C3 Mat Ma Khoa Cong Khai

7/22/2019 C3 Mat Ma Khoa Cong Khai

1/34

Chng 3. Mt m kho cng khai


2/34

3. Mt s h mt kho cng khai

3.1 H mt RSA

3.2 H mt Merkle Hellman 3.3 H mt McEliece

3.4 H mt ElGamal

3.5 H mt Chor- Rivest 3.6 H mt trn ng cong Elliptic


3/34

3.1 H mt RSA

RSA l m cng khai c sng to bi Rivest, Shamir & Adleman MIT (Trng i hc Cng ngh Massachusetts) vo nm 1977.

RSA l m cng khai c bit n nhiu nht v s dng rng rinht hin nay. RSA da trn cc php ton ly tha trong trng hu hn cc s

nguyn theo modulo nguyn t. C th, m ho hay gii m l ccphp ton lu tha theo modulo s rt ln.

Vic thm m, tc l tm kho ring khi bit kho cng khai, da trnbi ton kh lphn tch mt s rt ln ra tha s nguyn t. Nukhng c thng tin g, th ta phi ln lt kim tra tnh chia ht ca s cho tt c cc s nguyn t nh hn cn ca n. y l vic lmkhng kh thi!


4/34

3.1 H mt RSA

Ngi ta chng minh c rng, php ly thacn O((log n)3) php ton, nn c th coi ly thal bi ton d.

Cn ch rng y ta s dngcc s rt lnkhong 1024 bit, tc l c 10350.

Tnh an ton da vo kh ca bi ton phntch ra tha s cc s ln. Bi ton phn tch ratha s yu cu O(elogn log logn) php ton, y l

bi ton kh.


5/34

3.1 H mt RSA

Khi to kho RSA Mi ngi s dng to mt cp kho cng khai ring nh sau:

Chn ngu nhin 2 s nguyn t ln p v q Tnh s lm modulo ca h thng: N = p.q

Ta bit (N)=(p-1)(q-1) V c th dng nh l Trung Hoa gim bt tnh ton Chn ngu nhin kho m e Trong 1


6/34

3.1 H mt RSA

S dng RSA

m ho mu tin, ngi gi: Ly kho cng khai ca ngi nhn KU={e,N}

Tnh C=Memod N, trong 0M


7/34

3.1 H mt RSA

C s ca RSA Theo nh l Ole

a(n) mod N = 1 trong gcd(a,N)=1 Ta c N=p.q

(N)=(p-1)(q-1) e.d=1 mod (N)

e.d=1+k.(N) i vi mt gi tr k no . Suy ra

Cd= (Me)d= M1+k.(N)= M1.(M(n))k suy ra

CdmodN = M1.(1)kmodN = M1modN = M modN


8/34

3.1 H mt RSA

V d Chn cc s nguyn t: p=17& q=11. Tnh n =pq, n = 1711=187

Tnh (n)=(p1)(q-1)=1610=160 Chn e: gcd(e,160)=1; Ly e=7 Xc nh d: de=1 mod 160 v d < 160 Gi tr cn tm l d=23, v 237=161= 10160+1 In kho cng khai KU={7,187}

Gi kho ring b mt KR={23,17,11}


9/34

3.1 H mt RSA

V d p dng m RSA trn nh sau: Cho mu tin M = 88 (vy 88


10/34

3.1 H mt RSA

M hiu qu: M s dng ly tha ca kho cng khai e, nu gi tr ca e nh th

tnh ton s nhanh, nhng d b tn cng. Thng chn e nh hnhoc bng 65537 (216-1), tc l di kho cng khai l 16 bit.Chng hn trong v d trn ta c th la chn e = 23 hoc e = 7.

Ta c th tnh m ho nhanh, nu bit n=pq v s dng nh l phnd Trung Hoa vi mu tin M theo cc Modulo p v q khc nhau. Nukho cng khai e c nh th cn tin tng rng khi chn n ta lun cgcd(e,(n)) = 1. Loi b mi p, q m lm cho (n) khng nguyn tcng nhau vi e.


11/34

3.1 H mt RSA

Gii m hiu qu:

C th s dng nh l phn d Trung Hoa tnhtheo mod p v q, sau kt hp li tm ra bn r.V y ngi s dng kho ring bit c p v q,do c th s dng k thut ny.

Nu s dng nh l phn d Trung Hoa gii m thhiu qu l nhanh gp 4 ln so vi gii m tnh trctip.


12/34

3.1 H mt RSA

Sinh kho RSA

Ngi s dng RSA cn phi xc nh ngu nhin 2 snguyn t rt ln p, q thng thng khong 512 bit.

Sau khi chn c mt kho e hoc d nguyn t cngnhau vi (n), d dng tnh c kho kia chnh l snghch o ca n qua thut ton Euclide m rng.


13/34

3.1 H mt RSA

An ton ca RSA Trn thc t c nhiu cch tn cng khc nhau i vi

m cng khai RSA nh sau: Tm kim kho bng phng php vt cn, phng php ny

khng kh thi vi kch thc ln ca cc s Tn cng bng ton hc da vo kh vic tnh (n) bng

cch phn tch n thnh hai s nguyn t p v q hoc tm cch

tnh trc tip (n). Trong qu trnh nghin cu vic thm m ngi ta xut kiu

tn cng thi gian trong khi gii m, tc l cn c vo tc mho v gii m cc mu tin cho trc m phn on cc thngtin v kho.


14/34

3.1 H mt RSA

im bt ng

nh l:Nu cc thng bo c m bng h mtRSA vi cp kha cng khai (e,n) vi n = p.q th scc thng bo khng th che du c l

1q,1dUCLN11p,1eUCLN1N


15/34

3.2 H mt Merkle Hellman

H mt Merkle Hellman

- Dy siu tng: Dy s ngdng tmn

vi

- Bi ton xp ba l:Cho tp cc gi tr

v mt tng S. Hy tnh cc gi tr bi:vi

n21

a,,a,a

1i

1jji aa ni2,i

n21 M,,M,M

nn2211 MbMbMbS 1,0bi


16/34


TT gii bton xp ba l trong trng hp dy siu tng:


17/34


- To kho:


18/34


M ho


19/34

3. Mt s h mt kho cng khai (15)

Gii m


20/34


Chng minh


21/34


3.1 H mt RSA (Ron Rivest, Adi Shamir v Len Adleman)

- To kho:


22/34


- M ho: Bn m l B, bn nhn l A


23/34


Ch :

1. S m vn nng

thay cho

2. im bt ng

nh l: Nu cc thng bo c m bng h mt RSA vi

cp kha cng khai vith s cc thng bo khng th che du c l

1q,1pBCNN

1q1p

n,e q.pn

1q,1dUCLN11p,1eUCLN1N


24/34


3.2 H mt Rabin

- To kho+ To 2 s nguyn t ln, ngu nhin v phn

bit p v q c kch thc xp x nhau.

+ Tnh

+ Kho cng khai l n, kho b mt l cc cp s(p, q).

q.pn


25/34


- M ho:

+ Nhn kho cng khai ca A: n.+ Biu th bn tin di dng mt s nguyn mnm trong di

+ Tnh

+ Gi bn m c cho A

1n,0

nmodmc 2


26/34


- Gii m:

+ A phi thc hin cc bc sau:Tm 4 cn bchai ca l m1, m2, m3hoc m4

+ Thng bo cho ngi gi l mt trong 4 gi trm1, m2, m3hoc m4. Bng mt cch no A s

quyt nh m l gi tr no.

nmodc


27/34


Ch : Khi p, q l cc s nguyn Blum th ta c thtnh 4 cn bc 2 ca c mod n nh sau:+ Tm a,b nguyn tho mn:+Tnh cc gi tr sau:

4 gi tr cn bc 2 ca l x,, y v

1bqap

pmodcr

4/1p

qmodcs

4/1q

nmodbqrapsx nmodbqrapsy nmodc nmodx

nmody


28/34


3.3 H mt Elgamal

- To kho:+ To 1 s nguyn t p ln v mt phn t sinhca nhm nhn ca cc s nguyn mod p.

+ Chn mt s nguyn ngu nhin a,

v tnh

Kho cng khai l b 3 s , kho b mt l a.

*pZ

2pa1

pmodaa

,,p


29/34


- M ho:

+ Nhn kho cng khai ca A+ Biu th bn tin di dng mt s nguyn mtrong di

+ Chn s nguyn ngu nhin k,

+ Tnh v

+ Gi bn m cho A

a

,,p

1p,,1,0 2pk1

pmodk pmodm ka ,c


30/34


- Gii m:

+ S dng kho ring a tnh+ Khi phc bn r bng cch tnh

- Chng minh:

pmod

a1p

pmod

a

pmodmm. kakaa


31/34


3.5 H mt trn ng cong Elipptic


32/34



33/34



34/34


Documents

C3 Mat Ma Khoa Cong Khai