trt t rs - - PowerPoint PPT Presentation
trt t rs rt rst trs r
■♥tr♦❞✉❝t✐♦♥ t♦ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs
❇❛rt ▼❡♥♥✐♥❦ ❘❛❞❜♦✉❞ ❯♥✐✈❡rs✐t② ✭❚❤❡ ◆❡t❤❡r❧❛♥❞s✮
❙✉♠♠❡r s❝❤♦♦❧ ♦♥ r❡❛❧✲✇♦r❧❞ ❝r②♣t♦ ❛♥❞ ♣r✐✈❛❝② ❏✉♥❡ ✺✱ ✷✵✶✼
✶ ✴ ✺✸
❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥
← − − − − − − − − − − − − − − − − − − − − − − − − − − − − → ❇ ❊♥❝r②♣t✐♦♥ ◆♦ ♦✉ts✐❞❡r ❝❛♥ ❧❡❛r♥ ❛♥②t❤✐♥❣ ❛❜♦✉t ❞❛t❛ ❆✉t❤❡♥t✐❝❛t✐♦♥ ◆♦ ♦✉ts✐❞❡r ❝❛♥ ♠❛♥✐♣✉❧❛t❡ ❞❛t❛
✷ ✴ ✺✸
❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥
← − − − − − − − − − − − − − − − − − − − − − − − − − − − − → ❇ − − − − − → ← − − − − − ❊♥❝r②♣t✐♦♥ ◆♦ ♦✉ts✐❞❡r ❝❛♥ ❧❡❛r♥ ❛♥②t❤✐♥❣ ❛❜♦✉t ❞❛t❛ ❆✉t❤❡♥t✐❝❛t✐♦♥ ◆♦ ♦✉ts✐❞❡r ❝❛♥ ♠❛♥✐♣✉❧❛t❡ ❞❛t❛
✷ ✴ ✺✸
❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥
← − − − − − − − − − − − − − − − − − − − − − − − − − − − − → ❇ − − − − − → ← − − − − − ❊♥❝r②♣t✐♦♥
❆✉t❤❡♥t✐❝❛t✐♦♥ ◆♦ ♦✉ts✐❞❡r ❝❛♥ ♠❛♥✐♣✉❧❛t❡ ❞❛t❛
✷ ✴ ✺✸
❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥
← − − − − − − − − − − − − − − − − − − − − − − − − − − − − → ❇ − − − − − → ← − − − − − ❊♥❝r②♣t✐♦♥
❆✉t❤❡♥t✐❝❛t✐♦♥
✷ ✴ ✺✸
❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥
A, M N C, T
AE
k
◆♦♥❝❡ r❛♥❞♦♠✐③❡s t❤❡ s❝❤❡♠❡
✸ ✴ ✺✸
❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥
A, M N C, T
AE
k
✸ ✴ ✺✸
❈❆❊❙❆❘ ❈♦♠♣❡t✐t✐♦♥ ❈♦♠♣❡t✐t✐♦♥ ❢♦r ❆✉t❤❡♥t✐❝❛t❡❞ ❊♥❝r②♣t✐♦♥✿ ❙❡❝✉r✐t②✱ ❆♣♣❧✐❝❛❜✐❧✐t②✱ ❛♥❞ ❘♦❜✉st♥❡ss
▼❛r ✶✺✱ ✷✵✶✹✿ ✺✼ ✜rst r♦✉♥❞ ❝❛♥❞✐❞❛t❡s ❏✉❧ ✼✱ ✷✵✶✺✿ ✷✾✳✺ s❡❝♦♥❞ r♦✉♥❞ ❝❛♥❞✐❞❛t❡s ❆✉❣ ✶✺✱ ✷✵✶✻✿ ✶✻ t❤✐r❞ r♦✉♥❞ ❝❛♥❞✐❞❛t❡s ❄❄✿ ❛♥♥♦✉♥❝❡♠❡♥t ♦❢ ✜♥❛❧✐sts ❉❡❝ ✶✺✱ ✷✵✶✼✿ ❛♥♥♦✉♥❝❡♠❡♥t ♦❢ ✜♥❛❧ ♣♦rt❢♦❧✐♦ ✭❄✮
✹ ✴ ✺✸
❈❆❊❙❆❘ ❈♦♠♣❡t✐t✐♦♥✱ ◆♦t ❚♦ ❇❡ ❈♦♥❢✉s❡❞ ❲✐t❤✿
✺ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs
m c
E
k ❚✇❡❛❦✿ ✢❡①✐❜✐❧✐t② t♦ t❤❡ ❝✐♣❤❡r ❊❛❝❤ t✇❡❛❦ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥
✻ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs
m t c k
✻ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❖❈❇①
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T ˜ E
N,tA1 k
˜ E
N,tA2 k
˜ E
N,tAa k
˜ E
N,tM⊕ k
˜ E
N,tM1 k
˜ E
N,tM2 k
˜ E
N,tMd k
■♥t❡r♥❛❧❧② ❜❛s❡❞ ♦♥ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡r
❚✇❡❛❦ ✐s ✉♥✐q✉❡ ❢♦r ❡✈❡r② ❡✈❛❧✉❛t✐♦♥ ❉✐✛❡r❡♥t ❜❧♦❝❦s ❛❧✇❛②s tr❛♥s❢♦r♠❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t✇❡❛❦
❈❤❛♥❣❡ ♦❢ t✇❡❛❦ s❤♦✉❧❞ ❜❡ ❡✣❝✐❡♥t
✼ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❖❈❇①
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T ˜ E
N,tA1 k
˜ E
N,tA2 k
˜ E
N,tAa k
˜ E
N,tM⊕ k
˜ E
N,tM1 k
˜ E
N,tM2 k
˜ E
N,tMd k
E
❈❤❛♥❣❡ ♦❢ t✇❡❛❦ s❤♦✉❧❞ ❜❡ ❡✣❝✐❡♥t
✼ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❖❈❇①
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T ˜ E
N,tA1 k
˜ E
N,tA2 k
˜ E
N,tAa k
˜ E
N,tM⊕ k
˜ E
N,tM1 k
˜ E
N,tM2 k
˜ E
N,tMd k
E
✼ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❳❚❙
M1 M2 Md C1 C2 Cd ˜ E i,1
k
˜ E i,2
k
˜ E i,d
k
❈❤❛♥❣❡ ♦❢ t✇❡❛❦ s❤♦✉❧❞ ❜❡ ❡✣❝✐❡♥t ✭❛s ❜❡❢♦r❡✮ ■♥❝r❡♠❡♥t❛❧✐t②✿ ❝❤❛♥❣❡ ✐♥ ♦♥❡ ✭♦r ❢❡✇✮ ❜❧♦❝❦s
✽ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❳❚❙
M1 M2 Md C1 C2 Cd ˜ E i,1
k
˜ E i,2
k
˜ E i,d
k
■♥❝r❡♠❡♥t❛❧✐t②✿ ❝❤❛♥❣❡ ✐♥ ♦♥❡ ✭♦r ❢❡✇✮ ❜❧♦❝❦s
✽ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❳❚❙
M1 M′
2 = M2
Md C1 C′
2
Cd ˜ E i,1
k
˜ E i,2
k
˜ E i,d
k
✽ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ✐♥ ❙❦❡✐♥
config M1 Mℓ iv h
· · · · · · ˜ Econ ˜ Emsg ˜ Emsg ˜ Eout
✾ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡r ❉❡s✐❣♥s ✐♥ ❈❆❊❙❆❘
t
E
P
❉❡❞✐❝❛t❡❞ ❇❧♦❝❦❝✐♣❤❡r✲❇❛s❡❞ P❡r♠✉t❛t✐♦♥✲❇❛s❡❞ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ Prøst✱ ❏♦❧t✐❦✱ ▼❛r❜❧❡✱ ❖▼❉✱ P❖❊❚✱ ▼✐♥❛❧♣❤❡r ❙❈❘❊❆▼✱ ❙❍❊▲▲✱ ❆❊❩✱ ❈❖P❆✴ ❉❡♦①②s ❊▲♠❉✱ ❖❈❇✱ ❖❚❘
✶✵ ✴ ✺✸
✜rst r♦✉♥❞✱ s❡❝♦♥❞ r♦✉♥❞✱ t❤✐r❞ r♦✉♥❞
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡r ❉❡s✐❣♥s ✐♥ ❈❆❊❙❆❘
t
E
P
❉❡❞✐❝❛t❡❞ ❇❧♦❝❦❝✐♣❤❡r✲❇❛s❡❞ P❡r♠✉t❛t✐♦♥✲❇❛s❡❞ ❑■❆❙❯✱ ❈❇❆✱ ❈❖❇❘❆✱ ✐❋❡❡❞✱ Prøst✱ ❏♦❧t✐❦✱ ▼❛r❜❧❡✱ ❖▼❉✱ P❖❊❚✱ ▼✐♥❛❧♣❤❡r ❙❈❘❊❆▼✱ ❙❍❊▲▲✱ ❆❊❩✱ ❈❖P❆✴ ❉❡♦①②s ❊▲♠❉✱ ❖❈❇✱ ❖❚❘
✶✵ ✴ ✺✸
✜rst r♦✉♥❞✱ s❡❝♦♥❞ r♦✉♥❞✱ t❤✐r❞ r♦✉♥❞
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✶✶ ✴ ✺✸
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✶✷ ✴ ✺✸
❉❡❞✐❝❛t❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs
✶✸ ✴ ✺✸
❚❲❊❆❑❊❨ ❋r❛♠❡✇♦r❦
(k, t) m c
· · · · · · · · · · · ·
f f f g g g g h h h
❙❡❝✉r✐t② ♠❡❛s✉r❡❞ t❤r♦✉❣❤ ❝r②♣t❛♥❛❧②s✐s ❖✉r ❢♦❝✉s✿ ♠♦❞✉❧❛r ❞❡s✐❣♥
✶✹ ✴ ✺✸
❚❲❊❆❑❊❨ ❋r❛♠❡✇♦r❦
(k, t) m c
· · · · · · · · · · · ·
f f f g g g g h h h
✶✹ ✴ ✺✸
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✶✺ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡
✶ ❉❡t❡r♠✐♥❡ ❛♣♣r♦♣r✐❛t❡ s❡❝✉r✐t② ♠♦❞❡❧ ✷ ❉❡s✐❣♥ t❤❡ s❝❤❡♠❡ ✸ P❡r❢♦r♠ s❡❝✉r✐t② ❛♥❛❧②s✐s
✶✻ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧
m c k
t
✳
← − − − − →
✳ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② ❉✐✛❡r❡♥t t✇❡❛❦s ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❙tr♦♥❣ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② ❆❞✈❡rs❛r② ♠❛② ❤❛✈❡ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❛❝❝❡ss t♦
✶✼ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧
m c k
t
✳
← − − − − →
✳ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t②
Ek s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t
→ ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❙tr♦♥❣ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t② ❆❞✈❡rs❛r② ♠❛② ❤❛✈❡ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ ❛❝❝❡ss t♦
✶✼ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧
m c k
t
✳
← − − − − →
✳ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t②
Ek s❤♦✉❧❞ ❧♦♦❦ ❧✐❦❡ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t
→ ♣s❡✉❞♦✲✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❙tr♦♥❣ ❚✇❡❛❦❛❜❧❡ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ ❙❡❝✉r✐t②
E
✶✼ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✶✿ ❙❡❝✉r✐t② ▼♦❞❡❧
❊①❛♠♣❧❡
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T ˜ E
N,tA1 k
˜ E
N,tA2 k
˜ E
N,tAa k
˜ E
N,tM⊕ k
˜ E
N,tM1 k
˜ E
N,tM2 k
˜ E
N,tMd k
Ek ❡✈❛❧✉❛t❡❞ ✐♥ ❢♦r✇❛r❞ ❞✐r❡❝t✐♦♥ ♦♥❧②
Ek ❡✈❛❧✉❛t❡❞ ✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s
✶✽ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t ?
❍♦✇ t♦ ♠✐♥❣❧❡ t❤❡ t✇❡❛❦ ✐♥t♦ t❤❡ ❡✈❛❧✉❛t✐♦♥❄ ❜❧❡♥❞ ✐t ✇✐t❤ t❤❡ ❦❡② ❜❧❡♥❞ ✐t ✇✐t❤ t❤❡ st❛t❡
✶✾ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t ?
❍♦✇ t♦ ♠✐♥❣❧❡ t❤❡ t✇❡❛❦ ✐♥t♦ t❤❡ ❡✈❛❧✉❛t✐♦♥❄
← − − − ← − − −
❜❧❡♥❞ ✐t ✇✐t❤ t❤❡ ❦❡② ❜❧❡♥❞ ✐t ✇✐t❤ t❤❡ st❛t❡
✶✾ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t
❋♦r ✲♠✐①✐♥❣✱ ❦❡② ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ✐♥ ❡✈❛❧✉❛t✐♦♥s ❙❝❤❡♠❡ ✐s ✐♥s❡❝✉r❡ ✐❢ ✐s ❊✈❡♥✲▼❛♥s♦✉r ❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ✐s ♠♦r❡ ❛❞✈❛♥❝❡❞
✷✵ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t
❚❲❊❆❑❊❨ ❜❧❡♥❞✐♥❣ ✐s ♠♦r❡ ❛❞✈❛♥❝❡❞
✷✵ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t
✷✵ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t
❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦
❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r②
✷✶ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k t
Ek(t, m) = Ek(t ⊕ C, m ⊕ C)
❙♦♠❡ s❡❝r❡❝② r❡q✉✐r❡❞✿ ❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦
❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r②
✷✶ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k h ⊗ t
Ek(t, m) = Ek(t ⊕ C, m ⊕ C)
❙t✐❧❧ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ❛❞✈❡rs❛r② ❤❛s ❛❝❝❡ss t♦
❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r②
✷✶ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k h ⊗ t
Ek(t, m) = Ek(t ⊕ C, m ⊕ C)
E−1
k
❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r②
✷✶ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k h ⊗ t
Ek(t, m) = Ek(t ⊕ C, m ⊕ C)
E−1
k
E−1
k (t, c) ⊕
E−1
k (t ⊕ C, c) = h ⊗ C
❚✇♦✲s✐❞❡❞ ♠❛s❦✐♥❣ ♥❡❝❡ss❛r②
✷✶ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k h ⊗ t h ⊗ t
Ek(t, m) = Ek(t ⊕ C, m ⊕ C)
E−1
k
E−1
k (t, c) ⊕
E−1
k (t ⊕ C, c) = h ⊗ C
✷✶ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k h ⊗ t h ⊗ t
❱❛r✐❛t✐♦♥ ✐♥ ♠❛s❦✐♥❣❄ ❉❡♣❡♥❞s ♦♥ ❢✉♥❝t✐♦♥s ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♦ ♣r♦❜❧❡♠
✷✷ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k f(t) f(t)
❱❛r✐❛t✐♦♥ ✐♥ ♠❛s❦✐♥❣❄ ❉❡♣❡♥❞s ♦♥ ❢✉♥❝t✐♦♥s ❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♦ ♣r♦❜❧❡♠
✷✷ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
E
k f1(t) f2(t)
❘❡❧❡❛s✐♥❣ s❡❝r❡❝② ✐♥ ❄ ❯s✉❛❧❧② ♥♦ ♣r♦❜❧❡♠
✷✷ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
P
f1(t) f2(t)
✷✷ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✷✿ P❧❛②❣r♦✉♥❞
m c
P
f1(t) f2(t)
✷✷ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸✿ ❆♥❛❧②s✐s
m c
Ek/P
f1(t) f2(t)
Ek s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t
Ek ❙t❡♣ ✸❛✿
❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s ♥❡❡❞ ❛t ♠♦st❄
❙t❡♣ ✸❛✿
❇♦✐❧s ❞♦✇♥ t♦ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s
❙t❡♣ ✸❜✿
❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s ♥❡❡❞ ❛t ❧❡❛st❄
❙t❡♣ ✸❜✿
❇♦✐❧s ❞♦✇♥ t♦ ♣r♦✈❛❜❧❡ s❡❝✉r✐t②
✷✸ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸✿ ❆♥❛❧②s✐s
m c
Ek/P
f1(t) f2(t)
Ek s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t
Ek
❙t❡♣ ✸❛✿ • ❇♦✐❧s ❞♦✇♥ t♦ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s ❙t❡♣ ✸❜✿
❍♦✇ ♠❛♥② ❡✈❛❧✉❛t✐♦♥s ❞♦❡s ♥❡❡❞ ❛t ❧❡❛st❄
❙t❡♣ ✸❜✿
❇♦✐❧s ❞♦✇♥ t♦ ♣r♦✈❛❜❧❡ s❡❝✉r✐t②
✷✸ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸✿ ❆♥❛❧②s✐s
m c
Ek/P
f1(t) f2(t)
Ek s❤♦✉❧❞ ✏❧♦♦❦ ❧✐❦❡✑ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ❢♦r ❡✈❡r② t
Ek
❙t❡♣ ✸❛✿ • ❇♦✐❧s ❞♦✇♥ t♦ ✜♥❞✐♥❣ ❣❡♥❡r✐❝ ❛tt❛❝❦s
❙t❡♣ ✸❜✿ • ❇♦✐❧s ❞♦✇♥ t♦ ♣r♦✈❛❜❧❡ s❡❝✉r✐t②
✷✸ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦
m c
Ek/P
f1(t) f2(t) ❋♦r ❛♥② t✇♦ q✉❡r✐❡s ✱ ✿ ❯♥❧✐❦❡❧② t♦ ❤❛♣♣❡♥ ❢♦r r❛♥❞♦♠ ❢❛♠✐❧② ♦❢ ♣❡r♠✉t❛t✐♦♥s ■♠♣❧✐❝❛t✐♦♥ st✐❧❧ ❤♦❧❞s ✇✐t❤ ❞✐✛❡r❡♥❝❡ ①♦r❡❞ t♦
❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s
✷✹ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦
m c
Ek/P
f1(t) f2(t)
m ⊕ f1(t) = m′ ⊕ f1(t′) = ⇒ c ⊕ f2(t) = c′ ⊕ f2(t′) ❯♥❧✐❦❡❧② t♦ ❤❛♣♣❡♥ ❢♦r r❛♥❞♦♠ ❢❛♠✐❧② ♦❢ ♣❡r♠✉t❛t✐♦♥s ■♠♣❧✐❝❛t✐♦♥ st✐❧❧ ❤♦❧❞s ✇✐t❤ ❞✐✛❡r❡♥❝❡ ①♦r❡❞ t♦
❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s
✷✹ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦
m c
Ek/P
f1(t) f2(t)
m ⊕ f1(t) = m′ ⊕ f1(t′) = ⇒ c ⊕ f2(t) = c′ ⊕ f2(t′)
■♠♣❧✐❝❛t✐♦♥ st✐❧❧ ❤♦❧❞s ✇✐t❤ ❞✐✛❡r❡♥❝❡ ①♦r❡❞ t♦
❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s
✷✹ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦
m c
Ek/P
f1(t) f2(t)
m ⊕ f1(t) = m′ ⊕ f1(t′) = ⇒ c ⊕ f2(t) = c′ ⊕ f2(t′)
❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ❡✈❛❧✉❛t✐♦♥s
✷✹ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❛✿ ●❡♥❡r✐❝ ❆tt❛❝❦
m c
Ek/P
f1(t) f2(t)
m ⊕ f1(t) = m′ ⊕ f1(t′) = ⇒ c ⊕ f2(t) = c′ ⊕ f2(t′)
❙❝❤❡♠❡ ❝❛♥ ❜❡ ❜r♦❦❡♥ ✐♥ ≈ 2n/2 ❡✈❛❧✉❛t✐♦♥s
✷✹ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❜✿ ❙❡❝✉r✐t② Pr♦♦❢
m c
Ek/P
f1(t) f2(t)
❚②♣✐❝❛❧ ❛♣♣r♦❛❝❤✿
❈♦♥s✐❞❡r ❛♥② tr❛♥s❝r✐♣t ❛♥ ❛❞✈❡rs❛r② ♠❛② s❡❡ ▼♦st ✬s s❤♦✉❧❞ ❜❡ ❡q✉❛❧❧② ❧✐❦❡❧② ✐♥ ❜♦t❤ ✇♦r❧❞s ❖❞❞ ♦♥❡s s❤♦✉❧❞ ❤❛♣♣❡♥ ✇✐t❤ ✈❡r② s♠❛❧❧ ♣r♦❜❛❜✐❧✐t②
❆❧❧ ❝♦♥str✉❝t✐♦♥s ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥✿ s❡❝✉r❡ ✉♣ t♦ ❡✈❛❧✉❛t✐♦♥s
✷✺ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❜✿ ❙❡❝✉r✐t② Pr♦♦❢
m c
Ek/P
f1(t) f2(t)
❆❧❧ ❝♦♥str✉❝t✐♦♥s ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥✿ s❡❝✉r❡ ✉♣ t♦ ❡✈❛❧✉❛t✐♦♥s
✷✺ ✴ ✺✸
❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❙t❡♣ ✸❜✿ ❙❡❝✉r✐t② Pr♦♦❢
m c
Ek/P
f1(t) f2(t)
❆❧❧ ❝♦♥str✉❝t✐♦♥s ✐♥ t❤✐s ♣r❡s❡♥t❛t✐♦♥✿ s❡❝✉r❡ ✉♣ t♦ ≈ 2n/2 ❡✈❛❧✉❛t✐♦♥s
✷✺ ✴ ✺✸
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣
❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✷✻ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣
❇❧♦❝❦❝✐♣❤❡r✲❇❛s❡❞✳
m c tweak-based mask
Ek
t②♣✐❝❛❧❧② ✶✷✽ ❜✐ts
♣P❡r♠✉t❛t✐♦♥✲❇❛s❡❞✳♣
m c tweak-based mask
P
♠✉❝❤ ❧❛r❣❡r✿ ✷✺✻✲✶✻✵✵ ❜✐ts
✷✼ ✴ ✺✸
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣
❇❧♦❝❦❝✐♣❤❡r✲❇❛s❡❞✳
m c tweak-based mask
Ek
t②♣✐❝❛❧❧② ✶✷✽ ❜✐ts
♣P❡r♠✉t❛t✐♦♥✲❇❛s❡❞✳♣
m c tweak-based mask
P
♠✉❝❤ ❧❛r❣❡r✿ ✷✺✻✲✶✻✵✵ ❜✐ts
✷✼ ✴ ✺✸
❖r✐❣✐♥❛❧ ❈♦♥str✉❝t✐♦♥s
m c t
Ek Ek
m c h(t)
Ek
✷✽ ✴ ✺✸
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✭❳❊❳✮
m c 2α3β7γ · Ek(N)
Ek
❯s❡❞ ✐♥ ❖❈❇✷✱ ✶✹ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s✱ ❛♥❞ ❳❚❙ P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▼✐♥❛❧♣❤❡r ❛♥❞ Prøst ✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♦❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪✮
✷✾ ✴ ✺✸
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✭❳❊❳✮
m c 2α3β7γ · Ek(N)
Ek
P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ ✈❛r✐❛♥ts ✐♥ ▼✐♥❛❧♣❤❡r ❛♥❞ Prøst ✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♦❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪✮
✷✾ ✴ ✺✸
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✭❳❊❳✮
m c 2α3β7γ · Ek(N)
Ek
m c 2α3β7γ · (kN ⊕ P(kN))
P
✭❣❡♥❡r❛❧✐③❡❞ ❜② ❈♦❣❧✐❛t✐ ❡t ❛❧✳ ❬❈▲❙✶✺❪✮
✷✾ ✴ ✺✸
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T ˜ E
N,tA1 k
˜ E
N,tA2 k
˜ E
N,tAa k
˜ E
N,tM⊕ k
˜ E
N,tM1 k
˜ E
N,tM2 k
˜ E
N,tMd k
❯♣❞❛t❡ ♦❢ ♠❛s❦✿
❙❤✐❢t ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❳❖❘
❱❛r✐❛❜❧❡ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ❊①♣❡♥s✐✈❡ ♦♥ ❝❡rt❛✐♥ ♣❧❛t❢♦r♠s
✸✵ ✴ ✺✸
L = EK(N)
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T
2·32L 2232L 2a32L 2d3L 2L 22L 2dL 2L 22L 2dL
Ek Ek Ek Ek Ek Ek Ek
❯♣❞❛t❡ ♦❢ ♠❛s❦✿
❙❤✐❢t ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❳❖❘
❱❛r✐❛❜❧❡ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ❊①♣❡♥s✐✈❡ ♦♥ ❝❡rt❛✐♥ ♣❧❛t❢♦r♠s
✸✵ ✴ ✺✸
L = EK(N)
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T
2·32L 2232L 2a32L 2d3L 2L 22L 2dL 2L 22L 2dL
Ek Ek Ek Ek Ek Ek Ek
❯♣❞❛t❡ ♦❢ ♠❛s❦✿
❙❤✐❢t ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❳❖❘
❱❛r✐❛❜❧❡ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ❊①♣❡♥s✐✈❡ ♦♥ ❝❡rt❛✐♥ ♣❧❛t❢♦r♠s
✸✵ ✴ ✺✸
L = EK(N)
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T
2·32L 2232L 2a32L 2d3L 2L 22L 2dL 2L 22L 2dL
Ek Ek Ek Ek Ek Ek Ek
❯♣❞❛t❡ ♦❢ ♠❛s❦✿
❙❤✐❢t ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❳❖❘
❱❛r✐❛❜❧❡ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ❊①♣❡♥s✐✈❡ ♦♥ ❝❡rt❛✐♥ ♣❧❛t❢♦r♠s
✸✵ ✴ ✺✸
L = EK(N)
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T
2·32L 2232L 2a32L 2d3L 2L 22L 2dL 2L 22L 2dL
Ek Ek Ek Ek Ek Ek Ek
❯♣❞❛t❡ ♦❢ ♠❛s❦✿
❙❤✐❢t ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❳❖❘
❱❛r✐❛❜❧❡ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ❊①♣❡♥s✐✈❡ ♦♥ ❝❡rt❛✐♥ ♣❧❛t❢♦r♠s
✸✵ ✴ ✺✸
L = EK(N)
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T
2·32L 2232L 2a32L 2d3L 2L 22L 2dL 2L 22L 2dL
Ek Ek Ek Ek Ek Ek Ek
❯♣❞❛t❡ ♦❢ ♠❛s❦✿
❙❤✐❢t ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❳❖❘
❱❛r✐❛❜❧❡ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ❊①♣❡♥s✐✈❡ ♦♥ ❝❡rt❛✐♥ ♣❧❛t❢♦r♠s
✸✵ ✴ ✺✸
L = EK(N)
P♦✇❡r✐♥❣✲❯♣ ▼❛s❦✐♥❣ ✐♥ ❖❈❇✷
A1 A2 Aa M1 M2 Md ⊕Mi C1 C2 Cd T
2·32L 2232L 2a32L 2d3L 2L 22L 2dL 2L 22L 2dL
Ek Ek Ek Ek Ek Ek Ek
✸✵ ✴ ✺✸
L = EK(N)
m c
Ek
❙✐♥❣❧❡ ❳❖❘ ▲♦❣❛r✐t❤♠✐❝ ❛♠♦✉♥t ♦❢ ✜❡❧❞ ❞♦✉❜❧✐♥❣s ✭♣r❡❝♦♠♣✉t❡❞✮
▼♦r❡ ❡✣❝✐❡♥t t❤❛♥ ♣♦✇❡r✐♥❣✲✉♣ ❬❑❘✶✶❪
✸✶ ✴ ✺✸
m c
Ek
✸✶ ✴ ✺✸
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣
❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✸✷ ✴ ✺✸
▼❛s❦❡❞ ❊✈❡♥✲▼❛♥s♦✉r ✭MEM✮
m c ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 ◦ P(Nk)
P
❈♦♠❜✐♥❡s ❛❞✈❛♥t❛❣❡s ♦❢✿
P♦✇❡r✐♥❣✲✉♣ ♠❛s❦✐♥❣ ❲♦r❞✲❜❛s❡❞ ▲❋❙❘s
❙✐♠♣❧❡r✱ ❝♦♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♦r❡ ❡✣❝✐❡♥t
✸✸ ✴ ✺✸
▼❛s❦❡❞ ❊✈❡♥✲▼❛♥s♦✉r ✭MEM✮
m c ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 ◦ P(Nk)
P
❙✐♠♣❧❡r✱ ❝♦♥st❛♥t✲t✐♠❡ ✭❜② ❞❡❢❛✉❧t✮✱ ♠♦r❡ ❡✣❝✐❡♥t
✸✸ ✴ ✺✸
▼❛s❦❡❞ ❊✈❡♥✲▼❛♥s♦✉r ✭MEM✮
m c ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 ◦ P(Nk)
P
✸✸ ✴ ✺✸
MEM✿ ❉❡s✐❣♥ ❈♦♥s✐❞❡r❛t✐♦♥s
❙❛♠♣❧❡ ▲❋❙❘s ✭st❛t❡ s✐③❡ ❛s ✇♦r❞s ♦❢ ❜✐ts✮✿
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❲♦r❦ ❡①❝❡♣t✐♦♥❛❧❧② ✇❡❧❧ ❢♦r ❆❘❳ ♣r✐♠✐t✐✈❡s
✸✹ ✴ ✺✸
MEM✿ ❉❡s✐❣♥ ❈♦♥s✐❞❡r❛t✐♦♥s
b w n ϕ 128 8 16 (x1, . . . , x15, (x0 ≪ 1) ⊕ (x9 ≫ 1) ⊕ (x10 ≪ 1)) 128 32 4 (x1, . . . , x3, (x0 ≪ 5) ⊕ x1 ⊕ (x1 ≪ 13)) 128 64 2 (x1, (x0 ≪ 11) ⊕ x1 ⊕ (x1 ≪ 13)) 256 64 4 (x1, . . . , x3, (x0 ≪ 3) ⊕ (x3 ≫ 5)) 512 32 16 (x1, . . . , x15, (x0 ≪ 5) ⊕ (x3 ≫ 7)) 512 64 8 (x1, . . . , x7, (x0 ≪ 29) ⊕ (x1 ≪ 9)) 1024 64 16 (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13)) 1600 32 50 (x1, . . . , x49, (x0 ≪ 3) ⊕ (x23 ≫ 3)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❲♦r❦ ❡①❝❡♣t✐♦♥❛❧❧② ✇❡❧❧ ❢♦r ❆❘❳ ♣r✐♠✐t✐✈❡s
✸✹ ✴ ✺✸
MEM✿ ❉❡s✐❣♥ ❈♦♥s✐❞❡r❛t✐♦♥s
b w n ϕ 128 8 16 (x1, . . . , x15, (x0 ≪ 1) ⊕ (x9 ≫ 1) ⊕ (x10 ≪ 1)) 128 32 4 (x1, . . . , x3, (x0 ≪ 5) ⊕ x1 ⊕ (x1 ≪ 13)) 128 64 2 (x1, (x0 ≪ 11) ⊕ x1 ⊕ (x1 ≪ 13)) 256 64 4 (x1, . . . , x3, (x0 ≪ 3) ⊕ (x3 ≫ 5)) 512 32 16 (x1, . . . , x15, (x0 ≪ 5) ⊕ (x3 ≫ 7)) 512 64 8 (x1, . . . , x7, (x0 ≪ 29) ⊕ (x1 ≪ 9)) 1024 64 16 (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13)) 1600 32 50 (x1, . . . , x49, (x0 ≪ 3) ⊕ (x23 ≫ 3)) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✸✹ ✴ ✺✸
MEM✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣
ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 = ϕγ′ 2 ◦ ϕβ′ 1 ◦ ϕα′
❢♦r ❛♥② (α, β, γ) = (α′, β′, γ′)
✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹
s♦❧✈❡❞ ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪
✸✺ ✴ ✺✸
MEM✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣
ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 = ϕγ′ 2 ◦ ϕβ′ 1 ◦ ϕα′
❢♦r ❛♥② (α, β, γ) = (α′, β′, γ′)
✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹
s♦❧✈❡❞ ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪
✸✺ ✴ ✺✸
MEM✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣
ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 = ϕγ′ 2 ◦ ϕβ′ 1 ◦ ϕα′
❢♦r ❛♥② (α, β, γ) = (α′, β′, γ′)
✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹
s♦❧✈❡❞ ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪ r❡s✉❧ts ✐♠♣❧✐❝✐t❧② ✉s❡❞✱ ❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪
✸✺ ✴ ✺✸
MEM✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣
ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 = ϕγ′ 2 ◦ ϕβ′ 1 ◦ ϕα′
❢♦r ❛♥② (α, β, γ) = (α′, β′, γ′)
✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹
s♦❧✈❡❞ ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪
❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮ s♦❧✈❡❞ ❜② ●r❛♥❣❡r ❡t ❛❧✳ ❬●❏▼◆✶✻❪
✸✺ ✴ ✺✸
MEM✿ ❯♥✐q✉❡♥❡ss ♦❢ ▼❛s❦✐♥❣
ϕγ
2 ◦ ϕβ 1 ◦ ϕα 0 = ϕγ′ 2 ◦ ϕβ′ 1 ◦ ϕα′
❢♦r ❛♥② (α, β, γ) = (α′, β′, γ′)
✻✹ ✶✷✽ ✷✺✻ ✺✶✷ ✶✵✷✹
s♦❧✈❡❞ ❜② ❘♦❣❛✇❛② ❬❘♦❣✵✹❪
❡✳❣✳✱ ❜② Prøst ✭✷✵✶✹✮
✸✺ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❖PP
A0 A1 Aa–1 M0 M1 Md–1 ⊕Mi C1 C2 Cd T
ϕ0(L) ϕ0(L) ϕ1(L) ϕ1(L) ϕa–1(L) ϕa–1(L) ϕ2◦ϕ2
1◦ϕd–1(L)
ϕ2◦ϕ2
1◦ϕd–1(L)
ϕ2◦ϕ0(L) ϕ2◦ϕ1(L) ϕ2◦ϕd–1(L) ϕ2◦ϕ0(L) ϕ2◦ϕ1(L) ϕ2◦ϕd–1(L)
P P P P P P P
✸✻ ✴ ✺✸
L = P(Nk) ϕ1 = ϕ ⊕ id, ϕ2 = ϕ2 ⊕ ϕ ⊕ id
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ▼❘❖
A0 Aa–1 T0 Td–1 M0 Md–1
|A||M|
C1 Cd T
ϕ0(L) ϕ0(L) ϕa–1(L) ϕa–1(L) ϕ1◦ϕ0(L) ϕ1◦ϕ0(L) ϕ1◦ϕd–1(L) ϕ1◦ϕd–1(L) ϕ2
1(L)
ϕ2
1(L)
ϕ2(L) ϕ2(L) ϕ2(L)⊕M0 ϕ2(L)⊕Md–1
P P P P P P P
✸✼ ✴ ✺✸
L = P(Nk) ϕ1 = ϕ ⊕ id, ϕ2 = ϕ2 ⊕ ϕ ⊕ id
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣
❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✸✽ ✴ ✺✸
❳P❳
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
❙❡❝✉r✐t② ♦❢ str♦♥❣❧② ❞❡♣❡♥❞s ♦♥ ❝❤♦✐❝❡ ♦❢
✶ ✏❲❡❛❦✑
✐♥s❡❝✉r❡
✷ ✏◆♦r♠❛❧✑
s✐♥❣❧❡✲❦❡② s❡❝✉r❡
✸ ✏❙tr♦♥❣✑
r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✸✾ ✴ ✺✸
❳P❳
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
✶ ✏❲❡❛❦✑
✐♥s❡❝✉r❡
✷ ✏◆♦r♠❛❧✑
s✐♥❣❧❡✲❦❡② s❡❝✉r❡
✸ ✏❙tr♦♥❣✑
r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✸✾ ✴ ✺✸
❳P❳
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
✶ ✏❲❡❛❦✑ T
− → ✐♥s❡❝✉r❡
✷ ✏◆♦r♠❛❧✑
s✐♥❣❧❡✲❦❡② s❡❝✉r❡
✸ ✏❙tr♦♥❣✑
r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✸✾ ✴ ✺✸
❳P❳
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
✶ ✏❲❡❛❦✑ T
− → ✐♥s❡❝✉r❡
✷ ✏◆♦r♠❛❧✑ T
− → s✐♥❣❧❡✲❦❡② s❡❝✉r❡
✸ ✏❙tr♦♥❣✑
r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✸✾ ✴ ✺✸
❳P❳
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
✶ ✏❲❡❛❦✑ T
− → ✐♥s❡❝✉r❡
✷ ✏◆♦r♠❛❧✑ T
− → s✐♥❣❧❡✲❦❡② s❡❝✉r❡
✸ ✏❙tr♦♥❣✑ T
− → r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✸✾ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
m 0k ⊕ 0P(k) 0k ⊕ 0P(k)
P
(0, 0, 0, 0) ∈ T
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
m P(m) 0k ⊕ 0P(k) 0k ⊕ 0P(k)
P
(0, 0, 0, 0) ∈ T = ⇒ XPXk((0, 0, 0, 0), m) = P(m)
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
k 1k ⊕ 0P(k) 1k ⊕ 1P(k)
P
(0, 0, 0, 0) ∈ T = ⇒ XPXk((0, 0, 0, 0), m) = P(m) (1, 0, 1, 1) ∈ T = ⇒ XPXk((1, 0, 1, 1), 0) = k
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
3P(k) 1k ⊕ 0P(k) 0k ⊕ 2P(k)
P
(0, 0, 0, 0) ∈ T = ⇒ XPXk((0, 0, 0, 0), m) = P(m) (1, 0, 1, 1) ∈ T = ⇒ XPXk((1, 0, 1, 1), 0) = k (1, 0, 0, 2) ∈ T = ⇒ XPXk((1, 0, 0, 2), 0) = 3P(k)
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
3P(k) 1k ⊕ 0P(k) 0k ⊕ 2P(k)
P
(0, 0, 0, 0) ∈ T = ⇒ XPXk((0, 0, 0, 0), m) = P(m) (1, 0, 1, 1) ∈ T = ⇒ XPXk((1, 0, 1, 1), 0) = k (1, 0, 0, 2) ∈ T = ⇒ XPXk((1, 0, 0, 2), 0) = 3P(k) · · · · · · · · ·
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts ❚❡❝❤♥✐❝❛❧ ❞❡✜♥✐t✐♦♥ t♦ ❡❧✐♠✐♥❛t❡ ✇❡❛❦ ❝❛s❡s ✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
3P(k) 1k ⊕ 0P(k) 0k ⊕ 2P(k)
P
(0, 0, 0, 0) ∈ T = ⇒ XPXk((0, 0, 0, 0), m) = P(m) (1, 0, 1, 1) ∈ T = ⇒ XPXk((1, 0, 1, 1), 0) = k (1, 0, 0, 2) ∈ T = ⇒ XPXk((1, 0, 0, 2), 0) = 3P(k) · · · · · · · · ·
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts
✐♥✈❛❧✐❞ ✐♥s❡❝✉r❡ ✈❛❧✐❞ s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳✿ ❲❡❛❦ ❚✇❡❛❦s
3P(k) 1k ⊕ 0P(k) 0k ⊕ 2P(k)
P
(0, 0, 0, 0) ∈ T = ⇒ XPXk((0, 0, 0, 0), m) = P(m) (1, 0, 1, 1) ∈ T = ⇒ XPXk((1, 0, 1, 1), 0) = k (1, 0, 0, 2) ∈ T = ⇒ XPXk((1, 0, 0, 2), 0) = 3P(k) · · · · · · · · ·
✏❱❛❧✐❞✑ ❚✇❡❛❦ ❙❡ts
⇒ XPX ✐♥s❡❝✉r❡
⇒ XPX s✐♥❣❧❡✲ ♦r r❡❧❛t❡❞✲❦❡② s❡❝✉r❡
✹✵ ✴ ✺✸
❳P❳ ❈♦✈❡rs ❊✈❡♥✲▼❛♥s♦✉r
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
m c k k
P
❢♦r T = {(1, 0, 1, 0)} ❙✐♥❣❧❡✲❦❡② ❙❚P❘P s❡❝✉r❡ ✭s✉r♣r✐s❡❄✮
✱ ✐s ❛ ♥♦r♠❛❧ ❜❧♦❝❦❝✐♣❤❡r
✹✶ ✴ ✺✸
❳P❳ ❈♦✈❡rs ❊✈❡♥✲▼❛♥s♦✉r
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
m c k k
P
❢♦r T = {(1, 0, 1, 0)}
✱ ✐s ❛ ♥♦r♠❛❧ ❜❧♦❝❦❝✐♣❤❡r
✹✶ ✴ ✺✸
❳P❳ ❈♦✈❡rs ❊✈❡♥✲▼❛♥s♦✉r
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
m c k k
P
❢♦r T = {(1, 0, 1, 0)}
✹✶ ✴ ✺✸
❳P❳ ❈♦✈❡rs ❳❊❳ ❲✐t❤ ❊✈❡♥✲▼❛♥s♦✉r
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
m c (2α3β7γ ⊕ 1)k ⊕ 2α3β7γP(k)
P
❢♦r T = ( 2α3β7γ ⊕ 1 , 2α3β7γ , ( 2α3β7γ ⊕ 1 , 2α3β7γ )
❘❡❧❛t❡❞✲❦❡② ❙❚P❘P s❡❝✉r❡ ✭✐❢ ✮
✹✷ ✴ ✺✸
❳P❳ ❈♦✈❡rs ❳❊❳ ❲✐t❤ ❊✈❡♥✲▼❛♥s♦✉r
m c t11k ⊕ t12P(k) t21k ⊕ t22P(k)
P
m c (2α3β7γ ⊕ 1)k ⊕ 2α3β7γP(k)
P
❢♦r T = ( 2α3β7γ ⊕ 1 , 2α3β7γ , ( 2α3β7γ ⊕ 1 , 2α3β7γ )
✹✷ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
A1 A2 Aa−1 Aa M1 M2 Md M1⊕···⊕Md C1 C2 Cd T
33L 2·33L 2a-233L 2a-134L L 3L 2·3L 2d-13L 2d-132L 2L 22L 2dL 2d-17L
Ek Ek Ek Ek Ek Ek Ek Ek Ek Ek Ek Ek
Prøst✲❈❖P❆ ❜② ❑❛✈✉♥ ❡t ❛❧✳ ✭✷✵✶✹✮✿ ❈❖P❆ ❜❛s❡❞ ♦♥ ❳❊❳ ❜❛s❡❞ ♦♥ ❊✈❡♥✲▼❛♥s♦✉r
✹✸ ✴ ✺✸
L = EK(0)
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
A1 A2 Aa−1 Aa M1 M2 Md M1⊕···⊕Md C1 C2 Cd T
33L 2·33L 2a-233L 2a-134L L 3L 2·3L 2d-13L 2d-132L 2L 22L 2dL 2d-17L
Ek Ek Ek Ek Ek Ek Ek Ek Ek Ek Ek Ek
❈❖P❆ ❜❛s❡❞ ♦♥ ❳❊❳ ❜❛s❡❞ ♦♥ ❊✈❡♥✲▼❛♥s♦✉r
✹✸ ✴ ✺✸
L = EK(0)
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ ❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
s❦
✳ ✳ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ ❊①✐st✐♥❣ ♣r♦♦❢ ❣❡♥❡r❛❧✐③❡s ✳ ✳ ❈❖P❆
r❦
✳ ✳
r❦
✳ ✳
r❦
✳ ✳
✹✹ ✴ ✺✸
r❦
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
s❦
✳ ✳ P ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ ❊①✐st✐♥❣ ♣r♦♦❢ ❣❡♥❡r❛❧✐③❡s ✳ ✳ ❈❖P❆
r❦
✳ ✳
r❦
✳ ✳
r❦
✳ ✳
✹✹ ✴ ✺✸
r❦
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
O
2n
− − − →
s❦
✳ ✳ P ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ ❊①✐st✐♥❣ ♣r♦♦❢ ❣❡♥❡r❛❧✐③❡s ✳ ✳ ❈❖P❆
r❦
✳ ✳
r❦
✳ ✳
r❦
✳ ✳
✹✹ ✴ ✺✸
r❦
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
O
2n
− − − →
s❦
✳ ✳ P ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ ❈❖P❆
✳ ✳ ❈❖P❆
O
2n
− − − →
r❦
✳ ✳ XEX
O
2n
− − − →
r❦
✳ ✳ E
r❦
✳ ✳
✹✹ ✴ ✺✸
r❦
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
O
2n
− − − →
s❦
✳ ✳ P ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆
✳ ✳ ❈❖P❆
O
2n
− − − →
r❦
✳ ✳ XEX
O
2n
− − − →
r❦
✳ ✳ E
r❦
✳ ✳ P
✹✹ ✴ ✺✸
r❦
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
O
2n
− − − →
s❦
✳ ✳ P ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆
✳ ✳ ❈❖P❆
O
2n
− − − →
r❦
✳ ✳ XEX
O
2n
− − − →
r❦
✳ ✳ E
Ω
− − − →
r❦
✳ ✳ P
✹✹ ✴ ✺✸
r❦
❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ❈❖P❆ ❛♥❞ Prøst✲❈❖P❆
❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆ ✳ ✳ ❈❖P❆
O
2n
− − − →
s❦
✳ ✳ XEX
O
2n
− − − →
s❦
✳ ✳ E
O
2n
− − − →
s❦
✳ ✳ P ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t② ♦❢ Prøst✲❈❖P❆
✳ ✳ ❈❖P❆
O
2n
− − − →
r❦
✳ ✳ XEX
O
2n
− − − →
r❦
✳ ✳ E
Ω
− − − →
r❦
✳ ✳ P
✹✹ ✴ ✺✸
O
2n
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P k M1 M2 T P P P Md10∗ 4k 4k
Ek(m) = P(m ⊕ k) ⊕ k Ek(m) = P(m ⊕ 3k) ⊕ 2k Ek(m) = P(m ⊕ 5k) ⊕ 4k
❊q✉✐✈❛❧❡♥t t♦ ✇✐t❤ ✳ ✳ ❈❤❛s❦❡②
s❦
✳ ✳
s❦
✳ ✳
✹✺ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P k M1 M2 T P P P Md10∗ 4k 4k
Ek(m) = P(m ⊕ k) ⊕ k Ek(m) = P(m ⊕ 3k) ⊕ 2k Ek(m) = P(m ⊕ 5k) ⊕ 4k
✳ ✳ ❈❤❛s❦❡②
s❦
✳ ✳
s❦
✳ ✳
✹✺ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P k M1 M2 T P P P Md10∗ 4k 4k
Ek(m) = P(m ⊕ k) ⊕ k Ek(m) = P(m ⊕ 3k) ⊕ 2k Ek(m) = P(m ⊕ 5k) ⊕ 4k
✳ ✳ ❈❤❛s❦❡②
O
2n
− − − →
s❦
✳ ✳ XPX
O
2n
− − − →
s❦
✳ ✳ P
✹✺ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❆❞❥✉st❡❞ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P P k M1 M2 T P P P P Md10∗ 4k 4k
❇❛s❡❞ ♦♥ ✇✐t❤ ✳ ✳ ❈❤❛s❦❡②
r❦
✳ ✳
r❦
✳ ✳ ❆♣♣r♦❛❝❤ ❝❛♥ ❛❧s♦ ❜❡ ❛♣♣❧✐❡❞ t♦✿
❑❡②❡❞ ❙♣♦♥❣❡ ❛♥❞ ❉✉♣❧❡① ✶✵ ❙♣♦♥❣❡✲✐♥s♣✐r❡❞ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s
✹✻ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❆❞❥✉st❡❞ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P P k M1 M2 T P P P P Md10∗ 4k 4k
✳ ✳ ❈❤❛s❦❡②
r❦
✳ ✳
r❦
✳ ✳ ❆♣♣r♦❛❝❤ ❝❛♥ ❛❧s♦ ❜❡ ❛♣♣❧✐❡❞ t♦✿
❑❡②❡❞ ❙♣♦♥❣❡ ❛♥❞ ❉✉♣❧❡① ✶✵ ❙♣♦♥❣❡✲✐♥s♣✐r❡❞ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s
✹✻ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❆❞❥✉st❡❞ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P P k M1 M2 T P P P P Md10∗ 4k 4k
✳ ✳ ❈❤❛s❦❡②
O
2n
− − − →
r❦
✳ ✳ XPX
O
2n
− − − →
r❦
✳ ✳ P ❆♣♣r♦❛❝❤ ❝❛♥ ❛❧s♦ ❜❡ ❛♣♣❧✐❡❞ t♦✿
❑❡②❡❞ ❙♣♦♥❣❡ ❛♥❞ ❉✉♣❧❡① ✶✵ ❙♣♦♥❣❡✲✐♥s♣✐r❡❞ ❈❆❊❙❆❘ ❝❛♥❞✐❞❛t❡s
✹✻ ✴ ✺✸
❆♣♣❧✐❝❛t✐♦♥ t♦ ▼❆❈✿ ❆❞❥✉st❡❞ ❈❤❛s❦❡②
k 2k 2k M1 M2 Md T P P P P k M1 M2 T P P P P Md10∗ 4k 4k
✳ ✳ ❈❤❛s❦❡②
O
2n
− − − →
r❦
✳ ✳ XPX
O
2n
− − − →
r❦
✳ ✳ P
✹✻ ✴ ✺✸
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✹✼ ✴ ✺✸
❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs
m c
Ek/P
f1(t) f2(t)
■❢ ✐s ❧❛r❣❡ ❡♥♦✉❣❤ ♥♦ ♣r♦❜❧❡♠ ■❢ ✐s s♠❛❧❧ ✏❜❡②♦♥❞ ❜✐rt❤❞❛②✲❜♦✉♥❞✑ s♦❧✉t✐♦♥s
❈❛s❝❛❞✐♥❣ ❚✇❡❛❦✲r❡❦❡②✐♥❣
✹✽ ✴ ✺✸
❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs
m c
Ek/P
f1(t) f2(t)
→ ♥♦ ♣r♦❜❧❡♠
→ ✏❜❡②♦♥❞ ❜✐rt❤❞❛②✲❜♦✉♥❞✑ s♦❧✉t✐♦♥s
✹✽ ✴ ✺✸
❈❛s❝❛❞✐♥❣ ▲❘❲✬s
m c
· · · · · ·
h1(t) h1(t)⊕h2(t) hρ−1(t)⊕hρ(t) hρ(t)
Ek1 Ek2 Ekρ
✿ s❡❝✉r❡ ✉♣ t♦ q✉❡r✐❡s ❬▲❙❚✶✷✱Pr♦✶✹❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♦ q✉❡r✐❡s ❬▲❙✶✸❪ ❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ s❡❝✉r✐t②
✹✾ ✴ ✺✸
❈❛s❝❛❞✐♥❣ ▲❘❲✬s
m c
· · · · · ·
h1(t) h1(t)⊕h2(t) hρ−1(t)⊕hρ(t) hρ(t)
Ek1 Ek2 Ekρ
✹✾ ✴ ✺✸
❈❛s❝❛❞✐♥❣ ❚❊▼✬s
m c
· · · · · ·
h1(t) h1(t)⊕h2(t) hρ−1(t)⊕hρ(t) hρ(t)
P1 P2 Pρ
✿ s❡❝✉r❡ ✉♣ t♦ q✉❡r✐❡s ❬❈▲❙✶✺❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♦ q✉❡r✐❡s ❬❈▲❙✶✺❪ ❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ s❡❝✉r✐t②
✺✵ ✴ ✺✸
❈❛s❝❛❞✐♥❣ ❚❊▼✬s
m c
· · · · · ·
h1(t) h1(t)⊕h2(t) hρ−1(t)⊕hρ(t) hρ(t)
P1 P2 Pρ
✺✵ ✴ ✺✸
❚✇❡❛❦✲❘❡❦❡②✐♥❣
m c
E
k h ⊗ t h ⊗ t t
▼♦r❡ ♦♥ ✏❜❡②♦♥❞ ❜✐rt❤❞❛②✲❜♦✉♥❞ s❡❝✉r✐t②✑ ♦♥ ❚❤✉rs❞❛②
✺✶ ✴ ✺✸
❖✉t❧✐♥❡ ❉❡❞✐❝❛t❡❞ ❉❡s✐❣♥ ❇❛s✐❝ ●❡♥❡r✐❝ ❘❡❝✐♣❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❇❛s❡❞ ♦♥ ▼❛s❦✐♥❣ ❇❡②♦♥❞ ▼❛s❦✐♥❣✲❇❛s❡❞ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs ❈♦♥❝❧✉s✐♦♥
✺✷ ✴ ✺✸
❈♦♥❝❧✉s✐♦♥
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs✿ ❙✐♠♣❧❡ ❛♥❞ P♦✇❡r❢✉❧
❙❡❝✉r✐t② ▲❡✈❡❧ ❇✐rt❤❞❛②✲❜♦✉♥❞ s❡❝✉r✐t②✿ ♦❦❛② ✐❢ ✐s ❧❛r❣❡ ❡♥♦✉❣❤
P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡rs
❇❡②♦♥❞ ❜✐rt❤❞❛②✲❜♦✉♥❞ s❡❝✉r✐t② ♣♦ss✐❜❧❡
▼♦r❡ ♦♥ ❚❤✉rs❞❛②
❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦
✺✸ ✴ ✺✸
❈♦♥❝❧✉s✐♦♥
❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs✿ ❙✐♠♣❧❡ ❛♥❞ P♦✇❡r❢✉❧
❙❡❝✉r✐t② ▲❡✈❡❧
− → P❡r♠✉t❛t✐♦♥✲❜❛s❡❞ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡rs
− → ▼♦r❡ ♦♥ ❚❤✉rs❞❛②
❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦
✺✸ ✴ ✺✸
❙❯PP❖❘❚■◆● ❙▲■❉❊❙
✺✹ ✴ ✺✸
▼❊▼✿ ■♠♣❧❡♠❡♥t❛t✐♦♥
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
▼❛✐♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ r❡s✉❧ts✿
♥♦♥❝❡✲r❡s♣❡❝t✐♥❣ ♠✐s✉s❡✲r❡s✐st❛♥t P❧❛t❢♦r♠ ❆❊❙✲●❈▼ ❖❈❇✸ ❉❡♦①②s ❖PP ❖PP
❉❡♦①②s ▼❘❖ ▼❘❖ ❈♦rt❡①✲❆✽ ✸✽✳✻ ✷✽✳✾ ✲ ✹✳✷✻ ✺✳✾✶ ✲ ✲ ✽✳✵✼ ✶✶✳✸✷ ❙❛♥❞② ❇r✐❞❣❡ ✷✳✺✺ ✵✳✾✽ ✶✳✷✾ ✶✳✷✹ ✶✳✾✶ ✲ ✷✳✺✽ ✷✳✹✶ ✸✳✺✽ ❍❛s✇❡❧❧ ✶✳✵✸ ✵✳✻✾ ✵✳✾✻ ✵✳✺✺ ✵✳✼✺ ✶✳✶✼ ✶✳✾✷ ✶✳✵✻ ✶✳✸✾
✺✺ ✴ ✺✸
▼❊▼✿ ■♠♣❧❡♠❡♥t❛t✐♦♥
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
♥♦♥❝❡✲r❡s♣❡❝t✐♥❣ ♠✐s✉s❡✲r❡s✐st❛♥t P❧❛t❢♦r♠ ❆❊❙✲●❈▼ ❖❈❇✸ ❉❡♦①②s= ❖PP4 ❖PP6
❉❡♦①②s ▼❘❖ ▼❘❖ ❈♦rt❡①✲❆✽ ✸✽✳✻ ✷✽✳✾ ✲ ✹✳✷✻ ✺✳✾✶ ✲ ✲ ✽✳✵✼ ✶✶✳✸✷ ❙❛♥❞② ❇r✐❞❣❡ ✷✳✺✺ ✵✳✾✽ ✶✳✷✾ ✶✳✷✹ ✶✳✾✶ ✲ ✷✳✺✽ ✷✳✹✶ ✸✳✺✽ ❍❛s✇❡❧❧ ✶✳✵✸ ✵✳✻✾ ✵✳✾✻ ✵✳✺✺ ✵✳✼✺ ✶✳✶✼ ✶✳✾✷ ✶✳✵✻ ✶✳✸✾
✺✺ ✴ ✺✸
▼❊▼✿ ■♠♣❧❡♠❡♥t❛t✐♦♥
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
♥♦♥❝❡✲r❡s♣❡❝t✐♥❣ ♠✐s✉s❡✲r❡s✐st❛♥t P❧❛t❢♦r♠ ❆❊❙✲●❈▼ ❖❈❇✸ ❉❡♦①②s= ❖PP4 ❖PP6
❉❡♦①②s= ▼❘❖4 ▼❘❖6 ❈♦rt❡①✲❆✽ ✸✽✳✻ ✷✽✳✾ ✲ ✹✳✷✻ ✺✳✾✶ ✲ ✲ ✽✳✵✼ ✶✶✳✸✷ ❙❛♥❞② ❇r✐❞❣❡ ✷✳✺✺ ✵✳✾✽ ✶✳✷✾ ✶✳✷✹ ✶✳✾✶ ✲ ≈ ✷✳✺✽ ✷✳✹✶ ✸✳✺✽ ❍❛s✇❡❧❧ ✶✳✵✸ ✵✳✻✾ ✵✳✾✻ ✵✳✺✺ ✵✳✼✺ ✶✳✶✼ ≈ ✶✳✾✷ ✶✳✵✻ ✶✳✸✾
✺✺ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13)) ❇❡❣✐♥ ✇✐t❤ st❛t❡ ♦❢ ✲❜✐t ✇♦r❞s P❛r❛❧❧❡❧✐③❛❜❧❡ ✭❆❱❳✷✮ ❛♥❞ ✇♦r❞✲s❧✐❝❡❛❜❧❡
✺✻ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 P❛r❛❧❧❡❧✐③❛❜❧❡ ✭❆❱❳✷✮ ❛♥❞ ✇♦r❞✲s❧✐❝❡❛❜❧❡
✺✻ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16
P❛r❛❧❧❡❧✐③❛❜❧❡ ✭❆❱❳✷✮ ❛♥❞ ✇♦r❞✲s❧✐❝❡❛❜❧❡
✺✻ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17
P❛r❛❧❧❡❧✐③❛❜❧❡ ✭❆❱❳✷✮ ❛♥❞ ✇♦r❞✲s❧✐❝❡❛❜❧❡
✺✻ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18
P❛r❛❧❧❡❧✐③❛❜❧❡ ✭❆❱❳✷✮ ❛♥❞ ✇♦r❞✲s❧✐❝❡❛❜❧❡
✺✻ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19
P❛r❛❧❧❡❧✐③❛❜❧❡ ✭❆❱❳✷✮ ❛♥❞ ✇♦r❞✲s❧✐❝❡❛❜❧❡
✺✻ ✴ ✺✸
▼❊▼✿ P❛r❛❧❧❡❧✐③❛❜✐❧✐t②
ϕ(x0, . . . , x15) = (x1, . . . , x15, (x0 ≪ 53) ⊕ (x5 ≪ 13))
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19
✺✻ ✴ ✺✸
❳P❳✿ ❙✐♥❣❧❡✲❑❡② ❙❡❝✉r✐t②
✭❙tr♦♥❣✮ ❚✇❡❛❦❛❜❧❡ P❘P
IC
XPX(±)
k
P ±
P ±
distinguisher D
π ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ♣❡r♠✉t❛t✐♦♥
T ✐s ✈❛❧✐❞ = ⇒ XPX ✐s ✭❙✮❚P❘P ✉♣ t♦ O q2 + qr 2n
❳P❳✿ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t②
❘❡❧❛t❡❞✲❑❡② ✭❙tr♦♥❣✮ ❚✇❡❛❦❛❜❧❡ P❘P
IC
XPX(±)
ϕ(k)
P ±
P ±
distinguisher D
rkπ ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ r❡❧❛t❡❞✲❦❡② ♣❡r♠✉t❛t✐♦♥
✺✽ ✴ ✺✸
❳P❳✿ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t②
❑❡②✲❉❡r✐✈✐♥❣ ❋✉♥❝t✐♦♥s
✿ ❛❧❧ ❢✉♥❝t✐♦♥s ♦r ◆♦t❡✿ ♠❛s❦✐♥❣s ✐♥ ❛r❡ ❘❡s✉❧ts
✐❢ ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♦r ❛❧❧ t✇❡❛❦s✿ s❡❝✉r✐t② ❚P❘P ❛♥❞ ❙❚P❘P ❚P❘P ❙❚P❘P
✺✾ ✴ ✺✸
❳P❳✿ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t②
❑❡②✲❉❡r✐✈✐♥❣ ❋✉♥❝t✐♦♥s
◆♦t❡✿ ♠❛s❦✐♥❣s ✐♥ ❛r❡ ❘❡s✉❧ts
✐❢ ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♦r ❛❧❧ t✇❡❛❦s✿ s❡❝✉r✐t② ❚P❘P ❛♥❞ ❙❚P❘P ❚P❘P ❙❚P❘P
✺✾ ✴ ✺✸
❳P❳✿ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t②
❑❡②✲❉❡r✐✈✐♥❣ ❋✉♥❝t✐♦♥s
❘❡s✉❧ts
✐❢ ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♦r ❛❧❧ t✇❡❛❦s✿ s❡❝✉r✐t② ❚P❘P ❛♥❞ ❙❚P❘P ❚P❘P ❙❚P❘P
✺✾ ✴ ✺✸
❳P❳✿ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t②
❑❡②✲❉❡r✐✈✐♥❣ ❋✉♥❝t✐♦♥s
❘❡s✉❧ts
✐❢ T ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♦r ❛❧❧ t✇❡❛❦s✿ s❡❝✉r✐t② Φ t12 = 0 ❚P❘P Φ⊕ t12, t22 = 0 ❛♥❞ (t21, t22) = (0, 1) ❙❚P❘P Φ⊕ ❚P❘P ❙❚P❘P
✺✾ ✴ ✺✸
❳P❳✿ ❘❡❧❛t❡❞✲❑❡② ❙❡❝✉r✐t②
❑❡②✲❉❡r✐✈✐♥❣ ❋✉♥❝t✐♦♥s
❘❡s✉❧ts
✐❢ T ✐s ✈❛❧✐❞✱ ❛♥❞ ❢♦r ❛❧❧ t✇❡❛❦s✿ s❡❝✉r✐t② Φ t12 = 0 ❚P❘P Φ⊕ t12, t22 = 0 ❛♥❞ (t21, t22) = (0, 1) ❙❚P❘P Φ⊕ t11, t12 = 0 ❚P❘P ΦP ⊕ t11, t12, t21, t22 = 0 ❙❚P❘P ΦP ⊕
✺✾ ✴ ✺✸
❳P❳✿ ❙❡❝✉r✐t② Pr♦♦❢ ❚❡❝❤♥✐q✉❡s
P❛t❛r✐♥✬s ❍✲❝♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡
✲
❜❛❞ tr❛♥s❝r✐♣t ❢♦r ♣r♦❜✳ r❛t✐♦ ❢♦r ❣♦♦❞ tr❛♥s❝r✐♣ts ❚r❛❞❡✲♦✛✿ ❞❡✜♥❡ ❜❛❞ tr❛♥s❝r✐♣ts s♠❛rt❧②✦
✻✵ ✴ ✺✸
❳P❳✿ ❙❡❝✉r✐t② Pr♦♦❢ ❚❡❝❤♥✐q✉❡s
P❛t❛r✐♥✬s ❍✲❝♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡
Advrk✲(s)prp
XPX
(D) ≤ ε + Pr
rkπ, P)
❚r❛❞❡✲♦✛✿ ❞❡✜♥❡ ❜❛❞ tr❛♥s❝r✐♣ts s♠❛rt❧②✦
✻✵ ✴ ✺✸
❳P❳✿ ❙❡❝✉r✐t② Pr♦♦❢ ❚❡❝❤♥✐q✉❡s
P❛t❛r✐♥✬s ❍✲❝♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡
Advrk✲(s)prp
XPX
(D) ≤ ε + Pr
rkπ, P)
✻✵ ✴ ✺✸
❳P❳✿ ❙❡❝✉r✐t② Pr♦♦❢ ❚❡❝❤♥✐q✉❡s
❇❡❢♦r❡ t❤❡ ■♥t❡r❛❝t✐♦♥
❆❢t❡r t❤❡ ■♥t❡r❛❝t✐♦♥
❇♦✉♥❞✐♥❣ t❤❡ ❆❞✈❛♥t❛❣❡
✻✶ ✴ ✺✸
❳P❳✿ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ▼✐♥❛❧♣❤❡r
A1 A2 Aa−1 Aa M1 M2 Md−1 Md C1 C2 Cd−1 Cd T
2L′ 2L′ 22L′ 22L′ 2a-1L′ 2a-1L′ 2a-13L′ 2a-13L′ 2L 2L 23L 23L 22d-3L 22d-3L 22d-1L 22d-1L 22L 22L 24L 24L 22d-2L 22d-2L 22d-13L 22d-13L
P P P P P P P P P P P P
❇❛s❡❞ ♦♥ ✇✐t❤ ✳ ✳ ▼✐♥❛❧♣❤✳
r❦
✳ ✳
r❦
✳ ✳
✻✷ ✴ ✺✸
L′ = kflag0 ⊕ P(kflag0) L = kflagN ⊕ P(kflagN)
❳P❳✿ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ▼✐♥❛❧♣❤❡r
A1 A2 Aa−1 Aa M1 M2 Md−1 Md C1 C2 Cd−1 Cd T
2L′ 2L′ 22L′ 22L′ 2a-1L′ 2a-1L′ 2a-13L′ 2a-13L′ 2L 2L 23L 23L 22d-3L 22d-3L 22d-1L 22d-1L 22L 22L 24L 24L 22d-2L 22d-2L 22d-13L 22d-13L
P P P P P P P P P P P P
✳ ✳ ▼✐♥❛❧♣❤✳
r❦
✳ ✳
r❦
✳ ✳
✻✷ ✴ ✺✸
L′ = kflag0 ⊕ P(kflag0) L = kflagN ⊕ P(kflagN)
❳P❳✿ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❆❊✿ ▼✐♥❛❧♣❤❡r
A1 A2 Aa−1 Aa M1 M2 Md−1 Md C1 C2 Cd−1 Cd T
2L′ 2L′ 22L′ 22L′ 2a-1L′ 2a-1L′ 2a-13L′ 2a-13L′ 2L 2L 23L 23L 22d-3L 22d-3L 22d-1L 22d-1L 22L 22L 24L 24L 22d-2L 22d-2L 22d-13L 22d-13L
P P P P P P P P P P P P
✳ ✳ ▼✐♥❛❧♣❤✳
O
2n
− − − →
r❦
✳ ✳ XPX
O
2n
− − − →
r❦
✳ ✳ P
✻✷ ✴ ✺✸
L′ = kflag0 ⊕ P(kflag0) L = kflagN ⊕ P(kflagN)