SLIDE 1
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
t t rt t t tt - - PowerPoint PPT Presentation
t t rt t t tt - - PowerPoint PPT Presentation
Prs rs rt t tt s t r t t
SLIDE 2
SLIDE 3
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❖✉t❧✐♥❡
P❛✐r✐♥❣s Pr♦♣❡rt✐❡s ♦❢ ♣❛✐r✐♥❣s P❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ❈♦♥str✉❝t✐♦♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ Pr♦♣❡rt✐❡s ❈♦♥str✉❝t✐♦♥ ❋❛✉❧t ❛tt❛❝❦ ❉❡✜♥✐t✐♦♥ ❆❣❛✐♥st t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❈♦♥❝❧✉s✐♦♥
SLIDE 4
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❲❤❛t ✐s ❛ ♣❛✐r✐♥❣ ❄
▲❡t G✶✱ G✷ ❛♥❞ G✸ ❜❡ t❤r❡❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ♦r❞❡r r✳ ❆ ♣❛✐r✐♥❣ ✐s ❛ ♠❛♣ ✿ ❡ : G✶ × G✷ → G✸ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿
- ◆♦♥ ❞❡❣❡♥❡r❛t❡ ❀
- ❇✐❧✐♥❡❛r✐t② ❀
❈♦♥s❡q✉❡♥❝❡s
❥ ❡ ❥ P ◗ ❡ P ◗ ❥ ❡ P ❥ ◗
SLIDE 5
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❲❤❛t ✐s ❛ ♣❛✐r✐♥❣ ❄
▲❡t G✶✱ G✷ ❛♥❞ G✸ ❜❡ t❤r❡❡ ❣r♦✉♣s ✇✐t❤ t❤❡ s❛♠❡ ♦r❞❡r r✳ ❆ ♣❛✐r✐♥❣ ✐s ❛ ♠❛♣ ✿ ❡ : G✶ × G✷ → G✸ ✇❤✐❝❤ ✈❡r✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿
- ◆♦♥ ❞❡❣❡♥❡r❛t❡ ❀
- ❇✐❧✐♥❡❛r✐t② ❀
❈♦♥s❡q✉❡♥❝❡s
∀❥ ∈ N, ❡([❥]P, ◗) = ❡(P, ◗)❥ = ❡(P, [❥]◗)
SLIDE 6
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s
P❛rt ✶ ✲ ❈r②♣t❛♥❛❧②s❡
❚❤❡ ▼❖❱✴❋r❡② ❘ü❝❦ ❛tt❛❝❦ ❛❣❛✐♥st t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ✶✾✾✸✱ ✶✾✾✹ ✿ ✉s✐♥❣ ♣❛✐r✐♥❣s✱ t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❜❡❝♦♠❡s ❛ ❉▲P ♦♥ ✜♥✐t❡ ✜❡❧❞✳
- ✐✈❡♥ P ❛♥❞ ◗
P ❊
q ✱
t❤❡ ❉▲P ♦♥ ❊
q ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣
✳ ▲❡t ❙ ❊
q ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡ P ❙
✶✱ ❧❡t ❡ P ❙ ❣ ❛♥❞ ❡ ◗ ❙ ❤ ❊
q ✱ t❤❡♥
t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ s✉❝❤ t❤❛t ❤ ❣ ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳
SLIDE 7
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s
P❛rt ✶ ✲ ❈r②♣t❛♥❛❧②s❡
❚❤❡ ▼❖❱✴❋r❡② ❘ü❝❦ ❛tt❛❝❦ ❛❣❛✐♥st t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ✶✾✾✸✱ ✶✾✾✹ ✿ ✉s✐♥❣ ♣❛✐r✐♥❣s✱ t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❜❡❝♦♠❡s ❛ ❉▲P ♦♥ ✜♥✐t❡ ✜❡❧❞✳
- ●✐✈❡♥ P ❛♥❞ ◗ = αP ∈ ❊(Fq)✱
t❤❡ ❉▲P ♦♥ ❊(Fq) ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ α✳ ▲❡t ❙ ❊
q ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡ P ❙
✶✱ ❧❡t ❡ P ❙ ❣ ❛♥❞ ❡ ◗ ❙ ❤ ❊
q ✱ t❤❡♥
t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ s✉❝❤ t❤❛t ❤ ❣ ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳
SLIDE 8
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s
P❛rt ✶ ✲ ❈r②♣t❛♥❛❧②s❡
❚❤❡ ▼❖❱✴❋r❡② ❘ü❝❦ ❛tt❛❝❦ ❛❣❛✐♥st t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ✐♥ ✶✾✾✸✱ ✶✾✾✹ ✿ ✉s✐♥❣ ♣❛✐r✐♥❣s✱ t❤❡ ❉▲P ♦♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡s ❜❡❝♦♠❡s ❛ ❉▲P ♦♥ ✜♥✐t❡ ✜❡❧❞✳
- ●✐✈❡♥ P ❛♥❞ ◗ = αP ∈ ❊(Fq)✱
t❤❡ ❉▲P ♦♥ ❊(Fq) ❝♦♥s✐sts ✐♥ ✜♥❞✐♥❣ α✳
- ▲❡t ❙ ∈ ❊(Fq) ❜❡ ❛ ♣♦✐♥t s✉❝❤ t❤❛t ❡(P, ❙) = ✶✱
❧❡t ❡(P, ❙) = ❣ ❛♥❞ ❡(◗, ❙) = ❤ ∈ ❊(Fq)✱ t❤❡♥
- t❤❡ ❉▲P ❜❡❝♦♠❡s ✜♥❞✐♥❣ α s✉❝❤ t❤❛t ❤ = ❣α ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞✳
SLIDE 9
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s
P❛rt ✷ ✲ ❈r②♣t♦❣r❛♣❤②
P❛✐r✐♥❣s ❛❧❧♦✇ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ♥♦✈❡❧ ♣r♦t♦❝♦❧s ❛♥❞ s✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ ❡①✐st✐♥❣ ♣r♦t♦❝♦❧s✳
- ❚❤❡ tr✐ ♣❛rt✐t❡ ❉✐✣❡ ❍❡❧❧♠❛♥ ❦❡② ❡①❝❤❛♥❣❡ ♣r♦t♦❝♦❧ ✭❏♦✉①
✷✵✵✶✮
- ❚❤❡ ■❞❡♥t✐t② ❇❛s❡❞ ❊♥❝r②♣t✐♦♥ ✭❇♦♥❡❤ ❛♥❞ ❋r❛♥❦❧✐♥ ✷✵✵✶✮
- ❙❤♦rt s✐❣♥❛t✉r❡ s❝❤❡♠❡ ✭❇♦♥❡❤✱ ▲②♥♥✱ ❙❝❤❛❝❦❛♠♠ ✷✵✵✶✮
- ●r♦✉♣ s✐❣♥❛t✉r❡s s❝❤❡♠❡s ✭❇♦♥❡❤✱ ❙❝❤❛❝❦❛♠♠✱ ✷✵✵✹✮
SLIDE 10
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s
■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ✭■❇❈✮
❆♥ ■❞❡♥t✐t② ❇❛s❡❞ ♣r♦t♦❝♦❧ ✐s ❛♥ ❛s②♠❡tr✐❝ s②st❡♠ ✇❤❡r❡ ✿
- ❚❤❡ ♣✉❜❧✐❝ ❦❡② ♦❢ ❆❧✐❝❡ ✐ts ❤❡r ✐❞❡♥t✐t②✱
- ❤❡r ♣r✐✈❛t❡ ❑❡② ✐s ❝♦♥str✉❝t❡❞ ❜② ❛ ❚r✉st❡❞ ❛✉t❤♦r✐t②✳
❊①❛♠♣❧❡
❲❡ ✐❧❧✉str❛t❡ t❤❡ ♥♦t✐♦♥ ♦❢ ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ✇✐t❤ t❤❡ ❡①❛♠♣❧❡ ♦❢ ❛ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜✳
SLIDE 11
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
■❇❈
❙❡❝✉r❡ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜
SLIDE 12
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
■❇❈
❙❡❝✉r❡ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜
SLIDE 13
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
■❇❈
❙❡❝✉r❡ ❦❡② ❡①❝❤❛♥❣❡ ❜❡t✇❡❡♥ ❆❧✐❝❡ ❛♥❞ ❇♦❜
SLIDE 14
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊❧❧✐♣t✐❝ ❈✉r✈❡ ❈r②♣t♦❣r❛♣❤② ❛♥❞ ♣❛✐r✐♥❣s
P❛✐r✐♥❣s ✉s❡❞
❋♦✉r ♣❛✐r✐♥❣s ❛r❡ ♣r✐♥❝✐♣❛❧❧② ✉s❡❞ ✐♥ ❝r②♣t♦❣r❛♣❤② ✿
- t❤❡ ❲❡✐❧ ♣❛✐r✐♥❣✱
- t❤❡ ❚❛t❡ ♣❛✐r✐♥❣✱
- t❤❡ η❚ ♣❛✐r✐♥❣✱
- t❤❡ ❆t❡ ♣❛✐r✐♥❣✳
❚❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❝♦♥str✉❝t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❢r,P ✐s ❛ ❝❡♥tr❛❧ st❡♣ ❢♦r t❤❡ ❲❡✐❧✱ ❚❛t❡ ❛♥❞ ❆t❡ ♣❛✐r✐♥❣s✳
SLIDE 15
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❉❛t❛
❚♦ ❝♦♠♣✉t❡ ❛ ♣❛✐r✐♥❣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥ts ✿
- ❊ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Fq = {✵, ✶, ✷, . . . , q − ✶} ✿
❊ : ②✷ = ①✸ + ❛① + ❜✱ ✇❤❡r❡ ❛✱ ❜ ∈ Fq✳ r ❛ ♣r✐♠❡ ❞✐✈✐❞✐♥❣ ❝❛r❞ ❊
q
✱ ❝♦♥s✐❞❡r ❊ r ✿ ❊ r P ❊
q
r P P ✳ ❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✿ ♠✐♥✐♠❛❧ ✐♥t❡❣❡r s✉❝❤ t❤❛t r q❦ ✶ ✿ ■❢ ❣❝❞ r q ✶✱ t❤❡♥ ❊ r r r ✱ ■❢ ❦ ✶ t❤❡♥ ❊ r ❊
q❦
r ✳ ❆ ❢✉♥❝t✐♦♥ ❢r P ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳
SLIDE 16
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❉❛t❛
❚♦ ❝♦♠♣✉t❡ ❛ ♣❛✐r✐♥❣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥ts ✿
- ❊ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Fq = {✵, ✶, ✷, . . . , q − ✶} ✿
❊ : ②✷ = ①✸ + ❛① + ❜✱ ✇❤❡r❡ ❛✱ ❜ ∈ Fq✳
- r ❛ ♣r✐♠❡ ❞✐✈✐❞✐♥❣ ❝❛r❞(❊(Fq))✱
❝♦♥s✐❞❡r ❊[r] ✿ ❊[r] =
- P ∈ ❊(Fq), [r]P = P∞
- ✳
❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✿ ♠✐♥✐♠❛❧ ✐♥t❡❣❡r s✉❝❤ t❤❛t r q❦ ✶ ✿ ■❢ ❣❝❞ r q ✶✱ t❤❡♥ ❊ r r r ✱ ■❢ ❦ ✶ t❤❡♥ ❊ r ❊
q❦
r ✳ ❆ ❢✉♥❝t✐♦♥ ❢r P ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳
SLIDE 17
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❉❛t❛
❚♦ ❝♦♠♣✉t❡ ❛ ♣❛✐r✐♥❣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥ts ✿
- ❊ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Fq = {✵, ✶, ✷, . . . , q − ✶} ✿
❊ : ②✷ = ①✸ + ❛① + ❜✱ ✇❤❡r❡ ❛✱ ❜ ∈ Fq✳
- r ❛ ♣r✐♠❡ ❞✐✈✐❞✐♥❣ ❝❛r❞(❊(Fq))✱
❝♦♥s✐❞❡r ❊[r] ✿ ❊[r] =
- P ∈ ❊(Fq), [r]P = P∞
- ✳
- ❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✿ ♠✐♥✐♠❛❧ ✐♥t❡❣❡r s✉❝❤ t❤❛t
r|(q❦ − ✶) ✿ ■❢ ❣❝❞ r q ✶✱ t❤❡♥ ❊ r r r ✱ ■❢ ❦ ✶ t❤❡♥ ❊ r ❊
q❦
r ✳ ❆ ❢✉♥❝t✐♦♥ ❢r P ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳
SLIDE 18
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❉❛t❛
❚♦ ❝♦♠♣✉t❡ ❛ ♣❛✐r✐♥❣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥ts ✿
- ❊ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Fq = {✵, ✶, ✷, . . . , q − ✶} ✿
❊ : ②✷ = ①✸ + ❛① + ❜✱ ✇❤❡r❡ ❛✱ ❜ ∈ Fq✳
- r ❛ ♣r✐♠❡ ❞✐✈✐❞✐♥❣ ❝❛r❞(❊(Fq))✱
❝♦♥s✐❞❡r ❊[r] ✿ ❊[r] =
- P ∈ ❊(Fq), [r]P = P∞
- ✳
- ❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✿ ♠✐♥✐♠❛❧ ✐♥t❡❣❡r s✉❝❤ t❤❛t
r|(q❦ − ✶) ✿ ■❢ ❣❝❞(r, q) = ✶✱ t❤❡♥ ❊[r] ∼ = Z/rZ × Z/rZ✱ ■❢ ❦ > ✶ t❤❡♥ ❊[r] = ❊(Fq❦)[r]✳ ❆ ❢✉♥❝t✐♦♥ ❢r P ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳
SLIDE 19
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❉❛t❛
❚♦ ❝♦♠♣✉t❡ ❛ ♣❛✐r✐♥❣✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡❧❡♠❡♥ts ✿
- ❊ ❛♥ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ♦✈❡r Fq = {✵, ✶, ✷, . . . , q − ✶} ✿
❊ : ②✷ = ①✸ + ❛① + ❜✱ ✇❤❡r❡ ❛✱ ❜ ∈ Fq✳
- r ❛ ♣r✐♠❡ ❞✐✈✐❞✐♥❣ ❝❛r❞(❊(Fq))✱
❝♦♥s✐❞❡r ❊[r] ✿ ❊[r] =
- P ∈ ❊(Fq), [r]P = P∞
- ✳
- ❚❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ❦ ✿ ♠✐♥✐♠❛❧ ✐♥t❡❣❡r s✉❝❤ t❤❛t
r|(q❦ − ✶) ✿ ■❢ ❣❝❞(r, q) = ✶✱ t❤❡♥ ❊[r] ∼ = Z/rZ × Z/rZ✱ ■❢ ❦ > ✶ t❤❡♥ ❊[r] = ❊(Fq❦)[r]✳
- ❆ ❢✉♥❝t✐♦♥ ❢r,P ❝♦♥str✉❝t❡❞ ✇✐t❤ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳
SLIDE 20
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣
▲❡t P ∈ ❊(Fq)[r]✱ ◗ ∈ ❊(Fq❦)/r❊(Fq❦) ❛♥❞ ❦ ❜❡ t❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣ ✐s t❤❡ ❜✐❧✐♥❡❛r ♠❛♣ ✿ ❡❚ ❊
q r
❊
q❦
r❊
q❦ q❦
P ◗ ❢r P ◗
q❦ ✶ r
SLIDE 21
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣❛✐r✐♥❣s
❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣
▲❡t P ∈ ❊(Fq)[r]✱ ◗ ∈ ❊(Fq❦)/r❊(Fq❦) ❛♥❞ ❦ ❜❡ t❤❡ ❡♠❜❡❞❞✐♥❣ ❞❡❣r❡❡ ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡✳ ❚❤❡ ❚❛t❡ ♣❛✐r✐♥❣ ✐s t❤❡ ❜✐❧✐♥❡❛r ♠❛♣ ✿ ❡❚ : ❊(Fq)[r] × ❊(Fq❦)/r❊(Fq❦) → F∗
q❦
(P, ◗) → ❢r,P(◗)
q❦ −✶ r
SLIDE 22
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❖✉t❧✐♥❡
P❛✐r✐♥❣s Pr♦♣❡rt✐❡s ♦❢ ♣❛✐r✐♥❣s P❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ❈♦♥str✉❝t✐♦♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ Pr♦♣❡rt✐❡s ❈♦♥str✉❝t✐♦♥ ❋❛✉❧t ❛tt❛❝❦ ❉❡✜♥✐t✐♦♥ ❆❣❛✐♥st t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❈♦♥❝❧✉s✐♦♥
SLIDE 23
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ ❢✉♥❝t✐♦♥ ❢r,P
■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ♣❛✐r✐♥❣s✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❢✉♥❝t✐♦♥ ❢r,P✳ ❱✐❝t♦r ▼✐❧❧❡r✬s ❡st❛❜❧✐s❤❡❞ t❤❡ ▼✐❧❧❡r✬s ❡q✉❛t✐♦♥ ✿ ❢✐+❥,P = ❢✐,P × ❢❥,P × ❧[✐]P,[❥]P ✈[✐+❥]P ✇❤❡r❡ ❧ ✐ P
❥ P ✐s t❤❡ ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ♣♦✐♥ts ✐ P ❛♥❞ ❥ P✱
❛♥❞ ✈ ✐
❥ P ✐s t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t ✐
❥ P✳
SLIDE 24
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ ❢✉♥❝t✐♦♥ ❢r,P
■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ♣❛✐r✐♥❣s✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❢✉♥❝t✐♦♥ ❢r,P✳ ❱✐❝t♦r ▼✐❧❧❡r✬s ❡st❛❜❧✐s❤❡❞ t❤❡ ▼✐❧❧❡r✬s ❡q✉❛t✐♦♥ ✿ ❢✐+❥,P = ❢✐,P × ❢❥,P × ❧[✐]P,[❥]P ✈[✐+❥]P ✇❤❡r❡ ❧[✐]P+[❥]P ✐s t❤❡ ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ♣♦✐♥ts [✐]P ❛♥❞ [❥]P✱ ❛♥❞ ✈ ✐
❥ P ✐s t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t ✐
❥ P✳
SLIDE 25
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ ❢✉♥❝t✐♦♥ ❢r,P
■♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ t❤❡ ♣❛✐r✐♥❣s✱ ✇❡ ♥❡❡❞ t♦ ❝♦♠♣✉t❡ t❤❡ ❢✉♥❝t✐♦♥ ❢r,P✳ ❱✐❝t♦r ▼✐❧❧❡r✬s ❡st❛❜❧✐s❤❡❞ t❤❡ ▼✐❧❧❡r✬s ❡q✉❛t✐♦♥ ✿ ❢✐+❥,P = ❢✐,P × ❢❥,P × ❧[✐]P,[❥]P ✈[✐+❥]P ✇❤❡r❡ ❧[✐]P+[❥]P ✐s t❤❡ ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ♣♦✐♥ts [✐]P ❛♥❞ [❥]P✱ ❛♥❞ ✈[✐+❥]P ✐s t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ♣♦✐♥t [✐ + ❥]P✳
SLIDE 26
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿ ✼ ❂ ✻ ✰ ✶ ❢✼ P ❢✻ P ❢✶ P
❧ ✻ P P ✈ ✼ P
❢✶ P ✶ ❢✼ P ❢✻ P
❧ ✻ P P ✈ ✼ P
❢✻ P ❢✸ P ❢✸ P
❧ ✸ P ✸ P ✈ ✻ P
✇❤❡♥ ✐ ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t ✐ P ❢✻ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P
❢✼ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 27
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ✼ ❂ ✻ ✰ ✶
❢✼ P ❢✻ P ❢✶ P
❧ ✻ P P ✈ ✼ P
❢✶ P ✶ ❢✼ P ❢✻ P
❧ ✻ P P ✈ ✼ P
❢✻ P ❢✸ P ❢✸ P
❧ ✸ P ✸ P ✈ ✻ P
✇❤❡♥ ✐ ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t ✐ P ❢✻ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P
❢✼ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 28
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ✼ ❂ ✻ ✰ ✶
- ❢✼,P = ❢✻,P × ❢✶,P ×
❧[✻]P,P ✈[✼]P
❢✶ P ✶ ❢✼ P ❢✻ P
❧ ✻ P P ✈ ✼ P
❢✻ P ❢✸ P ❢✸ P
❧ ✸ P ✸ P ✈ ✻ P
✇❤❡♥ ✐ ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t ✐ P ❢✻ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P
❢✼ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 29
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ✼ ❂ ✻ ✰ ✶
- ❢✼,P = ❢✻,P × ❢✶,P ×
❧[✻]P,P ✈[✼]P
❢✶,P = ✶ ❢✼,P = ❢✻,P ×
❧[✻]P,P ✈[✼]P
❢✻ P ❢✸ P ❢✸ P
❧ ✸ P ✸ P ✈ ✻ P
✇❤❡♥ ✐ ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t ✐ P ❢✻ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P
❢✼ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 30
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ✼ ❂ ✻ ✰ ✶
- ❢✼,P = ❢✻,P × ❢✶,P ×
❧[✻]P,P ✈[✼]P
❢✶,P = ✶ ❢✼,P = ❢✻,P ×
❧[✻]P,P ✈[✼]P
- ❢✻,P = ❢✸,P × ❢✸,P ×
❧[✸]P,[✸]P ✈[✻]P
✇❤❡♥ ✐ = ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t [✐]P ❢✻ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P
❢✼ P ❢ ✷
✸ P ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 31
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ✼ ❂ ✻ ✰ ✶
- ❢✼,P = ❢✻,P × ❢✶,P ×
❧[✻]P,P ✈[✼]P
❢✶,P = ✶ ❢✼,P = ❢✻,P ×
❧[✻]P,P ✈[✼]P
- ❢✻,P = ❢✸,P × ❢✸,P ×
❧[✸]P,[✸]P ✈[✻]P
✇❤❡♥ ✐ = ❥✱ t❤❡ ❧✐♥❡ ❧ ✐s t❤❡ t❛♥❣❡♥t ❛t ♣♦✐♥t [✐]P
- ❢✻,P = ❢ ✷
✸,P × ❧[✸]P,[✸]P ✈[✻]P
❢✼,P = ❢ ✷
✸,P × ❧[✸]P,[✸]P ✈[✻]P
×
❧[✻]P,P ✈[✼]P
SLIDE 32
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ❢✼,P = ❢ ✷
✸,P × ❧[✸]P,[✸]P ✈[✻]P
×
❧[✻]P,P ✈[✼]P
❢✸ P ❢✷ P ❢✶ P
❧ ✷ P P ✈ ✸ P
❢✸ P ❢✷ P
❧ ✷ P P ✈ ✸ P
❢✷ P ❢✶ P ❢✶ P
❧P P ✈ ✷ P
❢✼ P
❧P P ✈ ✷ P ❧ ✷ P P ✈ ✸ P ✷ ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 33
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ❢✼,P = ❢ ✷
✸,P × ❧[✸]P,[✸]P ✈[✻]P
×
❧[✻]P,P ✈[✼]P
- ❢✸,P = ❢✷,P × ❢✶,P ×
❧[✷]P,P ✈[✸]P
❢✸,P = ❢✷,P ×
❧[✷]P,P ✈[✸]P
- ❢✷,P = ❢✶,P × ❢✶,P × ❧P,P
✈[✷]P
❢✼ P
❧P P ✈ ✷ P ❧ ✷ P P ✈ ✸ P ✷ ❧ ✸ P ✸ P ✈ ✻ P ❧ ✻ P P ✈ ✼ P
SLIDE 34
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❊①❛♠♣❧❡
❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ ❢✼,P ✿
- ❢✼,P = ❢ ✷
✸,P × ❧[✸]P,[✸]P ✈[✻]P
×
❧[✻]P,P ✈[✼]P
- ❢✸,P = ❢✷,P × ❢✶,P ×
❧[✷]P,P ✈[✸]P
❢✸,P = ❢✷,P ×
❧[✷]P,P ✈[✸]P
- ❢✷,P = ❢✶,P × ❢✶,P × ❧P,P
✈[✷]P
- ❢✼,P =
❧P,P
✈[✷]P × ❧[✷]P,P ✈[✸]P
✷ ×
❧[✸]P,[✸]P ✈[✻]P
×
❧[✻]P,P ✈[✼]P
SLIDE 35
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♠♣✉t❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s
❉❛t❛✿ r = (r♥ . . . r✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗
q❦ ❀
✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ✷ ❚ ❀ ✸ ✿ ❢✶ ❢✶
✷
❤✶ ◗ ❀ ✹ ✿ ❢✷ ❢✷
✷
✈✷ ◗ ❀ ✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ❚ P ❀ ❀ ❀ ❡♥❞ ❡♥❞ r❡t✉r♥
SLIDE 36
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♠♣✉t❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s
❉❛t❛✿ r = (r♥ . . . r✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗
q❦ ❀
✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ← [✷]❚ ❀ ✸ ✿ ❢✶ ❢✶
✷
❤✶ ◗ ❀ ✹ ✿ ❢✷ ❢✷
✷
✈✷ ◗ ❀ ✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ← ❚ + P ❀ ❀ ❀ ❡♥❞ ❡♥❞ r❡t✉r♥
SLIDE 37
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♠♣✉t❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s
❉❛t❛✿ r = (r♥ . . . r✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗
q❦ ❀
✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ← [✷]❚ ❀ ✸ ✿ ❢✶ ← − ❢✶
✷ × ❤✶(◗) ❀
✹ ✿ ❢✷ ← − ❢✷
✷ × ✈✷(◗) ❀
✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ← ❚ + P ❀ ❀ ❀ ❡♥❞ ❡♥❞ r❡t✉r♥
SLIDE 38
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♠♣✉t❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s
❉❛t❛✿ r = (r♥ . . . ❧✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗
q❦ ❀
✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ← [✷]❚❀ ✸ ✿ ❢✶ ← − ❢✶
✷ × ❤❞(◗) ❀
✹ ✿ ❢✷ ← − ❢✷
✷ × ✈❞(◗) ❀
✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ← ❚ + P ❀ ✻ ✿ ❢✶ ← − ❢✶ × ❤❛(◗) ❀ ✼ ✿ ❢✷ ← − ❢✷ × ✈❛(◗)❀ ❡♥❞ ❡♥❞ r❡t✉r♥
SLIDE 39
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♠♣✉t❛t✐♦♥ ♦❢ ♣❛✐r✐♥❣s
❉❛t❛✿ r = (r♥ . . . ❧✵)✷✱ P ∈ ❊(Fq) ❛♥❞ ◗ ∈ ❊(Fq❦) ❀ ❘❡s✉❧t✿ ❢r,P(◗) ∈ F∗
q❦ ❀
✶ ✿ ❚ ← P ✱ ❢✶ ← ✶✱ ❢✷ ← ✶ ❀ ❢♦r ✐ = ♥ − ✶ t♦ ✵ ❞♦ ✷ ✿ ❚ ← [✷]❚❀ ✸ ✿ ❢✶ ← − ❢✶
✷ × ❤❞(◗) ❀
✹ ✿ ❢✷ ← − ❢✷
✷ × ✈❞(◗) ❀
✐❢ r✐ = ✶ t❤❡♥ ✺ ✿ ❚ ← ❚ + P ❀ ✻ ✿ ❢✶ ← − ❢✶ × ❤❛(◗) ❀ ✼ ✿ ❢✷ ← − ❢✷ × ✈❛(◗)❀ ❡♥❞ ❡♥❞ r❡t✉r♥
❢✶ ❢✷
SLIDE 40
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❖✉t❧✐♥❡
P❛✐r✐♥❣s Pr♦♣❡rt✐❡s ♦❢ ♣❛✐r✐♥❣s P❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ❈♦♥str✉❝t✐♦♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ Pr♦♣❡rt✐❡s ❈♦♥str✉❝t✐♦♥ ❋❛✉❧t ❛tt❛❝❦ ❉❡✜♥✐t✐♦♥ ❆❣❛✐♥st t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❈♦♥❝❧✉s✐♦♥
SLIDE 41
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❙✐❞❡ ❈❤❛♥♥❡❧ ❆tt❛❝❦s
■♥ ❛♥ ■❞❡♥t✐t② ❇❛s❡❞ ♣r♦t♦❝♦❧ ✿
- ❛♥ ❛tt❛❝❦❡r ❦♥♦✇s t❤❡ ❛❧❣♦r✐t❤♠ ✉s❡❞ ❛♥❞
- t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s✳
- ❚❤❡ s❡❝r❡t ✐s ♦♥❧② ♦♥❡ ♦❢ t❤❡ ❛r❣✉♠❡♥ts ♦❢ t❤❡ ♣❛✐r✐♥❣✳
- ❚❤❡ s❡❝r❡t ❦❡② ✐♥✢✉❡♥❝❡s ♥❡✐t❤❡r t❤❡ ❡①❡❝✉t✐♦♥ t✐♠❡ ♥♦r t❤❡
♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳
SLIDE 42
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❙✐❞❡ ❈❤❛♥♥❡❧ ❆tt❛❝❦s
- ❙✐❞❡ ❈❤❛♥♥❡❧ ❆tt❛❝❦s ❛r❡ ❛tt❛❝❦s ❜❛s❡❞ ♦♥ ✐♥❢♦r♠❛t✐♦♥ ❣❛✐♥❡❞
❢r♦♠ t❤❡ ♣❤②s✐❝❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ❝r②♣t♦s②st❡♠✳
- ❋❛✉❧t ❛tt❛❝❦s ❛r❡ s✉❝❤ t❤❛t t❤❡ s❡❝r❡t ✐s ❞✐s❝♦✈❡r❡❞ ❜②
✐♥tr♦❞✉❝✐♥❣ ❢❛✉❧ts ❞✉r✐♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥✳
- ❚❤❡r❡ ✇❡r❡ ✜rst ✐♥tr♦❞✉✐❝❡❞ ✐♥ ✷✵✵✻ ❜② P❛❣❡ ❛♥❞ ❱❡r❝❛✉t❡r❡♥
❛❣❛✐♥st t❤❡ ❉✉✉rs♠❛ ❛♥❞ ▲❡❡ ❛❧❣♦r✐t❤♠✳
SLIDE 43
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❉❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢❛✉❧t ❛tt❛❝❦
- ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛✐r✐♥❣ ✐s ✉s❡❞ ❞✉r✐♥❣ ❛♥ ■❞❡♥t✐t② ❇❛s❡❞
Pr♦t♦❝♦❧✳
- ❚❤❡ s❡❝r❡t ✐s t❤❡ ♣♦✐♥t P✱ ✜rst ❛r❣✉♠❡♥t ♦❢ t❤❡ ♣❛✐r✐♥❣
❡(P, ◗)✳
❆✐♠ ♦❢ t❤❡ ❢❛✉❧t ❛tt❛❝❦
❚❤❡ ❛tt❛❝❦ ❝♦♥s✐sts ✐♥ ♠♦❞✐❢②✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✭❧♦❣✷(r)✮ ❞✉r✐♥❣ t❤❡ ❡①❡❝✉t✐♦♥ ♦❢ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ✐♥ ♦r❞❡r t♦ ✜♥❞ t✇♦ ❡①❡❝✉t✐♦♥s ✇✐t❤ ❝♦♥s❡❝✉t✐✈❡ ♥✉♠❜❡rs ♦❢ ✐t❡r❛t✐♦♥s ✿ ❧ ❛♥❞ ❧ + ✶ ❢♦r ❧ ∈ {✶, . . . , ❧♦❣✷(r) − ✶}✳ ❲❡ ♥♦t❡ ❋❧,P(◗) ❛♥❞ ❋❧+✶,P(◗) t❤❡ t✇♦ r❡s✉❧ts✳
SLIDE 44
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ r❛t✐♦ ❘ = ❋❧+✶,P(◗)
❋❧,P(◗)
❚❤❡ r❛t✐♦ ❘ = ❋❧+✶,P(◗)
❋❧,P(◗)
❣✐✈❡s ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ s❡❝r❡t✳
- ❆t t❤❡ ❧✲t❤ st❡♣✱ ❚ = [❥]P ✐♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠✳
- ❲❡ ❞❡♥♦t❡ [❥]P = (❳❥, ❨❥, ❩❥) t❤❡ s❡❝r❡t ♣♦✐♥t ❛♥❞
◗ = (①◗, ②◗) t❤❡ ❦♥♦✇♥ ❡♥tr② ♦❢ ❡(P, ◗)✳ ❲r✐t✐♥❣ ❞♦✇♥ t❤❡ ❡q✉❛t✐♦♥✱ t❤❡ r❛t✐♦ ❘ ❡q✉❛❧s ✿
- ❩✷❥❩ ✷
❥ ②◗ − ✷❨ ✷ ❥ − ✸(❳❥ − ❩ ✷ ❥ )(❳❥ + ❩ ✷ ❥ )(①◗❩ ✷ ❥ − ❳❥)
- .
❲✐t❤ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ t❤❡♦r❡t✐❝❛❧ ❞❡✈❡❧♦♣♣❡♠❡♥t ♦❢ ❘ ❛♥❞ ✐ts ✈❛❧✉❡✱ ✇❡ ♦❜t❛✐♥ ❛ s②st❡♠ ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ [❥]P✳
SLIDE 45
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ r❛t✐♦ ❘ = ❋❧+✶,P(◗)
❋❧,P(◗)
❚❤❡ s②st❡♠ ✐s ✿ ❨❥❩ ✸
❥ = λ✷
✭✶✮ ❩ ✷
❥ (❳ ✷ ❥ − ❩ ✹ ❥ ) = λ✶
✭✷✮ ✸❳❥(❳ ✷
❥ − ❩ ✹ ❥ ) + ✷❨ ✷ ❥ = λ✵
✭✸✮ ❲❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ λ✵✱ λ✶ ❛♥❞ λ✷✳ ❚❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤✐s s②st❡♠ ❝♦✉❧❞ ❜❡ ❢♦r ❡①❛♠♣❧❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ✶✳ ❯s❡ ❡q✉❛t✐♦♥ ✶ t♦ ♦❜t❛✐♥ ❨❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✷✳ ❯s❡ ❡q✉❛t✐♦♥ ✷ t♦ ♦❜t❛✐♥ ✸ ❳ ✷
❥
❩ ✹
❥
❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✸✳ ❙✉❜st✐t✉t❡ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥ ❡q✉❛t✐♦♥ ✸ t♦ ♦❜t❛✐♥ ❳❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✹✳ ❙✉❜st✐t✉t❡ ❳❥ ✐♥ ❡q✉❛t✐♦♥ ✷ t♦ ♦❜t❛✐♥ ❛ ❞❡❣r❡❡ ✶✷ ❡q✉❛t✐♦♥ ✐♥ ❩❥✳
✷ ✵
✾ ✷
✶ ❩ ✶✷
✹ ✵
✷ ✷
✾ ✸
✶ ❩ ✻
✹ ✹
✶
✵
SLIDE 46
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ r❛t✐♦ ❘ = ❋❧+✶,P(◗)
❋❧,P(◗)
❚❤❡ s②st❡♠ ✐s ✿ ❨❥❩ ✸
❥ = λ✷
✭✶✮ ❩ ✷
❥ (❳ ✷ ❥ − ❩ ✹ ❥ ) = λ✶
✭✷✮ ✸❳❥(❳ ✷
❥ − ❩ ✹ ❥ ) + ✷❨ ✷ ❥ = λ✵
✭✸✮ ❲❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ λ✵✱ λ✶ ❛♥❞ λ✷✳ ❚❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤✐s s②st❡♠ ❝♦✉❧❞ ❜❡ ❢♦r ❡①❛♠♣❧❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ✶✳ ❯s❡ ❡q✉❛t✐♦♥ ✶ t♦ ♦❜t❛✐♥ ❨❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✷✳ ❯s❡ ❡q✉❛t✐♦♥ ✷ t♦ ♦❜t❛✐♥ ✸ ❳ ✷
❥
❩ ✹
❥
❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✸✳ ❙✉❜st✐t✉t❡ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥ ❡q✉❛t✐♦♥ ✸ t♦ ♦❜t❛✐♥ ❳❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✹✳ ❙✉❜st✐t✉t❡ ❳❥ ✐♥ ❡q✉❛t✐♦♥ ✷ t♦ ♦❜t❛✐♥ ❛ ❞❡❣r❡❡ ✶✷ ❡q✉❛t✐♦♥ ✐♥ ❩❥✳
✷ ✵
✾ ✷
✶ ❩ ✶✷
✹ ✵
✷ ✷
✾ ✸
✶ ❩ ✻
✹ ✹
✶
✵
SLIDE 47
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ r❛t✐♦ ❘ = ❋❧+✶,P(◗)
❋❧,P(◗)
❚❤❡ s②st❡♠ ✐s ✿ ❨❥❩ ✸
❥ = λ✷
✭✶✮ ❩ ✷
❥ (❳ ✷ ❥ − ❩ ✹ ❥ ) = λ✶
✭✷✮ ✸❳❥(❳ ✷
❥ − ❩ ✹ ❥ ) + ✷❨ ✷ ❥ = λ✵
✭✸✮ ❲❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ λ✵✱ λ✶ ❛♥❞ λ✷✳ ❚❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤✐s s②st❡♠ ❝♦✉❧❞ ❜❡ ❢♦r ❡①❛♠♣❧❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ✶✳ ❯s❡ ❡q✉❛t✐♦♥ (✶) t♦ ♦❜t❛✐♥ ❨❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✷✳ ❯s❡ ❡q✉❛t✐♦♥ (✷) t♦ ♦❜t❛✐♥ ✸(❳ ✷
❥ − ❩ ✹ ❥ ) ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳
✸✳ ❙✉❜st✐t✉t❡ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥ ❡q✉❛t✐♦♥ (✸) t♦ ♦❜t❛✐♥ ❳❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✹✳ ❙✉❜st✐t✉t❡ ❳❥ ✐♥ ❡q✉❛t✐♦♥ (✷) t♦ ♦❜t❛✐♥ ❛ ❞❡❣r❡❡ ✶✷ ❡q✉❛t✐♦♥ ✐♥ ❩❥✳
✷ ✵
✾ ✷
✶ ❩ ✶✷
✹ ✵
✷ ✷
✾ ✸
✶ ❩ ✻
✹ ✹
✶
✵
SLIDE 48
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ r❛t✐♦ ❘ = ❋❧+✶,P(◗)
❋❧,P(◗)
❚❤❡ s②st❡♠ ✐s ✿ ❨❥❩ ✸
❥ = λ✷
✭✶✮ ❩ ✷
❥ (❳ ✷ ❥ − ❩ ✹ ❥ ) = λ✶
✭✷✮ ✸❳❥(❳ ✷
❥ − ❩ ✹ ❥ ) + ✷❨ ✷ ❥ = λ✵
✭✸✮ ❲❤❡r❡ ✇❡ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ λ✵✱ λ✶ ❛♥❞ λ✷✳ ❚❤❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤✐s s②st❡♠ ❝♦✉❧❞ ❜❡ ❢♦r ❡①❛♠♣❧❡ t❤❡ ❢♦❧❧♦✇✐♥❣✳ ✶✳ ❯s❡ ❡q✉❛t✐♦♥ (✶) t♦ ♦❜t❛✐♥ ❨❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✷✳ ❯s❡ ❡q✉❛t✐♦♥ (✷) t♦ ♦❜t❛✐♥ ✸(❳ ✷
❥ − ❩ ✹ ❥ ) ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳
✸✳ ❙✉❜st✐t✉t❡ t❤✐s ❡①♣r❡ss✐♦♥ ✐♥ ❡q✉❛t✐♦♥ (✸) t♦ ♦❜t❛✐♥ ❳❥ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ❩❥✳ ✹✳ ❙✉❜st✐t✉t❡ ❳❥ ✐♥ ❡q✉❛t✐♦♥ (✷) t♦ ♦❜t❛✐♥ ❛ ❞❡❣r❡❡ ✶✷ ❡q✉❛t✐♦♥ ✐♥ ❩❥✳ (λ✷
✵ − ✾λ✷ ✶)❩ ✶✷ − (✹λ✵λ✷ ✷ + ✾λ✸ ✶)❩ ✻ + ✹λ✹ ✶ = ✵
SLIDE 49
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❖✉t❧✐♥❡
P❛✐r✐♥❣s Pr♦♣❡rt✐❡s ♦❢ ♣❛✐r✐♥❣s P❛✐r✐♥❣ ❜❛s❡❞ ❝r②♣t♦❣r❛♣❤② ■❞❡♥t✐t② ❇❛s❡❞ ❈r②♣t♦❣r❛♣❤② ❈♦♥str✉❝t✐♦♥ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ Pr♦♣❡rt✐❡s ❈♦♥str✉❝t✐♦♥ ❋❛✉❧t ❛tt❛❝❦ ❉❡✜♥✐t✐♦♥ ❆❣❛✐♥st t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❈♦♥❝❧✉s✐♦♥
SLIDE 50
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❈♦♥❝❧✉s✐♦♥
❚❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ✐s ✈✉❧♥❡r❛❜❧❡ t♦ ❛ ❢❛✉❧t ❛tt❛❝❦✳
❱✉❧♥❡r❛❜✐❧✐t② ♦❢ ♣❛✐r✐♥❣s ❜❛s❡❞ ♦♥ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠
❚❤❡ ❲❡✐❧ ♣❛✐r✐♥❣ ✐s ❞✐r❡❝t❧② s❡♥s✐t✐✈❡ t♦ t❤❡ ❛tt❛❝❦✳ ❚❤❡ ❚❛t❡ ❛♥❞ ❆t❡ ♣❛✐r✐♥❣s ❛r❡ ❝♦♥str✉❝t❡❞ ♦♥ t❤❡ s❛♠❡ ♠♦❞❡❧✱ ♦♥❡ ❡①❡❝✉t✐♦♥ ♦❢ t❤❡ ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ♣❧✉s ❛ ✜♥❛❧ ❡①♣♦♥❡♥t✐❛t✐♦♥✳ ❚❤❡ ✜♥❛❧ ❡①♣♦♥❡♥t✐❛t✐♦♥ ✐s ❛ ❝♦✉♥t❡r✲♠❡❛s✉r❡ t♦ t❤✐s ❢❛✉❧t ❛tt❛❝❦✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ♦✈❡rr✐❞❡ ✐t ✉s✐♥❣ ❛ s❝❛♥ ❛tt❛❝❦ ♦r t❤❡ ✉♥❞❡r ✈♦❧t❛❣❡ t❡❝❤♥✐q✉❡✳
SLIDE 51
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥
❚❤❡ ❡♥❞ ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳ ❆♥② q✉❡st✐♦♥ ❄
SLIDE 52
P❛✐r✐♥❣s ▼✐❧❧❡r✬s ❛❧❣♦r✐t❤♠ ❋❛✉❧t ❛tt❛❝❦ ❈♦♥❝❧✉s✐♦♥