379b381675771f77b63347aa239a333b\
5c1bcc839109b0792571f46b2f831249ce107f2924c770a3b3502673d34371d1f980e\
a6165e5f2f2b0cff2bd93e53fa9bf1009991530352efcbd0314899512ca67f07801a7\
e3846a055fd28f4f84effbbdc37e934ae58fd205a37e0fdb3b050f0cc6b534687c991\
58a7d309249bb3a882120abf6b77612ba56a016e8451397f6353632011f1258d54abc\
6728795e3cc7b08760d20b44566ea116e55fe95d15819ca6824e77714dff9b6497240\
8553822f7232630c577084b0e2eee3102bd681897fa357918ce256bfdff7ed7f10175\
ecdf91c68617947591004bb0d48eed50357c0ec1d47654cd4f2bfe46beb702c070b61\
eb6475887b43304a1fc229cca9e98b55dcc47afb416c749c9759470afe119300dab77\
6d0a64adc7dd36224fc0b938088b7968ab485ac497e358b631f540b5c21174acdf0c0\
fa553e55f19b9889c5676ea13fddfadc2a5757c91b9e87d4efbcd1338aeb47fef35bd\
9883515e6b44e3b0077895d70ed735d3a6562b1d8a382cbbf515918bfc412829bebaa\
f66704ce74474a1eb85a92ca705d9d368da840add1e5500d59895860ed457e35ad13f\
a7c2cb452b276c8c50d4403eb538b8177e9942184fc0744c75e72335c49fb410e0730\
797c5a0c3edfebb7a2c22659add398eff4fdc9016b7fab6ebcd6c4597301975e62c0c\
06d68104f4d39e1877a2f811853fceb33840ef3b78114174c84486399f5157ba46948\
ffcbb56413d707f5f70b0aae90a93f917c1a43d426db24f98c15360d999376b3e92e1\
1aaf0cdd2b35189d04e82caccf2c02a22ac686aac4e0e9cc8f7e7c9658682f361fb77\
a60455fe273e2d2f6a6e71435f975e7ebe0abf4cdb3756d99c91d76cd68e4b9c15a53\
d45de1b641f9e2b88bba35387ef9a2594484c85fb09befa8f9a9134af687b8f7c8be7\
eec426dd3f776bc25b4776558b93f44ef39dad3766ceade7f00485192bb555349bb9b\
aa183462e457e962801455054304b66bbce2e82a7d5d508ac3d298a97bd9c502ac99b\
07236de11ad541723bcaabc3436f5301c6ca565d7900a1d32bad50fd67095804d3ae9\
3c3397c4464e2a1a12c2498d9f32272e2f87a7cb7214a6c16b6eaa7d3ad46fbcb6ae1\
114736395bd45695cc8678b3b94a68ffbaa85cce2db86e2aa9f216d795f9da70b3f29\
72b79db28c1f45bbd9bb3697ad33fbb4941d64c14f1fb484570698adfa97f9974e589\
7034974ae9e8b2c1b612e4cc4534e6181c4ddcc2474a5984971eee822af5a353748db\
d4c32ed0148afc464682f699fc974140eb1f580cb0cf7cb8d66a653bd97f4706b1cca\
a3bd3c85e492f48f379d5263f72a51d93fceef135b5a3d9a722f4f9c4b3661a46a901\
2ae2a77a2f3759e4ce43330ccd0e38ae97b8d445f3fd7f5f9a4ddb93d48bbc0e41e7f\
1f2e5c31bb430c9ee9b80be2acf9766f3cc6f1d82f9f84168e65d3cff88cefcb09422\
54ec9ec3248678e71acd4ab69974e7cab027b42c239c725e024e8c86a65a58251432f\
c9c1f0a0d0daa05fbaf15fb4dfc7818c3980df32a59b5c991864617c728623a31783d\
e8ee1d98fa30557d946f365b34bf91740617d5225542159a2560a0c6abaf1ad195f8c\
0f68d9df953583c13e7ce9586aee437a9fd7170fd7829454554ab3b98868cae23d9a3\
ea7b7439d5a331207e2fc11d4313235c196580f6acaa27ecb830ab71f0a73d9b735d7\
4489f7a99e0c9ce2c6e5b1ce5f7a420ed9addaaf4013a29bfd7e9349ec341cdff9fe9\
3c9b90c325b68fd838b803697f9e86eb9573b7dfc39d159bd1edd3ba8a1757326e5fb\
19f31b636c834804f6467831d309ca058875f8e10d214102ae494ad61228feb67bffd\
ee7914853a82443c5db758eaf646ca9f969f580e1df3ea096a92ee76dbdd9f0eb1a1b\
03be11caeb6a67e41cd77d28d8f9f634f0ffb08dbce3524fad29cadade5bec090acb0\
09b037c7731869fd6ce247e6c86a8a8d936e1f2a0fca4402c5a2f4084340d50260ce4\
8c80445cdd549bead1b1b02eadc1422efaf5593d7058b74ab3398199fffe6023b7f4a\
54533ae1c9b8a669a02d98bd8d0a469418c82baa1e2b89de9baf2ea4d19f33dcaefc3\
60c8d6bff7db9fe14d04255054716a0ebd7b3a7613480ba548d6477a7257de2b9fca2\
aea36672fa6d9f67879a8cd2d6290ac1ee298ebe488e6359d2c7a957470256ddbec67\
f82625366b6f9d5f5ee1c8d83fb4a63340284a022bbdada306cc478e6571330991789\
d56bda33293008282ab67896a3dbfd6bd2b2c9b0577ece3266b1bfccf50617ee23246\
3f9944cb9fca269dd3565f7d23ce06ace9facfff509ef1ab08bef3fdb19b1545a7e61\
ec9f1118fcb68ffcad477bcbbe10dfe4b58404f9931341dc634c467bee2447ce30a12\
f1044408578fb5716eb97ba8b43a7ae6a0f96a73b67db92c395b9e85c3be352fb36a4\
8bd8b309eed5937972a29494f81e386d9e17a682425b36cf14773abe072e993768d7b\
084b93c664a879e3c12586968231c19693786c2c6431f7415645c3d1affc832829d5a\
d070611929eaa2f49f1786ce44ecd7c771,

then the number of points of E is equal to

 2^65538+1-c

with c = 0x2648d05a48f636041688386855273d33f6a8c594ed02093e6e6438eee2\
410b32ab7e595b52dee2b100aa046fcc96bbb006a0860bf87b89bf8f79b331ea5a433\
659c20721e12e21f9e12b396da16b7c32b775e6cc50513968bd65e4ae451b828aa5d4\
c02f8221dbcdfc54bc106e2e283b1d286c29bb2e6adf9792459941d96bc14abb2037b\
cbee4889a46a9256fa1caaefa39be9a692b4f758c95b594d29f09a48ed4e0665e030c\
48faaa207e7506903b59073b4cbab42c9b1d93ed216f79df5f0427bafebe84f686d8d\
28068357dd4b02db730e49984617a28ae523867a596d9ba99c6d9b2ea6c58230bed65\
954b049d726fa6cbb05851c3b36c39f8958f8683b8a7c14673d7f813baf55581654e9\
ee0d7278597a66861ae247b8562bafb34196633325dc111f2d58668ca172c82bf3f6c\
f8d3b357409d343a422717325e4941e5ada93e9ffeab559b682d6dad0dd549d7c6d7d\
274cfcd1d2709a4ee64df21c00557538ec76d0408b8246b8a6112b85b5098a32e30de\
d2f8b9c2311293c7e1ba2a4110300b422da53c2964e6085ac0ea3d9559808901d2283\
dea6992a8e6eb9b3828711bc06b39d94759ed64e56094fd6d9ce84d41e97c48e4ba48\
1b8665269fd8e468adba960d0cfb13e843610b12f239ce58d805f778fd9ac1d9a92f6\
e5965970dda9e24a41fc2a765bbe672bba04b36c82f378c7bae4aac36b0abd24a96f3\
01224c0912eec4791aacced70742c4852eafb78f827258bc63acb134cbfbdb8cff69d\
6d87495a80933c9e93afccb38e3d557e1c15ef9b51e8247ef513979e7f396d73d602b\
c1ebf05a99f987351f94351518eec0beff143804b0e6035f0c59a8eb80444b37c821b\
425423d4e80749e73469acf849370765e5144b63ef8f77c85f4951298b4b1ca022255\
dc39b0b0569965a6430f31722ccd35f232ef9b2647b795a02018c5a30b9aa04e0a28e\
9e458f1955a0a301493d182ac7ef15d6d4cd8427d2c187078821f1927bfad0e4f07ef\
db2dc8d06f0743c791fc83054ea302ce9b7857560f36869164f4b8dc2c178eceb0c55\
ac79d8d58633939e4bd69042b7ca88aec66b0c949a0d062402804fc340a3347b9f873\
9a110f796e8e05348b489afaa7b8f650f9cce03482c5d45c4d24609971d5015befc9a\
9c6e0b8ed03cb7fd87aad07ccbb7e11b1f4a005c99adb5484239e30809e4941cc931c\
7d960a55320694d65c0c68d79e28b7f80112e729ce332588b50d3fad3590661a7d096\
e8d35301d549eb8c4ee698bc5dca50f80772b4c8627d803ce1f714b2affd822e2b272\
68c9c31e873ce9ceb8af31da292c98d01ef90b0fb97e2add989236f8418b1a64a30b0\
f713f73d53df5bc2666a5540dc61309598c0b978437b5c0502238fef6dc1391be120f\
dd3b77a933e4de1a11c422ab8976d61ccf8697f34d31119a6822ab039d02af5f757f9\
2540c78fa07fa112ced169812c516738e1c4275654b6f7bf1d6efdb10d9274114c314\
c877c04ec1994706d7ecd335da588c34a0c1c434680eb0ee82aa2bc3d657018cb37fd\
9bc43f545d7cbddbac5c98f858965f25fb921800cc04f7bf5a0c3efcae96ba34f7f78\
e1ee66c2a7d8f715210070597691ae063a58091cadcd068a4c6e58e8d81a1602d4696\
485640346ebe9480214efdde46642addbb59bcb85f1dbe6dabf28dbb479eb454cec8c\
7306faa4d43f48d111c5f19731fc8d62f44339bbc3862be1267e4e65340a10b25b074\
b070045633683090607856969aebd21eae5ca5a2c8eb9444cab756cfadf3f15a65992\
57c2938f0f270cab72ce740d1ef040dc023505a72a5df7449b650235ae5dfc2ee46a6\
ee117c8b93fbe0023649d23e1167390271b5225abbe71abbcfe760b137c20c556e9a1\
c1b5bfb76be46694abc2c7563b4cef9a86be5fae1fb797f473acfb72be4618d64d75c\
5eae12424703cd0734e7d627aa875f3c39969611ba06c72965745e0ac155142e7296f\
c4576a021a78d9cd7c1d5432b51d6a08e9f8a47523c5dd05191a4537d031da0e64b60\
01caf3e29ebdb43a7f7b45c7cb2cacb6cf9f5296eeb820bfab078fa5122da607bd47a\
44d8672cec71fb0ca84cbc8482a080620ebe4a9f3310f4f61b0b9624ed0139a143705\
2b3e7c0213e7a4207384f9c47d71b344b8a67a7f17d88692012860762aada8c33593e\
4df6c42f76242c567127d7ca12422bc9c51c78e8dd9e829c35f027a7aafbf09bf8a39\
27fb592b2c80bf72b443b0c084ad684f824d58a85b1facae80350ef9c4a3eeccf4ad8\
30aa6d15e94e07744b47f57443c3bda05f92d1b5fa58b402ab49d2607e7d7b99b6461\
42f548f9da6d52ae85a0cdb28f4201d6ca20e6d6f960848ccb5277f9e2ca46656778a\
bd1817844b6b264ac5ec76549c720e61862a49f4b97ec3494e6b197956ac582800b6e\
6bda28088bca3662ef3b2342ac6ffd627163b485107de5abfe0b2941a32c06cb0c462\
c3e0b0a8661a79b958f3b5b9d3503effd3986b005007d617718bdcbf1ff10185322c7\
af9be2e7edeccc3ce1122c993cc31f2a04aae3900989a25143d0b6ac58fa97da0da7f\
f84df4f6e68fa6657dcdfecf30970ea3b493883b534da384599d1d251498bc33eafd8\
18a6570ee72fb8286a8dacb4de1db5c2e8b1db2519c1e4c5ef561e204a5fefcc2df52\
a189296a1666d1ed7b74b5c20de5caff56ae9d9682ce903b8c0f97fa21b1a56a39288\
45fd3604705c44791a4942bf60b09d97f2c2365e6f42ead2fbf315e61a51c7e28a6cb\
ee3f9d122445c003a636a6cce7eee998045445358059cfa3168960cde6271c45d4032\
20c68192a69a57d03eafe1fdd88a3ca1ea63a6de284aea22a9a7c0a4cf7db6193ec86\
f041d8c37ad5260c8715cc4d413f60ec0a2c1bdca0bad86d11e3d10583d8fa2de8567\
3343c3e6ea667a6094f0c2ff6f67e01e72924b8e43c0cebca18de98fe5d0c784fc02f\
adba3b38e817da2584094981ee009b8056a79d5b1c2d716e15d77787f9558ae462661\
34d5204fe6b1a949e1d2f138463c906840fb1a1dea8b76ba4f48ffa41459767c57562\
5ba02b26b7871274fe7352210afd32d0a5236f88d283c81f665b7f2e09815e0cc60c0\
f397a1303802da47fb49f0032d4788f2d87513d3a1cf91637a6fa28b90e8278ab7df8\
1fa24af74f925026f93aa152c06b62d0d833dc600fabc71669f0995b2e407a3463a27\
df7eaaddc40b31a46319c2466c725ae634d3f7dd1127185f467b1c81fd64f4ba3ecd9\
7c606cb776ad513f999e40ca41676eaecab749ec1f804d17336471713b17afec031a7\
8f011ee99735d036ac1f9b5ca076faaa660333db6dcf9475b4df1366a2eb043ab09db\
2cadf46550c3d4e13b4d7a06e97bd7c92274923e5246f879be5e6c1fe869afba7b251\
2c27e44ec4f12cb2ab7c430ccefcf334af863f174a2f8c89c7aaf5de7cef1260b1bc1\
f0bdd10e33d70bc0afa0c116865d37664df9b2f0a05737a1e8a9186cdbc4f90d21300\
3f77da5e496887395eedc590877d41b6d67723d467ee0c510c37579734a01673d5b61\
cdab992dc503d11447da49be90fc5c674158c92cca5a83d2ed02728f68df5f8b0ced0\
ca2190faa11897174f4de6a9f12c7b1f3dcaa70384f8328337d5bb61467486434648e\
0094ab12b1a03126008a0514c3cc78fba6b82d75d5be4fc09fc9b3ba0dde3dbbe41ae\
dff46c8f426cd96016725d5f8ac69bd3313fb1dffd9240a8c036945cd35af5fd5b01f\
2f01b972710a9e379166104179a46e972ce6329edf1bd70b601e5c0653b70cbee94e0\
dc7b7ce648a4ec95ea587f9b48e9ef9c35293b7e77e95de92132d8b6d60974610cdf1\
7bb0e524e50d362f5932bb63ec8a2abb0bbb5a4b52e3c918fc6f7fd804efd5747b213\
ed0df8c055d25104434e37d245d530bc1707dfa92c237bbc89e0f55f35571bbdf18b3\
5553fccd558df28f86ebcef39ae71b1adfd8c96792443c7de9ab50e4487b81792997e\
4975d1e215bffea6351b1824ae9f4ff1a7d37b826714a5069af612d69b83f7ecb08c0\
c01fb7d3a2e19c9546f8c50108e1e82f7c87cd31c0c23fa8fb370a03ec0214cb9f27b\
cb054d3ece12cdc3b9f16ab5ba8a7cf98d0c0af283d08f707f832869ecb01e27d365b\
822164424005885cdf02fcc89374b7d2c899085a621097cc0b0144fb359ff6e768fe6\
5fd657859b36dbdf452adee9643a15d53b87b8b86131c000e6e98e9242c30891de81a\
8544e59c153723195afd0cc04b3a0bb43deabda276d33b6e8c9041e089954edf2ff11\
4e11f338e2c5aeae36d07e3b2b27d78b34b404cbd27c03899df3fe683782b5afcae2f\
c6c3dd80d37146617d58477b9856cf613a7dd2dbd7076643f18e3027624ad6ba318d6\
b3bcaaca1f2c9216f7d91105806083b5d822bdcf763923e6ff9714b3ca4ab868244b0\
ad0f552fbaccdac7c1ffa3e16070fe7e5177afe5573eefd9d26a1d17c1823f07e7e01\
1cefae1368d49d0dc6f85e7c9d4c2bfd64c39fd90533aecb4147e8007eb639cc42058\
44c35eb15bfa5f703ec4cf216226424d9e4fb555f6f1d174e692d9b4c69df88040780\
1a80b931955f7461a0a9baad795e5d0961394937b768ad88a5fcf59892de1d3af8e54\
3f4760a8251adb942bf32579d0e25057b7a04d53f1b91b98eff767dfe734ca8d5769b\
1a54fbdeaa7732c845efef369df5038cb49242a8b9171d872d92f5d5a9550b0fd6838\
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