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Nov 06, 2023 399 likes •982 views

t r rs rt st tr rt

slide-1
SLIDE 1

❖♣t✐♠❛❧❧② ❙❡❝✉r❡ ❚✇❡❛❦❛❜❧❡ ❇❧♦❝❦❝✐♣❤❡rs

❇❛rt ▼❡♥♥✐♥❦ ❑❯ ▲❡✉✈❡♥ ✭❇❡❧❣✐✉♠✮

❋❛st ❙♦❢t✇❛r❡ ❊♥❝r②♣t✐♦♥ ▼❛r❝❤ ✶✵✱ ✷✵✶✺

✶ ✴ ✷✵

slide-2
SLIDE 2

■♥tr♦❞✉❝t✐♦♥

m c

E

k

❚✇❡❛❦✿ ✢❡①✐❜✐❧✐t② t♦ t❤❡ ❝✐♣❤❡r ❊❛❝❤ t✇❡❛❦ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥ ✲ ❉❡❞✐❝❛t❡❞ ❝♦♥str✉❝t✐♦♥s✿

❍❛st② P✉❞❞✐♥❣ ❈✐♣❤❡r ❬❙❝❤✾✽❪ ▼❡r❝② ❬❈r♦✵✶❪ ❚❤r❡❡✜s❤ ❬❋▲❙✰✵✼❪

✷ ✴ ✷✵

slide-3
SLIDE 3

■♥tr♦❞✉❝t✐♦♥

m t c k

  • E
  • ❚✇❡❛❦✿ ✢❡①✐❜✐❧✐t② t♦ t❤❡ ❝✐♣❤❡r
  • ❊❛❝❤ t✇❡❛❦ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥

✲ ❉❡❞✐❝❛t❡❞ ❝♦♥str✉❝t✐♦♥s✿

❍❛st② P✉❞❞✐♥❣ ❈✐♣❤❡r ❬❙❝❤✾✽❪ ▼❡r❝② ❬❈r♦✵✶❪ ❚❤r❡❡✜s❤ ❬❋▲❙✰✵✼❪

✷ ✴ ✷✵

slide-4
SLIDE 4

■♥tr♦❞✉❝t✐♦♥

m t c k

  • E
  • ❚✇❡❛❦✿ ✢❡①✐❜✐❧✐t② t♦ t❤❡ ❝✐♣❤❡r
  • ❊❛❝❤ t✇❡❛❦ ❣✐✈❡s ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥

✲

  • ❉❡❞✐❝❛t❡❞ ❝♦♥str✉❝t✐♦♥s✿
  • ❍❛st② P✉❞❞✐♥❣ ❈✐♣❤❡r ❬❙❝❤✾✽❪
  • ▼❡r❝② ❬❈r♦✵✶❪
  • ❚❤r❡❡✜s❤ ❬❋▲❙✰✵✼❪

✷ ✴ ✷✵

slide-5
SLIDE 5

■♥tr♦❞✉❝t✐♦♥✿ ▼♦❞✉❧❛r ❉❡s✐❣♥s

  • LRW1 ❛♥❞ LRW2 ❜② ▲✐s❦♦✈ ❡t ❛❧✳ ❬▲❘❲✵✷❪✿

m c

E E

t k k m c

E

h(t) h(t) k

  • h ✐s ❳❖❘✲✉♥✐✈❡rs❛❧ ❤❛s❤
  • ❘❡❧❛t❡❞✿ XEX
  • ❙❡❝✉r❡ ✉♣ t♦ 2n/2 q✉❡r✐❡s

✸ ✴ ✷✵

slide-6
SLIDE 6

■♥tr♦❞✉❝t✐♦♥✿ ▼♦❞✉❧❛r ❉❡s✐❣♥s

m c

· · · · · ·

h1(t) h1(t)⊕h2(t) hρ−1(t)⊕hρ(t) hρ(t)

E E E

k1 k2 kρ

  • LRW2[ρ]✿ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ρ LRW2✬s
  • k1, . . . , kρ ❛♥❞ h1, . . . , hρ ✐♥❞❡♣❡♥❞❡♥t

✲ ✿ s❡❝✉r❡ ✉♣ t♦ q✉❡r✐❡s ❬▲❙❚✶✷✱Pr♦✶✹❪ ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♦ q✉❡r✐❡s ❬▲❙✶✸❪ ❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ s❡❝✉r✐t②

✹ ✴ ✷✵

slide-7
SLIDE 7

■♥tr♦❞✉❝t✐♦♥✿ ▼♦❞✉❧❛r ❉❡s✐❣♥s

m c

· · · · · ·

h1(t) h1(t)⊕h2(t) hρ−1(t)⊕hρ(t) hρ(t)

E E E

k1 k2 kρ

  • LRW2[ρ]✿ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ρ LRW2✬s
  • k1, . . . , kρ ❛♥❞ h1, . . . , hρ ✐♥❞❡♣❡♥❞❡♥t

✲

  • ρ = 2✿ s❡❝✉r❡ ✉♣ t♦ 22n/3 q✉❡r✐❡s ❬▲❙❚✶✷✱Pr♦✶✹❪
  • ρ ≥ 2 ❡✈❡♥✿ s❡❝✉r❡ ✉♣ t♦ 2ρn/(ρ+2) q✉❡r✐❡s ❬▲❙✶✸❪
  • ❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ 2ρn/(ρ+1) s❡❝✉r✐t②

✹ ✴ ✷✵

slide-8
SLIDE 8

■♥tr♦❞✉❝t✐♦♥✿ ❙t❛t❡ ♦❢ t❤❡ ❆rt

s❝❤❡♠❡ s❡❝✉r✐t② ✭log2✮ ❦❡② ❧❡♥❣t❤ ❝♦st E ⊗/h LRW1 n/2 n ✷ ✵ LRW2 n/2 2n ✶ ✶ XEX n/2 n ✷ ✵ LRW2[2] 2n/3 4n ✷ ✷ LRW2[ρ] ρn/(ρ+2) 2ρn ρ ρ max{n/2, n−|t|}

❖♣t✐♠❛❧ 2n s❡❝✉r✐t② ♦♥❧② ✐❢ ❦❡② ❧❡♥❣t❤ ❛♥❞ ❝♦st → ∞❄

✺ ✴ ✷✵

slide-9
SLIDE 9

■♥tr♦❞✉❝t✐♦♥✿ ❚✇❡❛❦✲❉❡♣❡♥❞❡♥t ❑❡②s

✳ ✳ ❊✣❝✐❡♥❝② t✇❡❛❦ s❝❤❡❞✉❧❡ ❧✐❣❤t❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡ ✳ ✳ ❙❡❝✉r✐t② t✇❡❛❦ s❝❤❡❞✉❧❡ str♦♥❣❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡

❚✇❡❛❦ ❛♥❞ ❦❡② ❝❤❛♥❣❡ ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧❧② ❡①♣❡♥s✐✈❡

❚❲❊❆❑❊❨ ❬❏◆P✶✹❪ ❦❡② s❝❤❡❞✉❧✐♥❣ ❜❧❡♥❞s ❦❡② ❛♥❞ t✇❡❛❦

✻ ✴ ✷✵

slide-10
SLIDE 10

■♥tr♦❞✉❝t✐♦♥✿ ❚✇❡❛❦✲❉❡♣❡♥❞❡♥t ❑❡②s

✳ ✳ ❊✣❝✐❡♥❝② t✇❡❛❦ s❝❤❡❞✉❧❡ ❧✐❣❤t❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡ ✳ ✳ ❙❡❝✉r✐t② t✇❡❛❦ s❝❤❡❞✉❧❡ str♦♥❣❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡

❚✇❡❛❦ ❛♥❞ ❦❡② ❝❤❛♥❣❡ ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧❧② ❡①♣❡♥s✐✈❡

❚❲❊❆❑❊❨ ❬❏◆P✶✹❪ ❦❡② s❝❤❡❞✉❧✐♥❣ ❜❧❡♥❞s ❦❡② ❛♥❞ t✇❡❛❦

✻ ✴ ✷✵

slide-11
SLIDE 11

■♥tr♦❞✉❝t✐♦♥✿ ❚✇❡❛❦✲❉❡♣❡♥❞❡♥t ❑❡②s

✳ ✳ ❊✣❝✐❡♥❝② t✇❡❛❦ s❝❤❡❞✉❧❡ ❧✐❣❤t❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡ ✳ ✳ ❙❡❝✉r✐t② t✇❡❛❦ s❝❤❡❞✉❧❡ str♦♥❣❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡

❚✇❡❛❦ ❛♥❞ ❦❡② ❝❤❛♥❣❡ ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧❧② ❡①♣❡♥s✐✈❡

❚❲❊❆❑❊❨ ❬❏◆P✶✹❪ ❦❡② s❝❤❡❞✉❧✐♥❣ ❜❧❡♥❞s ❦❡② ❛♥❞ t✇❡❛❦

✻ ✴ ✷✵

                                              

slide-12
SLIDE 12

■♥tr♦❞✉❝t✐♦♥✿ ❚✇❡❛❦✲❉❡♣❡♥❞❡♥t ❑❡②s

✳ ✳ ❊✣❝✐❡♥❝② t✇❡❛❦ s❝❤❡❞✉❧❡ ❧✐❣❤t❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡ ✳ ✳ ❙❡❝✉r✐t② t✇❡❛❦ s❝❤❡❞✉❧❡ str♦♥❣❡r t❤❛♥ ❦❡② s❝❤❡❞✉❧❡

❚✇❡❛❦ ❛♥❞ ❦❡② ❝❤❛♥❣❡ ❛♣♣r♦①✐♠❛t❡❧② ❡q✉❛❧❧② ❡①♣❡♥s✐✈❡

  • ❚❲❊❆❑❊❨ ❬❏◆P✶✹❪ ❦❡② s❝❤❡❞✉❧✐♥❣ ❜❧❡♥❞s ❦❡② ❛♥❞ t✇❡❛❦

✻ ✴ ✷✵

                                              

slide-13
SLIDE 13

■♥tr♦❞✉❝t✐♦♥✿ ❚✇❡❛❦✲❉❡♣❡♥❞❡♥t ❑❡②s

  • ▼✐♥❡♠❛ts✉ ❬▼✐♥✵✾❪✿

m c

E E

k t0n−|t|

  • ❙❡❝✉r❡ ✉♣ t♦ max{2n/2, 2n−|t|} q✉❡r✐❡s
  • ❇❡②♦♥❞ ❜✐rt❤❞❛② ❜♦✉♥❞ ❢♦r |t| < n/2

✼ ✴ ✷✵

slide-14
SLIDE 14

■♥tr♦❞✉❝t✐♦♥✿ ❙t❛t❡ ♦❢ t❤❡ ❆rt

s❝❤❡♠❡ s❡❝✉r✐t② ✭log2✮ ❦❡② ❧❡♥❣t❤ ❝♦st E ⊗/h t❞❦ LRW1 n/2 n ✷ ✵ ✵ LRW2 n/2 2n ✶ ✶ ✵ XEX n/2 n ✷ ✵ ✵ LRW2[2] 2n/3 4n ✷ ✷ ✵ LRW2[ρ] ρn/(ρ+2) 2ρn ρ ρ ✵ Min max{n/2, n−|t|} n ✷ ✵ ✶

✽ ✴ ✷✵

slide-15
SLIDE 15

❖✉r ●♦❛❧

  • ✐✈❡♥ ❛ ❜❧♦❝❦❝✐♣❤❡r E✱

❝♦♥str✉❝t ♦♣t✐♠❛❧❧② s❡❝✉r❡ t✇❡❛❦❛❜❧❡ ❜❧♦❝❦❝✐♣❤❡r E m t c

E

  • E

k

✾ ✴ ✷✵

  • ❛❧❧ ✇✐r❡s

❝❛rr② n ❜✐ts

slide-16
SLIDE 16
  • ❡♥❡r✐❝ ❉❡s✐❣♥

m m m m c

E E E A1 A2 A3 A4 B1 B2 B3

k, t k, t k, t, y1 k, t, y1, y2 l1 x1 y1 l2 x2 y2 l3 x3 y3

  • E[ρ] ✭❢♦r ρ ≥ 1✮

▼✐①✐♥❣ ❢✉♥❝t✐♦♥s

s❤♦✉❧❞ ❜❡ s✉❝❤ t❤❛t ✐s ✐♥✈❡rt✐❜❧❡ ❜✉t ❝❛♥ ❜❡ ❛♥②t❤✐♥❣ ♦t❤❡r✇✐s❡

✶✵ ✴ ✷✵

slide-17
SLIDE 17
  • ❡♥❡r✐❝ ❉❡s✐❣♥

m m m m c

E E E A1 A2 A3 A4 B1 B2 B3

k, t k, t k, t, y1 k, t, y1, y2 l1 x1 y1 l2 x2 y2 l3 x3 y3

  • E[ρ] ✭❢♦r ρ ≥ 1✮
  • ▼✐①✐♥❣ ❢✉♥❝t✐♦♥s Ai, Bi
  • s❤♦✉❧❞ ❜❡ s✉❝❤ t❤❛t

E[ρ] ✐s ✐♥✈❡rt✐❜❧❡

  • ❜✉t ❝❛♥ ❜❡ ❛♥②t❤✐♥❣ ♦t❤❡r✇✐s❡

✶✵ ✴ ✷✵

slide-18
SLIDE 18

❙❡❝✉r✐t② ▼♦❞❡❧

IC

  • E[ρ]±

k

E±

  • π±

E±

distinguisher D

  • ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

π ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r

  • E ✐❞❡❛❧ ❝✐♣❤❡r

❈♦♠♣❧❡①✐t②✲t❤❡♦r❡t✐❝ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②❄

✶✶ ✴ ✷✵

slide-19
SLIDE 19

❙❡❝✉r✐t② ▼♦❞❡❧

IC

  • E[ρ]±

k

E±

  • π±

E±

distinguisher D

  • ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②

π ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r

  • E ✐❞❡❛❧ ❝✐♣❤❡r
  • ❈♦♠♣❧❡①✐t②✲t❤❡♦r❡t✐❝ ✐♥❞✐st✐♥❣✉✐s❤❛❜✐❧✐t②❄

✶✶ ✴ ✷✵

slide-20
SLIDE 20

❖♥❡ E✲❈❛❧❧ ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣

m m c

E A1 A2 B1

k, t l1 x1 y1

❚❤❡♦r❡♠ ■❢ ❛r❡ ❧✐♥❡❛r✱ ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✐♥ ❛t ♠♦st ❛❜♦✉t q✉❡r✐❡s Pr♦♦❢ ✐❞❡❛ ❘❡❧❛t✐♦♥ ❛♠♦♥❣ q✉❡r✐❡s t♦ ❄ ❈❛s❡ ❞✐st✐♥❝t✐♦♥ ❜❛s❡❞ ♦♥ ❤♦✇ ❛r❡ ♣r♦❝❡ss❡❞

✶✷ ✴ ✷✵

slide-21
SLIDE 21

❖♥❡ E✲❈❛❧❧ ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣

m m c

E A1 A2 B1

k, t l1 x1 y1

❚❤❡♦r❡♠

  • ■❢ A1, B1, A2 ❛r❡ ❧✐♥❡❛r✱

E[1] ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠

  • π ✐♥ ❛t ♠♦st ❛❜♦✉t 2n/2 q✉❡r✐❡s

Pr♦♦❢ ✐❞❡❛ ❘❡❧❛t✐♦♥ ❛♠♦♥❣ q✉❡r✐❡s t♦ ❄ ❈❛s❡ ❞✐st✐♥❝t✐♦♥ ❜❛s❡❞ ♦♥ ❤♦✇ ❛r❡ ♣r♦❝❡ss❡❞

✶✷ ✴ ✷✵

slide-22
SLIDE 22

❖♥❡ E✲❈❛❧❧ ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣

m m c

E A1 A2 B1

k, t l1 x1 y1

❚❤❡♦r❡♠

  • ■❢ A1, B1, A2 ❛r❡ ❧✐♥❡❛r✱

E[1] ❝❛♥ ❜❡ ❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠

  • π ✐♥ ❛t ♠♦st ❛❜♦✉t 2n/2 q✉❡r✐❡s

Pr♦♦❢ ✐❞❡❛

  • ❘❡❧❛t✐♦♥ ❛♠♦♥❣ q✉❡r✐❡s t♦

E[1]❄

  • ❈❛s❡ ❞✐st✐♥❝t✐♦♥ ❜❛s❡❞ ♦♥ ❤♦✇ k, t, m ❛r❡ ♣r♦❝❡ss❡❞

✶✷ ✴ ✷✵

slide-23
SLIDE 23

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣

m c k t z

E

  • F[1](k, t, m) = c

■❞❡❛

  • ❙✉❜❦❡② k ⊕ t
  • ▼❛s❦✐♥❣ k ⊗ t

❙❡❝✉r✐t② ❯♣ t♦ q✉❡r✐❡s ❈♦st ❖♥❡ ✲❝❛❧❧ ❖♥❡ ✲❡✈❛❧✉❛t✐♦♥ ❖♥❡ r❡✲❦❡②

✶✸ ✴ ✷✵

slide-24
SLIDE 24

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣

m c k t z

E

  • F[1](k, t, m) = c

■❞❡❛

  • ❙✉❜❦❡② k ⊕ t
  • ▼❛s❦✐♥❣ k ⊗ t

❙❡❝✉r✐t②

  • ❯♣ t♦ 22n/3 q✉❡r✐❡s

❈♦st ❖♥❡ ✲❝❛❧❧ ❖♥❡ ✲❡✈❛❧✉❛t✐♦♥ ❖♥❡ r❡✲❦❡②

✶✸ ✴ ✷✵

slide-25
SLIDE 25

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣

m c k t z

E

  • F[1](k, t, m) = c

■❞❡❛

  • ❙✉❜❦❡② k ⊕ t
  • ▼❛s❦✐♥❣ k ⊗ t

❙❡❝✉r✐t②

  • ❯♣ t♦ 22n/3 q✉❡r✐❡s

❈♦st

  • ❖♥❡ E✲❝❛❧❧
  • ❖♥❡ ⊗✲❡✈❛❧✉❛t✐♦♥
  • ❖♥❡ r❡✲❦❡②

✶✸ ✴ ✷✵

slide-26
SLIDE 26

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

  • ❑❡② k ✐s s❡❝r❡t

❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② ▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② ❛♥❞ ❛♥❞ ♦r ♦r ❛♥❞ ❛♥❞

✶✹ ✴ ✷✵

slide-27
SLIDE 27

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)

▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② ❛♥❞ ❛♥❞ ♦r ♦r ❛♥❞ ❛♥❞

✶✹ ✴ ✷✵

slide-28
SLIDE 28

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t m ⊕ k ⊗ t c ⊕ k ⊗ t k ⊕ t

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)

▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② ❛♥❞ ❛♥❞ ♦r ♦r ❛♥❞ ❛♥❞

✶✹ ✴ ✷✵

slide-29
SLIDE 29

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t m ⊕ k ⊗ t c ⊕ k ⊗ t k ⊕ t x y l

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)
  • ▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y)

❛♥❞ ❛♥❞ ♦r ♦r ❛♥❞ ❛♥❞

✶✹ ✴ ✷✵

slide-30
SLIDE 30

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t m ⊕ k ⊗ t c ⊕ k ⊗ t k ⊕ t x y l

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)
  • ▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y)

k ⊕ t = l ❛♥❞ m ⊕ k ⊗ t = x ❛♥❞ ♦r ♦r ❛♥❞ ❛♥❞

✶✹ ✴ ✷✵

slide-31
SLIDE 31

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t m ⊕ k ⊗ t c ⊕ k ⊗ t k ⊕ t x y l

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)
  • ▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y)

k ⊕ t = l ❛♥❞ m ⊕ k ⊗ t = x ❛♥❞ ♦r ♦r k ⊕ t = l ❛♥❞ c ⊕ k ⊗ t = y ❛♥❞

✶✹ ✴ ✷✵

slide-32
SLIDE 32

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t m ⊕ k ⊗ t c ⊕ k ⊗ t k ⊕ t x y l

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)
  • ▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y)

k ⊕ t = l ❛♥❞ m ⊕ k ⊗ t = x ⇐ ⇒ k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x ♦r ♦r k ⊕ t = l ❛♥❞ c ⊕ k ⊗ t = y ⇐ ⇒ k = l ⊕ t ❛♥❞ c ⊕ (l ⊕ t) ⊗ t = y

✶✹ ✴ ✷✵

slide-33
SLIDE 33

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E

m c t m ⊕ k ⊗ t c ⊕ k ⊗ t k ⊕ t x y l

  • ❑❡② k ✐s s❡❝r❡t
  • ❈♦♥s✐❞❡r ❛♥② ❝♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c)
  • ▼❛② ✏❤✐t✑ ❛♥② ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y)

k ⊕ t = l ❛♥❞ m ⊕ k ⊗ t = x ⇐ ⇒ k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x ♦r ♦r k ⊕ t = l ❛♥❞ c ⊕ k ⊗ t = y ⇐ ⇒ k = l ⊕ t ❛♥❞ c ⊕ (l ⊕ t) ⊗ t = y

✶✹ ✴ ✷✵

slide-34
SLIDE 34

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x

❙③❡♠❡ré❞✐✲❚r♦tt❡r t❤❡♦r❡♠ ❬❙❚✽✸❪

❈♦♥s✐❞❡r ❛ ✜♥✐t❡ ✜❡❧❞ ✳ ▲❡t ❜❡ ❛ s❡t ♦❢ ❧✐♥❡s ❜❡ ❛ s❡t ♦❢ ♣♦✐♥ts ★ ♣♦✐♥t✲❧✐♥❡ ✐♥❝✐❞❡♥❝❡s

❈♦♥str✉❝t✐♦♥ q✉❡r✐❡s ❂ ❧✐♥❡s Pr✐♠✐t✐✈❡ q✉❡r✐❡s ❂ ♣♦✐♥ts ❆❜♦✉t s♦❧✉t✐♦♥s t♦ ❊✈❡r② s♦❧✉t✐♦♥ ✜①❡s ♦♥❡ ✐s r❛♥❞♦♠ ✲❜✐t ❦❡②

✶✺ ✴ ✷✵

slide-35
SLIDE 35

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x

❙③❡♠❡ré❞✐✲❚r♦tt❡r t❤❡♦r❡♠ ❬❙❚✽✸❪

❈♦♥s✐❞❡r ❛ ✜♥✐t❡ ✜❡❧❞ F✳ ▲❡t

  • L ⊆ F2 ❜❡ ❛ s❡t ♦❢ ❧✐♥❡s
  • P ⊆ F2 ❜❡ ❛ s❡t ♦❢ ♣♦✐♥ts

★ ♣♦✐♥t✲❧✐♥❡ ✐♥❝✐❞❡♥❝❡s ≤ min{|L|1/2|P|+|L|, |L||P|1/2 +|P|}

❈♦♥str✉❝t✐♦♥ q✉❡r✐❡s ❂ ❧✐♥❡s Pr✐♠✐t✐✈❡ q✉❡r✐❡s ❂ ♣♦✐♥ts ❆❜♦✉t s♦❧✉t✐♦♥s t♦ ❊✈❡r② s♦❧✉t✐♦♥ ✜①❡s ♦♥❡ ✐s r❛♥❞♦♠ ✲❜✐t ❦❡②

✶✺ ✴ ✷✵

slide-36
SLIDE 36

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x

❙③❡♠❡ré❞✐✲❚r♦tt❡r t❤❡♦r❡♠ ❬❙❚✽✸❪

❈♦♥s✐❞❡r ❛ ✜♥✐t❡ ✜❡❧❞ F✳ ▲❡t

  • L ⊆ F2 ❜❡ ❛ s❡t ♦❢ ❧✐♥❡s
  • P ⊆ F2 ❜❡ ❛ s❡t ♦❢ ♣♦✐♥ts

★ ♣♦✐♥t✲❧✐♥❡ ✐♥❝✐❞❡♥❝❡s ≤ min{|L|1/2|P|+|L|, |L||P|1/2 +|P|}

  • ❈♦♥str✉❝t✐♦♥ q✉❡r✐❡s ❂ ❧✐♥❡s
  • Pr✐♠✐t✐✈❡ q✉❡r✐❡s ❂ ♣♦✐♥ts

❆❜♦✉t s♦❧✉t✐♦♥s t♦ ❊✈❡r② s♦❧✉t✐♦♥ ✜①❡s ♦♥❡ ✐s r❛♥❞♦♠ ✲❜✐t ❦❡②

✶✺ ✴ ✷✵

slide-37
SLIDE 37

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x

❙③❡♠❡ré❞✐✲❚r♦tt❡r t❤❡♦r❡♠ ❬❙❚✽✸❪

❈♦♥s✐❞❡r ❛ ✜♥✐t❡ ✜❡❧❞ F✳ ▲❡t

  • L ⊆ F2 ❜❡ ❛ s❡t ♦❢ ❧✐♥❡s
  • P ⊆ F2 ❜❡ ❛ s❡t ♦❢ ♣♦✐♥ts

★ ♣♦✐♥t✲❧✐♥❡ ✐♥❝✐❞❡♥❝❡s ≤ min{|L|1/2|P|+|L|, |L||P|1/2 +|P|}

  • ❈♦♥str✉❝t✐♦♥ q✉❡r✐❡s ❂ ❧✐♥❡s
  • Pr✐♠✐t✐✈❡ q✉❡r✐❡s ❂ ♣♦✐♥ts
  • ❆❜♦✉t q3/2 s♦❧✉t✐♦♥s t♦ m ⊕ (l ⊕ t) ⊗ t = x

❊✈❡r② s♦❧✉t✐♦♥ ✜①❡s ♦♥❡ ✐s r❛♥❞♦♠ ✲❜✐t ❦❡②

✶✺ ✴ ✷✵

slide-38
SLIDE 38

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x

❙③❡♠❡ré❞✐✲❚r♦tt❡r t❤❡♦r❡♠ ❬❙❚✽✸❪

❈♦♥s✐❞❡r ❛ ✜♥✐t❡ ✜❡❧❞ F✳ ▲❡t

  • L ⊆ F2 ❜❡ ❛ s❡t ♦❢ ❧✐♥❡s
  • P ⊆ F2 ❜❡ ❛ s❡t ♦❢ ♣♦✐♥ts

★ ♣♦✐♥t✲❧✐♥❡ ✐♥❝✐❞❡♥❝❡s ≤ min{|L|1/2|P|+|L|, |L||P|1/2 +|P|}

  • ❈♦♥str✉❝t✐♦♥ q✉❡r✐❡s ❂ ❧✐♥❡s
  • Pr✐♠✐t✐✈❡ q✉❡r✐❡s ❂ ♣♦✐♥ts
  • ❆❜♦✉t q3/2 s♦❧✉t✐♦♥s t♦ m ⊕ (l ⊕ t) ⊗ t = x
  • ❊✈❡r② s♦❧✉t✐♦♥ ✜①❡s ♦♥❡ l ⊕ t

✐s r❛♥❞♦♠ ✲❜✐t ❦❡②

✶✺ ✴ ✷✵

slide-39
SLIDE 39

❖♥❡ E✲❈❛❧❧ ✇✐t❤ P♦❧②♥♦♠✐❛❧ ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k = l ⊕ t ❛♥❞ m ⊕ (l ⊕ t) ⊗ t = x

❙③❡♠❡ré❞✐✲❚r♦tt❡r t❤❡♦r❡♠ ❬❙❚✽✸❪

❈♦♥s✐❞❡r ❛ ✜♥✐t❡ ✜❡❧❞ F✳ ▲❡t

  • L ⊆ F2 ❜❡ ❛ s❡t ♦❢ ❧✐♥❡s
  • P ⊆ F2 ❜❡ ❛ s❡t ♦❢ ♣♦✐♥ts

★ ♣♦✐♥t✲❧✐♥❡ ✐♥❝✐❞❡♥❝❡s ≤ min{|L|1/2|P|+|L|, |L||P|1/2 +|P|}

  • ❈♦♥str✉❝t✐♦♥ q✉❡r✐❡s ❂ ❧✐♥❡s
  • Pr✐♠✐t✐✈❡ q✉❡r✐❡s ❂ ♣♦✐♥ts
  • ❆❜♦✉t q3/2 s♦❧✉t✐♦♥s t♦ m ⊕ (l ⊕ t) ⊗ t = x
  • ❊✈❡r② s♦❧✉t✐♦♥ ✜①❡s ♦♥❡ l ⊕ t
  • k ✐s r❛♥❞♦♠ n✲❜✐t ❦❡②

✶✺ ✴ ✷✵

slide-40
SLIDE 40

❚✇♦ E✲❈❛❧❧s ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣

m c k t z

E E

  • F[2](k, t, m) = c

■❞❡❛

  • ❙✉❜❦❡② k ⊕ t
  • ▼❛s❦✐♥❣ E(k, t)

❙❡❝✉r✐t② ❯♣ t♦ q✉❡r✐❡s ❈♦st ❚✇♦ ✲❝❛❧❧s ❩❡r♦ ✲❡✈❛❧✉❛t✐♦♥s ❖♥❡ r❡✲❦❡②

✶✻ ✴ ✷✵

slide-41
SLIDE 41

❚✇♦ E✲❈❛❧❧s ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣

m c k t z

E E

  • F[2](k, t, m) = c

■❞❡❛

  • ❙✉❜❦❡② k ⊕ t
  • ▼❛s❦✐♥❣ E(k, t)

❙❡❝✉r✐t②

  • ❯♣ t♦ 2n q✉❡r✐❡s

❈♦st ❚✇♦ ✲❝❛❧❧s ❩❡r♦ ✲❡✈❛❧✉❛t✐♦♥s ❖♥❡ r❡✲❦❡②

✶✻ ✴ ✷✵

slide-42
SLIDE 42

❚✇♦ E✲❈❛❧❧s ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣

m c k t z

E E

  • F[2](k, t, m) = c

■❞❡❛

  • ❙✉❜❦❡② k ⊕ t
  • ▼❛s❦✐♥❣ E(k, t)

❙❡❝✉r✐t②

  • ❯♣ t♦ 2n q✉❡r✐❡s

❈♦st

  • ❚✇♦ E✲❝❛❧❧s
  • ❩❡r♦ ⊗✲❡✈❛❧✉❛t✐♦♥s
  • ❖♥❡ r❡✲❦❡②

✶✻ ✴ ✷✵

slide-43
SLIDE 43

❚✇♦ E✲❈❛❧❧s ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E E

❈♦♥str✉❝t✐♦♥ q✉❡r② ✏❤✐ts✑ ♣r✐♠✐t✐✈❡ q✉❡r② ✐❢ ❛♥❞ ♦r ❛♥❞ ✐s r❛♥❞♦♠ ❦❡②✱ ✐s ❛❧♠♦st✲r❛♥❞♦♠ s✉❜❦❡②

✶✼ ✴ ✷✵

slide-44
SLIDE 44

❚✇♦ E✲❈❛❧❧s ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E E

z m c t z ⊕ m z ⊕ c k ⊕ t x y l

  • ❈♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c) ✏❤✐ts✑ ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y) ✐❢

k ⊕ t = l ❛♥❞ z ⊕ m = x ♦r k ⊕ t = l ❛♥❞ z ⊕ c = y ✐s r❛♥❞♦♠ ❦❡②✱ ✐s ❛❧♠♦st✲r❛♥❞♦♠ s✉❜❦❡②

✶✼ ✴ ✷✵

slide-45
SLIDE 45

❚✇♦ E✲❈❛❧❧s ✇✐t❤ ▲✐♥❡❛r ▼✐①✐♥❣✿ Pr♦♦❢ ■❞❡❛

k

E E

z m c t z ⊕ m z ⊕ c k ⊕ t x y l

  • ❈♦♥str✉❝t✐♦♥ q✉❡r② (t, m, c) ✏❤✐ts✑ ♣r✐♠✐t✐✈❡ q✉❡r② (l, x, y) ✐❢

k ⊕ t = l ❛♥❞ z ⊕ m = x ♦r k ⊕ t = l ❛♥❞ z ⊕ c = y

  • k ✐s r❛♥❞♦♠ ❦❡②✱ z ✐s ❛❧♠♦st✲r❛♥❞♦♠ s✉❜❦❡②

✶✼ ✴ ✷✵

slide-46
SLIDE 46

❈♦♠♣❛r✐s♦♥

s❝❤❡♠❡ s❡❝✉r✐t② ✭log2✮ ❦❡② ❧❡♥❣t❤ ❝♦st E ⊗/h t❞❦ LRW1 n/2 n ✷ ✵ ✵ LRW2 n/2 2n ✶ ✶ ✵ XEX n/2 n ✷ ✵ ✵ LRW2[2] 2n/3 4n ✷ ✷ ✵ LRW2[ρ] ρn/(ρ+2) 2ρn ρ ρ ✵ Min max{n/2, n−|t|} n ✷ ✵ ✶

  • F[1]

2n/3 ⋆ n ✶ ✶ ✶

  • F[2]

n ⋆ n ✷ ✵ ✶

✶✽ ✴ ✷✵

⋆ ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ♠♦❞❡❧

slide-47
SLIDE 47

❚♦✇❛r❞s ❈♦♠♣❧❡①✐t②✲❚❤❡♦r❡t✐❝ ▼♦❞❡❧

✳ ✳ ✇✐t❤ ❛♥② ❝✐♣❤❡r ✳ ✳

  • F[α] ✇✐t❤

✐❞❡❛❧ ❝✐♣❤❡r E ✳ ✳ ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r π ✲r❦ s❡❝✉r✐t② ♦❢ ❝✉rr❡♥t ♣r♦♦❢ ✳ ❋✐rst st❡♣ ✉♥♥❡❝❡ss❛r✐❧② ❧♦♦s❡ ❚✇❡❛❦ ❝❤❛♥❣❡ ✐♥✢✉❡♥❝❡s ❦❡② ❛♥❞ ♠❡ss❛❣❡ ✐♥♣✉t ❉❡t❛✐❧s ✐♥ ♣❛♣❡r

✶✾ ✴ ✷✵

slide-48
SLIDE 48

❚♦✇❛r❞s ❈♦♠♣❧❡①✐t②✲❚❤❡♦r❡t✐❝ ▼♦❞❡❧

✳ ✳

  • F[α] ✇✐t❤

❛♥② ❝✐♣❤❡r E ✳ ✳

  • F[α] ✇✐t❤

✐❞❡❛❧ ❝✐♣❤❡r E ✳ ✳ ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r π ✲r❦ s❡❝✉r✐t② ♦❢ ❝✉rr❡♥t ♣r♦♦❢ ✳ ❋✐rst st❡♣ ✉♥♥❡❝❡ss❛r✐❧② ❧♦♦s❡ ❚✇❡❛❦ ❝❤❛♥❣❡ ✐♥✢✉❡♥❝❡s ❦❡② ❛♥❞ ♠❡ss❛❣❡ ✐♥♣✉t ❉❡t❛✐❧s ✐♥ ♣❛♣❡r

✶✾ ✴ ✷✵

slide-49
SLIDE 49

❚♦✇❛r❞s ❈♦♠♣❧❡①✐t②✲❚❤❡♦r❡t✐❝ ▼♦❞❡❧

✳ ✳

  • F[α] ✇✐t❤

❛♥② ❝✐♣❤❡r E ✳ ✳

  • F[α] ✇✐t❤

✐❞❡❛❧ ❝✐♣❤❡r E ✳ ✳ ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r π ✲r❦ s❡❝✉r✐t② ♦❢ E ❝✉rr❡♥t ♣r♦♦❢ ✳ ❋✐rst st❡♣ ✉♥♥❡❝❡ss❛r✐❧② ❧♦♦s❡ ❚✇❡❛❦ ❝❤❛♥❣❡ ✐♥✢✉❡♥❝❡s ❦❡② ❛♥❞ ♠❡ss❛❣❡ ✐♥♣✉t ❉❡t❛✐❧s ✐♥ ♣❛♣❡r

✶✾ ✴ ✷✵

slide-50
SLIDE 50

❚♦✇❛r❞s ❈♦♠♣❧❡①✐t②✲❚❤❡♦r❡t✐❝ ▼♦❞❡❧

✳ ✳

  • F[α] ✇✐t❤

❛♥② ❝✐♣❤❡r E ✳ ✳

  • F[α] ✇✐t❤

✐❞❡❛❧ ❝✐♣❤❡r E ✳ ✳ ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r π ⊕✲r❦ s❡❝✉r✐t② ♦❢ E ❝✉rr❡♥t ♣r♦♦❢ ✳ ❋✐rst st❡♣ ✉♥♥❡❝❡ss❛r✐❧② ❧♦♦s❡ ❚✇❡❛❦ ❝❤❛♥❣❡ ✐♥✢✉❡♥❝❡s ❦❡② ❛♥❞ ♠❡ss❛❣❡ ✐♥♣✉t ❉❡t❛✐❧s ✐♥ ♣❛♣❡r

✶✾ ✴ ✷✵

slide-51
SLIDE 51

❚♦✇❛r❞s ❈♦♠♣❧❡①✐t②✲❚❤❡♦r❡t✐❝ ▼♦❞❡❧

✳ ✳

  • F[α] ✇✐t❤

❛♥② ❝✐♣❤❡r E ✳ ✳

  • F[α] ✇✐t❤

✐❞❡❛❧ ❝✐♣❤❡r E ✳ ✳ ✐❞❡❛❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r π ⊕✲r❦ s❡❝✉r✐t② ♦❢ E ❝✉rr❡♥t ♣r♦♦❢ ✳

  • ❋✐rst st❡♣ ✉♥♥❡❝❡ss❛r✐❧② ❧♦♦s❡
  • ❚✇❡❛❦ ❝❤❛♥❣❡ ✐♥✢✉❡♥❝❡s ❦❡② ❛♥❞ ♠❡ss❛❣❡ ✐♥♣✉t
  • ❉❡t❛✐❧s ✐♥ ♣❛♣❡r

✶✾ ✴ ✷✵

slide-52
SLIDE 52

❈♦♥❝❧✉s✐♦♥s

  • F [1] ❛♥❞

F [2]

  • ❙✐♠♣❧❡ ❛♥❞ ❢❡✇ ♣r✐♠✐t✐✈❡ ❝❛❧❧s
  • ❍✐❣❤ s❡❝✉r✐t② ❧❡✈❡❧
  • ❊✣❝✐❡♥t ✐❢ ❦❡② r❡♥❡✇❛❧ ✐s r❡❧❛t✐✈❡❧② ❝❤❡❛♣

❋✉t✉r❡ ❘❡s❡❛r❝❤ ❖♥❡✲❝❛❧❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r ✇✐t❤ ✐♠♣r♦✈❡❞ s❡❝✉r✐t②❄ ❆✈♦✐❞✐♥❣ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t② ❝♦♥❞✐t✐♦♥❄ ■♠♣❧❡♠❡♥t❛t✐♦♥s❄

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✷✵ ✴ ✷✵

slide-53
SLIDE 53

❈♦♥❝❧✉s✐♦♥s

  • F [1] ❛♥❞

F [2]

  • ❙✐♠♣❧❡ ❛♥❞ ❢❡✇ ♣r✐♠✐t✐✈❡ ❝❛❧❧s
  • ❍✐❣❤ s❡❝✉r✐t② ❧❡✈❡❧
  • ❊✣❝✐❡♥t ✐❢ ❦❡② r❡♥❡✇❛❧ ✐s r❡❧❛t✐✈❡❧② ❝❤❡❛♣

❋✉t✉r❡ ❘❡s❡❛r❝❤

  • ❖♥❡✲❝❛❧❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r ✇✐t❤ ✐♠♣r♦✈❡❞ s❡❝✉r✐t②❄
  • ❆✈♦✐❞✐♥❣ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t② ❝♦♥❞✐t✐♦♥❄
  • ■♠♣❧❡♠❡♥t❛t✐♦♥s❄

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✷✵ ✴ ✷✵

slide-54
SLIDE 54

❈♦♥❝❧✉s✐♦♥s

  • F [1] ❛♥❞

F [2]

  • ❙✐♠♣❧❡ ❛♥❞ ❢❡✇ ♣r✐♠✐t✐✈❡ ❝❛❧❧s
  • ❍✐❣❤ s❡❝✉r✐t② ❧❡✈❡❧
  • ❊✣❝✐❡♥t ✐❢ ❦❡② r❡♥❡✇❛❧ ✐s r❡❧❛t✐✈❡❧② ❝❤❡❛♣

❋✉t✉r❡ ❘❡s❡❛r❝❤

  • ❖♥❡✲❝❛❧❧ t✇❡❛❦❛❜❧❡ ❝✐♣❤❡r ✇✐t❤ ✐♠♣r♦✈❡❞ s❡❝✉r✐t②❄
  • ❆✈♦✐❞✐♥❣ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t② ❝♦♥❞✐t✐♦♥❄
  • ■♠♣❧❡♠❡♥t❛t✐♦♥s❄

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✷✵ ✴ ✷✵

slide-55
SLIDE 55

❙✉♣♣♦rt✐♥❣ ❙❧✐❞❡s ❙❯PP❖❘❚■◆● ❙▲■❉❊❙

✷✶ ✴ ✷✵

slide-56
SLIDE 56
  • ❡♥❡r✐❝ ❉❡s✐❣♥✿ ■♥✈❡rs❡

❱❛❧✐❞ ▼✐①✐♥❣ ❋✉♥❝t✐♦♥s ✭✐♥❢♦r♠❛❧✮ Ai, Bi ❛r❡ ✈❛❧✐❞ ✐❢ t❤❡r❡ ✐s ♦♥❡ Ai∗ t❤❛t ♣r♦❝❡ss❡s m✱ s✳t✳

  • ✜rst i∗ − 1 r♦✉♥❞s ❝♦♠♣✉t❛❜❧❡ ✐♥ ❢♦r✇❛r❞ ❞✐r❡❝t✐♦♥
  • ❧❛st ρ − (i∗ − 1) r♦✉♥❞s ❝♦♠♣✉t❛❜❧❡ ✐♥ ✐♥✈❡rs❡ ❞✐r❡❝t✐♦♥

❜♦t❤ ✇✐t❤♦✉t ✉s❛❣❡ ♦❢ m ❊①❛♠♣❧❡ ❢♦r i∗ = 2

m c

E E E

  • A1

A-1

2

  • A-1

3

  • A-1

4

B1 B2

  • B3

k, t k, t k, t, y1 k, t, y1 l1 x1 y1 l2 x2 y2 l3 x3 y3

✷✷ ✴ ✷✵

slide-57
SLIDE 57

❇♦t❤ ❉❡s✐❣♥s ♦♥ ❖♥❡ ❙❧✐❞❡

m c k t z

E

  • F[1](k, t, m) = c

m c k t z

E E

  • F[2](k, t, m) = c

✷✸ ✴ ✷✵