Tap chf Tin hpc vk Dieu khien hgc, T.27, S.l (2011), 253-262 MO RONG Lirgc D6 NGLTOTNG CUA SHAMIR CHO VIEC CHIA SE DONG THOTl NHIEU Bl MAT VU HUY H O A N G ^ HO THUAN^ ^ Cue Co yeu 893 Ban Ca yeu Chinh phu ^ Vien Cong nghe thong tin, Vien Khoa hgc vd Cong nghe Viet Nam Abstract Secret sharing schemes are important techniques in the key management in cryptography and distributed computation In 1979, Shamir [6] proposed a threshold secret sharing scheme, in which one secret is divided into w pieces (shares) and delivered to w users such that only groups of t or more users {t < w) could cooperately reconstruct the secret In this paper, the main idea of the proposed secret sharing scheme is the use of the secrets to be shared as the coefficients of the polynomial a{x) Numerical examples are given not only to illustrate the approach, but also show deeper comprehension of the method presented in [9] Tdm tat Cac lirgc chia s^ bi mat la nhirng ky thuat quan de qudn li khoa mat ma va tinh toan phan tan Nam 1979, Shamir [6] de xuat mot lugc dd chia s^ bi mat nguang, theo mot bl mat dugc chia lam w mdnh va trao cho w ngudi dimg, cho chi nhiing nhdm t ngudi diing hay nhieu horn (t < ty) cd the hgp tac vdi de khoi phuc lai bi mat Y tudng chinh ciia lugc dd chia sk bi mat dugc de xuat bai bao la viec sii dung cac bl mat can dugc chia sk lam cac he sd ciia da thiic a{x) Nhiing vi du dugc dua day ngoai viec minh hoa cho chinh each tiep can mdi ciia bai bao nay, co y nghia giiip hieu sau sac han ve phuang phap dugc trinh bay [9] D A T V A N D E Chia sl bi mat la mgt cdng cu quan trgng vd dirge nghien ciiu rgng rai mat ma vd tinh toan phan tan Mgt lurgrc phan chia bi mat la mgt giao thiic dd ngurdi dieu hanh D (dieu phdi) phan chia mgt bi mat thdnh nhiing mdnh nhd durgc ggi la cac phan chia va phan phdi mgt each bi mat cho mgt tap ngurdi tham gia he thdng (thanh vien) cho chi mgt sd tap xdc dinh (ggi la cdc tap dugrc quyen) la cd the khdi phuc durgrc bi mat da phan chia Vi vay, cung vdi su phdt trien ciia cdc ky thuat mat ma thi lugc chia sl bi mat dugrc nghien ciiu va phdt trien manh me nhu cac lugc durg-c trinh bdy trong: L Csirmaz [1], Farras O, Marti-Farre J, Padrd C [2], J Marti-Farre, C Padrd [3], hay phuong phap day dgc lap durgc dugc gidi thieu bdi C Blundo, A De Santis, R De Simone, U Vaccaro [4] vd dugc tong quat hod bdi Padrd vd Saez [5] Vao nam 1979, Shamir [6] de xuat lugc dd chia sl bi mat (hay cdn dugc ggi la phan chia bl mat) va dugc ggi la lugc nguong ciia Shamir Lugc ngudrng Shamir cho mgt each vii HUY HOANG, II6 THUXN 254 chuyen mdt bf mat, dan w i)han (-hia, Ttiy nliion trudng hgp phan chia dong thdi t bl mat, ma ta dp dung / lan hr(.re Shamir rd rdng Id khong hieu qud, bdi vi vdi moi mgt khda K deu phdi thiet lap lai vice chia sl, dan den tdng khong gian luu trii va khdi lugng tinh todn Bai bao de xuat sii dung f bf mat kj {0 < j < t - 1), de xay dung mgt da thiic a{x) co bac tdi da la / - va sau dd tao w phan chia bang each tim gia tri ciia da thiic tai w diem mdi Do dd lugc se hieu qud ve khong gian luu trii vd khdi lugng tfnh todn Moi vien Pi se eo mgt diem (:»:,, yi) tren da thiic Viec siV dung them mgt ham bdm vd he mat RSA an toan vdi sd mil gidi ma ldn [8] d day nham lam tang an todn ciia lugc P H U O N G P H A P C H I A S NHIEU Bf M A T Sl> D U N G HE P H U O N G T R t N H DAI SO T U Y E N T I N H • • * Giai doan kh6i tao D chgn it; phan tii khdc nhau, khac khong Zp vd kf hieu Id x^ (1 < z < w) Voi i