TD 3

x L(x) = E(log2(x)) + 1 x a
b a > b > 0r0=a r1=b
a b
a=r0=r1q1+r2
r1=r2q2+r3
rn−2=rn−1qn−1+rn
rn−1=rnqn
rn= pgcd(a, b)
F0= 0, F1= 1 Fi=Fi−1+Fi−2
i≥2 pgcd(Fn+2, Fn+1)
pgcd(a, b)n b ≥Fn+1
n≥2Fn≥Φn−2Φ = 1+√5
2
a b O(L(b))
a
q1q2. . . qn≤a
x y O(L(y)L(q)) q
O(L(a)L(b))
n=pq p q
m∈(Z/nZ)×me∈(Z/nZ)×c∈(Z/nZ)×cd∈(Z/nZ)×
d∈Zed ≡1 (mod ϕ(n))

(Z/nZ)×x
Z/nZx∈(Z/nZ)×
c m
c m
p q
n= 221
ϕ(n)
7
m= 3
c= 198
n=pq
d∈Nϕ(n)
cd(mod n)
mp≡cd(mod p)mq≡cd(mod q)dp=dmod (p−1) dq=dmod (q−
1) mp≡cdp(mod p)mq≡cdq(mod q)
Z/nZ
(m≡mp(mod p),
m≡mq(mod q).
m m ≡cd(mod n)
c= 198 n= 221 d= 55
ϕ(n)p q
n=pq
pq p +q n ϕ(n)
p q n ϕ(n)
n= 17063 ϕ(n) = 16800 p q

k B1, . . . , Bke= 3
ni1≤i≤k
ni1≤i≤k
m Bi
m3P:= Qk
i=1 nim P > m3
k
n eBeC
eBeC
m
m
(221,11) (221,7)
210 58
m
n p q p > q
p q  := p−q t =p+q
2
s=p−q
2
n=t2−s2
s t √n
n
11598781
p n
p−√n < 4
√4n
d p q
n p q
e ϕ(n)n p
q d
d n

e d n B ϕ(n)
λ= ppcm(p−1, q −1) a∈(Z/nZ)×, aλ= 1
aλ/2n
H={a∈(Z/nZ)×, aλ/2≡ ±1}H
(Z/nZ)×
b∈(Z/nZ)×b p −1p
(q−1)/2q
p−1=2vpp0q−1=2vqq0p0, q0
vp≥vqλ/2vpp0, q0
b H x (Z/nZ)×
x H 1/2
λ B x ∈(Z/nZ)×k
xλ/2=xB/2k+1
n
n e d
n= 77 e= 7 d= 43