Tap chi so 16. thang 02-2016. trudng Dai hpc Tay Nguyen ISSN 185j-4glp K Y T H U A T P H \ N T I C H R A T H U A S O N G U Y E N T O
T R E N N E N T I N H T O A N S O N G S O N G
Vu Anh -Hiiii Ngay nhan bai 03/11/2015; Xgay phan bien thong qua- 11/11/2016; Ngay duyet dang: 02/01/2016
T O M T A T
Mac dii chuan ma hoa khoa cong khai RSA duoc gidi thieu each day hon 35 nam, dira trfin bai loan kho phan lich ra thua so nguyen to, nhung hien lai van con an toan. Bdi viil ndy se lim hieu mot sfi phucmg phap phan lich ra thira so nguyen to vd tbit nghiem vdi plnrong phdp maj]h nhk hiennaylasaim Imdng so Iren nen linh loan song song.
Tir khda. phdn tich ra thira sd, linh lodn song song, sang tnrdng so 1. .MO DAU
Chuan ma hoa cong khai RSA ra ddi nam 1977, dua tren do kho ciia bai loan phan tich ra thira sd nguyen to (Prime Factorization Problem -* PFP), duoc thuc hien nhu sau [1]:
(1) Chon p.<^ la hai so nguyen to ral ldn va phan biel
(2) Tinh n = pq va <l>{n)^[p~\){q~\) (3) Chon ngau nhicn mot s6 nguyen ei\<e<^[n)) thoa gcd(e4[n))^\
(4) Tinh d = e-'(mod,j>{n)) {5)C6ngb6 (/;.e) vagiirbi mat ( / j , ^ , ^ ) . Trong do, n dug-c gpi la modulo, e la sd mu cong khai (hay con goi !a s6 mu ma hoa) va d la so mu bi mat (hay con goi la s6 mu giai ma)
phuong phap p - 1 , "rho" p ciia Pollard, phuong phap p + \ ciia Williams, dudng cong elliptic (Elliptic Curve Method - ECM) ciia Lenstra.,.
Nhom thuat loan phan tich nay hieu qua khi cac thua sd nguyen Id duoc chon dd lap ma la nh6, vi CO mol so diem dac biet,
Nhdm cac Ihuat toan phan tich t6ng quit:
bao gdm cac phuong phap sang bac hai (Quadratic Sieve - QS), cac biin ih^ ciia nd va phucmg phap sang trudng sd t6ng quai (General Number Field Sieve - GNFS). Sy: hieu qua ciia nhom nay phy ihuQc vao chinh kich thude s6 cin phan tich chii khdng phy Ihugc vao tinh chdt cy thS n^o do ciia no.
Trong Ihuc IS, cac so nguyen id RSA dirpc d£
xual la cac so nguyen to manh, co gia lii rit Ion do do nhdm thuat loan phan tich d^c bi?l khong kha Ihl Nhom Ihu^t loan tong quat dang \k huong Bp (".fJ goi la khoa cong khai, con {p,q,d) U chinh da giai quyai bai loan phan tieh ra thira s6 khoa bl mat nguyen to. Tuy nhien, ban chi vh t6c dp ciia miy KhiRm=ii,A-,i,- . - u . , t'nh lam cho Ihdi gian Ihuc thi la khong chip nh?n Khi B ma hoa thong dicp cho A ihi B phai linh dupc. DS giai quyil vin d^ nay, ngoai vi?c nghiL giJtn 1 - £ - \ ( / ; ; ) - / „ - ( m o d / ; ) , v d i k ^{n,e) '^'^ ^^ai'hicn thuat toan, con phai lang tic dp linh
Khi A mu6r giii ma lim m tii thi A linh
" ; - ' U r ) v r ^ ( m o d ; 0 vdi k- = d hay
^ -i'tp.q) neu sir dung dmh ly s i du Trung Hoa de giai ma nhanh
toan. Viec lan dung moi Irudng song song IA giai phap hicu qua, do cac y i u t i sau:
• Toc dp ciia cac bd xir ly theo kiiu Neumann da dan lien ldi gidi ban.
• Gia thanh ciia cac bo xu ly phin cirng
TliQl
Do kho cua no khong phai do khong co each "^^"^' ^^° '^'^^ '''^" ^^ ^^y dung nhung ha'thing giai. ma do ihdi gian thuc hicn bing miy tinh la " ^ ' ^ " ^^ ^'^ '>"-
qua lau. khCng ihi chip nhan diroc Cac ihuai toan phan lich ra ihira so nguyen l6 CO Ihe diroc chia lam hai nhom
Nhom cac ihuai loan phan (ich dac biel bao gom c^pjnrong phap nhu phuong phap chia thir.
'Tli\, KboaKlioaboc Tunluin A- r,;„r, MI-
»Anl, Tuan . Dicn thoai. 0983.717.389: Em^/mLi...
Su phal Irian ciia cong nghe mach tich h^p cho phep lao ra nhung he t h i n g da nhan tren mpi chip,
Bai viel nghmn cuu cac phuong phap phan lich thua so nguyen l i va lap Uung vao phuang phap sang irudng so. Ngoai ra cung nghien cuu ky thu|l
389; Email: toibanlan^gmaU.com
'
I S P f ^ s o 16, thang 02-2016, trudng Dai hoc Tay Nguyen
ISSN 1859^611 thiet lap he tinh loan song spng nhim lam giam
thoi gian phan tich va xay dung he thuc nghiem.
Qua do xac thuc do bao mat ciia ma hoa RSA 2, CAC PHiroxG PHAP PHA,N TICH DAC BIET
2.1. Phmmg phdp chta thir (Trivial Division)
Thong thucmg khi cin phan tich s6 nguyen la thiia so nguyen 16, tmoc khi phai thuc hien voi cac kT thual manh hon, chiing u dimg phuong phap chia thit voi cac so nguyen 16 "nho". d day "nho"
so voi gia tri eiia " . Cac s6 nguyen 16 "nho" nay co gid tri Hi 2 dJn I V;rj [1]. Tuy nhien thay vi chia ihii tit 2 din j ^ X j , ta chi cin chia thit cho 2, 3 va cacs6c6dangp = 6*- + I viA k sZ',p<\ ./J, \ (do cac s6 dang 6k, 64 + 2, 6*: + 3, a + 4 khong the la s6 nguyen 16). Thuat loan ciia phuong phap chia Ihu nhu sau:
dupc Iim thiy Dac biet, voi i = kA>fi. thi lupn co.v, ^x.,. Do do, thuat tpan bit diu vth cap (-Tp .r,) va tinh loan min Iu cap (-V,, c,,) tir cac cap ( j ,, jT^i^J cho loi khi-V, =-v,,. Sau do thuat loan se Ian lupt tim gia trj dupc lap lai diu lien x^, va chieu dai ngan nhit ciia chu trinh /,.
Phuong phap p i,p dung thual loan Flpyd voi day s6 .Vo,.v,,.v,....vp!.v„ = 2, - ' ' - i = / ( - v ) = J r , ' + I ( m o d p ) V / > 0 dii tim
^.,--'^;.(m°t'/') [IJ
input: s6 nguyen n cin phan lich nutput- mpt thiia so nguyen 16 ciia n l.il tis0fmoti2) return p = 2 2. if n«0(mod3J return p=3 3. set p=3
6=2
4. while p<^
set p = p + b
if/7sO{mod/j) return /;
set 6 = 6 -b 5. return 0
input: sp nguyen a cin phan Iich output: met diiia so nguyen t6 cua 17 1 Seta = 2, 6 = 2
2. For t = 1.2,... do:
2.1 Tinh(7 = « - + l ( m o d / ; ) , 6 = 6'+l(mod/i) , 6 = 6'+l(modH)
2.2 Tmh d = gcA[a~b,n) 2 3U l<d <n then return d 2 4 If d = n then Khong lim dupc kgt qua
Thuat loan 2, Thuat loan P Pollard Phupng phap phan lich thita s6 nguyen 16 P ciia Ppllard ap dung thuat loan Floyd vpi tap 5 gpm p phan hi (la uoc eua n va chua biSt) do do 16c do thual loan phu Ihuoc vac p . Vi vay phuong phap nay ehi dimg dupc vdi nhimg s6 n co thita s6 nguyen 16 nh6.
2.3. Phuang phdp p-\
Phupng phap P - I ciia Pollard dua vap dmh Iy Fermat nhp (Linlc Theorem of Feimal) va khai Thuat loan I, Thual tpan chia thii
Uu diem cua phuong phap nay la dpn gian, tuy Iv Fermat nlio n ,„i TI, nhi«n nhupc diem la thpi gian Ihuc thi riflon khi l^/„",° ° ' ' n li, tich ciia hai s6 nguyen 16 gin bing nhan. Do ' "" '" '"^ ' ^""'°"''
do, phuong phap nay chi Ihich hop khi a cc thua Bmh ly Fcimat nhp- N6u gcd(a, p) = 1 SP nguyen 16 nhp ,,,, /,-i , / , ,
, , „ , , , " • I ( r a o d p
2.2. Phuang phap a . r ,Phumig phap phan tieh thita s6 nguyen 16 mang , • ' ^ ^ ' "''!" " " ° ° " ' ' ' = ° " " » " ' ° ' ^^ •'"""E '-" ' ' dp PpIIard c6ng bo nam 1975 Y tuong cpa ' ^'': ^' '^° *° ^S^yca Q dupc goi la B -smopih phucmg phap nay dua Ircn thuat tpan lim chu trinh
cua Floyd.
Bai loan lim chu Irinh: Gia su co ham bai k y / : 5 - > S , vol 5 la mol tap him han Lay X, la mot phan tti bit ky ciia S, la luon c6 mpt day cac sP .f„, x,, .r,, . . dinh nghia boi
•'"-. I = / ( v,) V/ > 0. Vi 5 la hihi han do do chic chan 16n Iai mot chu trinh.
Thual loan Floyd tim chu trinh. Floyd nhan thay voi bai ky so nguyen ,>^i va A>0
•', = >,.., trong do X la chieu dai ciia chu uinh
neu lat cS cac thita s6 nguyen 16 cua Q nho hon hoac bang B
Y mong ciia phuong phap p-l n ehpn mot gia tn B lam bien xel smooth. Tinh gia tn Q la boi Chung nhP nhit ciia lit ca cac luy thua co so nguyen 16 nho hon 11 ma nhP hon „. Do do:
~ i ; ' - >-'cu /' la mpt thita sp nguyen 16 cna n ma /> - 1 la mpt s6 S-smoolh thi /) - I,' {3.
nen thep dinh Iy Fcmial nho ta cP a'' B I (mod p) Dnd6nlurf = g c d ( a ' ' - I , „ ) , h i p'd [I].
47
Tap chi so 16. thang 02-2016. uudng Bai hpc Tay Nguyen ISSN t S S g j g P
Input so n can phan tich Output- thira so nguyen to «* ciia /;
1 Chon ngudng smooth B
2 Chon so nguyen ngau nhian 2<a <n-], vaiinh d = gc(i[a,n)
If d>2 then Return (Z 3 For cach so nguyen lo q.< B do.
3 1 T i n h / = i ^
[ln<,J
3 2 Tinh a <— a'' ( m o d u ) 4, Tinh rf = g c d ( u - I , H ) 5 If t/ = I hpac d = I, then
Ket thlic ma kh6ng Iim dupc kel qua Else
return d
Khi do gcd[x-y,ii) va g c d ( . r + _f,j,l ^ nhiing thita so kh6og lam thupng ciia " [1],
Thuat loan 3. Thuat loan p~l ciia PpIIard Phucmg phap phan lich thira s6 nguyen to p-l eua Pollard dua vao diiu kien P - 1 la mpt so 13 -smpplh Vi vay phupng phap nay chi dimg dupc voi nhirng so n co thita s6 nguyen l6p ,,-na
P - I thod B -s,nooth
2.4. Phuongphap duang cong Elliptic (ElTipllc Curve Melhod - ECM)
Phupng phap P - 1 ciia PpIIard thanh cpng khi p-l la mol sn S-smpoth vol P la mpt thua s6 nguyen 16 cua " . Nhan thiy P - 1 cOng la cip cua nhomZj,', dp do phuong phap ECM t6ng quill hoa thual loan p~l bing each thay nhom Z , , ' bing mpt nhom dupng cong Elliptic bit ky tren
**,• Neu cap ciia nhom dupc chon smooth irong pham VI B ,i,i t | , j „j„g ^^„ ^ i j ^ ^ . ^ ^^^ ^^^^
ngupe Iai Ihi cin phai chpn m6l nhom duong cpng Elliplie khac.
Miic dil ECM kha manh, luy nhien no vln thuoc nhom cac phupi^g ph.ip phan tich dac biel nen khong the g,a, quyit ba, loan mot each ling qiiat Bai vii-l kh6ng lap imng vao phuong phap khaopi' " ' ' * ' " ' " " ' ' ^'^ ' " " ' = '*' ''™ ' h ™ 3, CAC P I U / O N C PIIAP SANG
-*•'• Phuongphap Fermat
^',2%Zt''"'"'""'"'"''"'^'''-"^'
nguven ,1 \ a \ saoeho
1. Ifin 2. .c-\
3. Tron;
Of mod 2) return 2
^]
*.x-nkhi z khong la thang du biic hai moti/7 va x < ( u + l ) / 2
3.1.
3.2.
4. If.v<
Else
Tinh J = z-i-2*.v-f 1 Tinh x = .i- + i (/7 + l}/2 return .v
return 0 -V.
T h u a t toan 4. Thuat loan Fermat Trong trudng hop t6ng quai, n rk ldn thi r^
kho de tim dupc x, y s a o e h o x^ = v^(modn).
3.2. Phuongphap sdng hac hai Y hrdng dua Iren khai niem thira s6 tron - smooth da dupc nhSc tdi trong phuong phap p - 1 , _ Dung kt Ihuat sang da tim ra cac gia In A - ( m o d / ? ) thoa fl-smooth. Tich mpi phan cic eia trj lim duoc cd kha nang d ^ g y^ ("mod rt) [3].
' = . i " ( n i o d n l va ;'(mod/j)
1. Tinh tpan dii lieu khtn d6ng Chpn can tren cho e o so thifa s6 nguyen to B.
K h p i l a o c p s p l h i t a s 6 n g u y e n t 6 ( / ' i ' Pi^---^Pi) vpi A = - l , V i e [ 2 , r ] : p , < f l vaf—
^Voi m6i /;, ^ giai p h u o n g iiinh
"I = n ( m o d / i , ) bing Ihuat loan Shank- Tonclti luu lai
2 Sang lim lap U g6m I +1 phin hi Vol m 6 i x e [ - M , M ] , linh ?(,i:) = 2 ( l ) Voi in6i /', xem xet tit ca X = ,S„ + k'p, vi
x = S,,+k*p. vol ksl
TmM{x) = q(x)l(p^)' ^ ^ j , y ^ j ^ ^
nhat cp the co cua <;'(x) d i i vpi P,*(•>')= I thi Q(x) la S-srapoth vadua Qyx] vao trong U
3 Dai s6 tuyin tinh
Tim ra trong tap f/, lap 7" g 6 m c a c 5 ( x ) tich la mot binh phucmg f T ^ / x ) = v*
4. Tinh ,-, ' .
"=n(-4^J) " # ( ^ -
K.ai qua ra ve la gcd{£/ + v, /;]
T h u a t toan 5. Thuat toan sang b?ic hai
Tap chi SP 16, thang 02-2016, tmimg Dai hpe Tay Nguyen Vol thuat loan sang bac hai nhu Iren thi la thay gta In 0 ( x ) tang rit nhanh, dp d6 kha niing thoa S-smooth rat thip, nen khi Q{x) tang lot mpt gia n-i nap dp thi cc thi thay da thitc bpi mot da thuc khae, tit do ta co phuong phap sang bac hai nhieu da thic vpi Q{x) = (ax-tbf-n (.Multiple Polynomial Quadralic Sieve - MPQS) Tuy nhien phuong phap MPQS phai tin chi phi kha Ion 6 m6i buoc khpi tao Iai da thiic Theo [4]
chi phi chuyen doi da thitc chiim khoang 25%-^
30% t6ng chi phi chupng trinh. Phuong phap sang bac hai tu khoi tap (Self-Initializing Quadratic Sieve - SIQS) giup lam giam chi phi chuyin doi da thiic bang each vin sii dung da thitc dang l)(x)-(ax-^h} -n nhung gia tri " duoc c6 dmh trong nhieu da thiic va co gia tri bing lich ciia
" so nguyen 16 lc thupc cp so thita s6, Khi dp se CO 2 giatri 6 la nghiem cua 6^ =«(modrj),
3.3. Phuang phdp sdng trudng soPhuong phap sang hirong si trong bai viit nay la phuong phap sang ttuong si ting quat (Geneial Number Field Sieve - GNFS), phan biet vol phuong phap sang trupng s6 dac biet (Special Number Field Sieve - SNFS).
Trong phuong phap sang bac hai, vice tim - . y sao cho x ^ B / f m o d n ) dua tren mpt ding ciu vanh / : Z - ^ z / „ Z vpi
/ (xJ - X n, anh xa m6t binh phupng trong Z tpi mpt binh phuong trong Z / n Z . Ting quat hon ta co tin tai m6t vanh K va mpt dpng ciu vanh (« : 72 -^ Z / nZ . Niu ,9 E 7J va 4>(P')^y'-(moin) vi, x = ^(/?)(modt,) thi taeo: x'^4,(fif .,j,(p')^y-- (,„od„)
ISSN 1859-4611 I. Chpn da thiic (Pplynpmia! Selection)
Chon so nguyen n, , chpn da thitc bat kha quy / ( x j s Z f x ] co bac -/ thoa ./•(m)sO (modn). Gpi BGC I mpt nghiem ciia / ( x ) = 0.
2. Khoi lap CO so thiia si (Set up Factor Base)
Cp sp thita si RFB dupc luu trit la cae cap (/•,/') vpi p nguyen t i v a t - s 7/7(mod/;) Cp sp thira si dai si AFB se dupc ltm trii la cac cap (r. p) vfn p nguyen li va r e Z / n Z sapcho/(7-)aO(mod/)).
3. Sang ttiyin Iinh (Line Sieving) Tim mot luong can thiit cac cap si nguyen (a.b) nguyen li ciing nhau sap cho cac gia tri (a + b,n) smooth tren RFB va
N{a-hbe) smtjolh tren AFB 4, Dai so luyen tinh (Linear Algebra)
Xay dung ma pan trpng do mil cpt ciia ma Iran tuong ung vpi mpt s6 nguyen li trong cac co so RFB, AFB. M6i d6ng ttrong ling vol mpt cap (i7,6).
Gia tri cac phan tit Peng ma triin la si mu cua cac thira si nguyen 16 trpng co sp khi phan lich gia in {a + b,n), N(a-^be). Ap dung Ihuat tpan Lanczps de lim ra mot quan he phu Ihupc tuyen tinh tren cac dnng ciia ma Iran
5. Tinh can bac hai (Square Root)
Ttnh voi
loan ciia Shanks - TpncIIi va dinh ly si du Trung Hoa (Chinese Remainder Theorem - CRT)
Tinh.,, voi y'-^r(,nf [ 7 ("+6/7.)
( . . * ) T
Thua SP cua 77 se la: gcd{x± v ,77) Thuat loan 6, Thuat loan sang inrpng si . Sang ttucmg si la thuat loan phan lich ra thua so nguyen 16 manh nhit hien nay, nhung dupe diing dc phiin lich cac si ril Ion nen viec cai liin lot mi cai dai la mol yeu ciu bit buoc, Trong phiii liep theo sc trinh bay mpt si ky thual dupc sU dung trong qua trinh hien thuc buoc sang, la buoc cc thoi gian tinh lean Ion nhit
4, KY THUAT TANG TOC Bl/OC SANG TREN NEN TINH TOAN SONG SONG 4.1. Lallice Sieve
Hien nay, Irong budc sang, Lattice Sieve (sang
Tap chi so 16. thang 02-2016, Irudng Dai hoc Tay Nguyen ISSN 1859-4611 ludi) duoc cong nlian la phuong phap hieu qua
nhdl Cac s6 RSA duoc phan lich gan day dku sir dung phucmg phap nay [5],
Lallice Sieve duoc gidi thieu uong lai lieu [6], chia CO sd thira so thanh 3 phao^
-*" cac so nguyen to nho p ^ B,, M ; cac so nguyen to trung binh B^, < p < B, L. cac so nguyen lo Idn B, < p < B , ( B; Ion hon ° i ral nhieu)
1. Chpn vimg R cac cap [o.b) duac sang 2 Vdi mol so q Irong M , chi sang nhiing cap {a,b) irong R vdi a + hin ^Q[modg)
Sang gia Iri a + bin vdi cac so p <q Sang gia Iri i\'[a,b) vdi tai ca cac so nguyen to Irong S va M
Thuat toan 7. Thuat toan Lattice Sieve Cac cap ( o - ^ ) thoa a-i-bm = O[mo(i q'j se lao nen mol ludi ^ , Irong mat phang {a,b). D£
dang lim duoc hai veclor co sd cua ludi nay la:
' ^ = ( ^ 1 ' ^ ) va K - ( « , , / ) , ) Khi do mpt mpt diam [c,d\ ciia ludi co dang C>F| -ft/.C, urong ling
{a,b)^{c-a^ +d'a.,C'h, +d'b^) Phuong phap Lallice Sieve hiau qua hon phuong phap sang luyan linh (Line Sieving) do phuang phap sang tuyan tinh dya Iren mang mot chicu de duyet qua lit ca cac gia tri ciia cap (a, b) trong khi Lattice sieve sang mot each ril chii dong tren sang hai chiau,
4.2. Tang loc sdng dua tren tinh lodn song song Budc sang co thai gian ihirc thi rk ldn, nen vipc hi?n Ihuc song song ircn nhiau may tinh la can ihict Viec song song d6i vdi budc sang co tha theo hai each
Cach 1 Nhicu may iinh sang mot gia In Cach 2 Moi may linh sang mot gia tri c/
Vdi each 1 thi cac may linh phai chia se viing nha. moi may se sang mol phdn ludi h, tuong img vo, kich thude bp nhd Ll-cache. Tuy nhicn vai cach nay ih, vice quan ly phirc lap va Ihdi gian trao doi dir hcu g,ua cac may !dn do do rk it khi duac su dung
Song song hoa budc sang theo each 2 hmu qua hon each 1, ^ a duoc su dung irong hdu h6t cac ha ong Ihuc nghiem Vdi cdch song song hod ndy hi mci may tram (clicrl) se duoc giao sane mot sl Iirong ihich hap gid in ^/. sau khi sdng Kong cdc
so <? dupc giao, gui ket qua cho may chu (masW va masier se tdng hop cac kel qua sang. Niu s^
luong kel qua sang chua du thi masier tilp tyc gJao cac gia tri q cho cac client.
4.3. Kit qud thtrc nghiem
Thuc nghiera song song bang chucmg irinh ma ngu6n m d MSIEVE va GGNFS vdi 32 core CPU 2 20GHz, he dieu hanh Linux 64 bit vdi cac trudng hop nhu Irong Bang 1 dugc kk qua vk m^t thdi gian thyc thi va dung Iuong luu tru nhu trona Hinh 1 va Hinh 2.
Bang 1. Cac trudng hop thuc nghiem phan tich ra thira so nguyen to So ki t y ciia So ki tu S6 ki tir
ciia p cua q
100
105
no
115 120 126 129 (RSA-129)
45 47 50 53 56 51 64
55 58 60 62 64 76 65
Hinh I. So sanh thdi gian phan tich theo d?
Idn cua n
50
I g P c h i so 16, thang 02-2016, truong S a i hpc Tay Nguyen
ISSN 1859-4611
!, So sanh d u n g lupTig luu trii then do Ifi'n eua n
5, K E T LL'AX
Ket qua Ihuc nghiem ta thiy thoi gian cua buoc sang chiem phin Ion thcri gian phan tich va tang nhanh khi kieh thuoc 77 lang len. V61 si, RSA-129, CO 129 ki tu thap phan (426 bit), he t h i n g 32 core da phai phan lich trong 5 11 gio, thoi gian sang la 4.16 gio va dung luong dia cimg cin thiit 14 4.5GB. Dp do de Ihuc hien voi eac ii, Ion hon nhieu thi can phai song song hpa vpi nhiing he lh6ng cn s6 luong may Ion hon ril nhiiu. Va vol cac so RSA-1024 thi he t h i n g m i hpa khoa cong khai RSA v l n an tpan.
Vice spng spng tren cac may u a m lam cho l i c d6 tinh loan tang Ien. Tuy nhien v i n co t h i Iam lang hon nira t6c dp linh loan bing each tan dung them sue manh ciia GPU. Do do huong phiit t n i n la can cai dat thu vien luong tu GMP tren GPU, sau do song spng hpa Ihuat loan sang Uuong si, tren GPU de lang hieu qua thuc hien.
I N T E G E R F A C T O R I Z A T I O N T E C H N I Q U E BASED ON P A R A L L E L C O M P U T I N G
Received • a l c : 0 3 / , l / 2 0 , 5 ; Revised Dale:,l/1172016; Accepted for Publication: 0 2 / o ' ; / 2 0 1 6 " " " ' S U M M A R Y
and implemem parallel c o m ; t u ; r ' r l ? k ? d ' s f c l X r h r ^ ° ' " = ' " = ' ' ° * ' " ' ' ' ^ ' ' ™ " ™ Key wards prime fac,artza„o„. parallel compaang. aamber filed sieve
, , , „ TAI L I E U T H A M K H A O
H o T d r p p 286 9 ° o ! , t ' " " ' ' ' ' ' " ^ ' ° " = ' " ' " • """"'""' "f'^"-^" ^ ^ " ' " ^ ' - " ^ " . v . CRC Press, H.W Lenstra Jr( 1987), Factoring integers wilh elliplie curves. The Annals ofMalhemaacs ,26. pp. 649-
' ' o l T d r p p ' ^ r o " ' ' " " ' ' ' ' ' ^ " " ' ' ' ™ ^ " ' ^ " " " " ' ' ^ ' " ' ^ ^ ^ ^ ^ ^ - ^ » - ' = ^-".v-silM Dannstad., ' ' ' ^ S : : ^ , ^ ; : ^ : : ^ : : : ^ ^ -"• •"' ^"f-'-""--"^ e-7»^.7-c S,eve, ^ . e . Thesis,
!. M. Pollard (1993), The laiace s,eve. Lecture Notes in Mathematics 1554, pp. 43-49.
Phan hicn 1 ThS, T r u o n g Thi liluong Giang Khoa KHTN & CN, Pupng DH Tay Nguyen
Dien dicai: 0942509609 Email. lthuonggiang@gmail com
Phan bien 2 ThS, Nguyen Q u i c C u t m g Khoa KHTN- & CN. iiuPng DH Tay Nguyen
Dien ihoai: 0973303109 Email. quoccuong.dhtn(ggmail com 'Ma..,e,: Faciiln-of Na,ural Sciences and Technology. Tay Nguven Unneisilv
Corresponding Author Vu Anh Tuan. Cellphone 09S3 7,7 3,9. Email loiba„.angfa,gmail earn
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