Content uploaded by Filip Zagorski
Author content
All content in this area was uploaded by Filip Zagorski on Sep 02, 2014
Content may be subject to copyright.
Content uploaded by Filip Zagorski
Author content
All content in this area was uploaded by Filip Zagorski on Sep 02, 2014
Content may be subject to copyright.
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Vi
σi· · ·
N=pq
g G ⊆Z∗
N
GmodN
ZN
e
ϕ(N)d e ·d= 1 mod (p−1, q −1) d
e
ˆg=gd
λ
1≤j≤λ yj
j xjyj=gxj
ˆyi=yd
im k1
(α, β, γ , δ) := (m·(y1·...·yλ)k1, gk1, md
·(ˆy1·...·ˆyλ)k1,ˆgk1).
i
(αi, βi, γi, δi) = (m·(yi·...·yλ)ki, g ki, md·(ˆyi·...·ˆyλ)ki,ˆgki).
ri
(αi+1, βi+1 , γi+1, δi+1 ) :=
(αi/βxi
i
·(yi+1 ·...·yλ)ri, βi·gri, γi/δxi
i
·(ˆyi+1 ·...·ˆyλ)ri, δi·ˆgri).
ki+1 =ki+ri
(m·(yi+1 ·...·yλ)ki+1 , g ki+1, md·(ˆyi+1 ·...·ˆyλ)ki+1 ,ˆgki+1 ).
k α =m·ykγ=md·ˆykγ=αdδ=βd
α=γeβ=δe
(α, β)
y1·. . . ·yλ
ue(m)m e(m)
m ue(m)
l
ue(m)l= (ml·(y1·...·yλ)k·l, g k·l, md·l·(ˆy1·...·ˆyλ)k·l,ˆgk·l).
ue(ml)ml
k·l
(α, β, γ , δ)
k1
m1m2
(m1·αk, m2·βk, gk), k α, β
αhβh
ghh k
m1m2
β
α
y1, . . . , yλ
m
(m·(y1·...·yλ)k, g k)
k
k
5
6
= 1 −2
4·1
3m
(1
4W)m
W
N
N
i i
s
s
i cic0
i
ci, c0
iG
cic0
ig
i
N
s1s2
s=s1+s2s
s1s2
s, s1, s2
s1s2
z
z B.2
N
i cic0
i
i did0
i
di, d0
icic0
i
ci, c0
i
r
K r
K
r
K
AA X X
A A
1
2X
X
γ χ
(m·(y1· ·yλ)k, gk)
(m·(y1·...·yλ)k, M1·γk, M2·χk, g k)
M1M2
A A
X X
K
A A
A
A
1
2
1
2
2k
k
2−k
c0, c1, . . . , cN−1, c0
0, c0
1, . . . , c0
N−1
cNcN+1, . . . , c2N, c0
N, c0
N+1, . . . , c0
2N
cN+mc0
N+m
m
c0= = c0
0
s1, s2π(j) =
j+smod N s =s1+s2
(id, id0) (rid, rid0)
id0=BGS (id)rid0=B GS (rid)
j
A A X X
N TATXt∈ {A, X}fu,t,j =t
Tt(j)=1 fu,t,j =t fl,t,j ∈ {t, t}
fu,t,j
x∈ {u, l}u, l i
px,i = [ueγ(fx,A,i)ueχ(fx,X,i )uey(cπ(i))uey(c0
π(i))uey(id)uey(id0)]
γ χ y =y1·. . . ·yk
apx,i
[ueγ(fx,A,i)ueχ(fx,X,i )uey(cπ(K+i))uey(c0
π(K+i))uey(rid)uey(rid0)]
t j
v= [pt,j ]av =
[apt,j ;pt,j ]
n≥2
n
A, A
s1s2s
s
s1, s2TAid
i
z
Alice(z)
Alice(z)
t
w
j w
cjπ(w) = j
v= [pt,w]
av = [apt,w;pt,w ]
v av
r
Alice(r)
K:= R(Alice(r)) R
v EncK(av)
v v
ueγ(ft,A,i)
TA
ueχ(ft,X,i)
v av
v av
1≤i≤λ i
i−1
i
λ
λ
r
2
N
N
id, id0, ci, c0
i)
A/A, X/X
4K100N
10N
K= 20 200