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slide-1
SLIDE 1

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s✐s

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤

Pr❡s❡♥t❡❞ ❜②✿ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ❉❡♣❛rt♠❡♥t ♦❢ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣ ❲❡st❡r♥ ❯♥✐✈❡rs✐t②✱ ▲♦♥❞♦♥✱ ❖♥t❛r✐♦✱ ❈❛♥❛❞❛

✷✷♥❞ ■❊❊❊ ❙②♠♣♦s✐✉♠ ♦♥ ❈♦♠♣✉t❡r ❆r✐t❤♠❡t✐❝✱ ✷✵✶✺

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✴✷✽

slide-2
SLIDE 2

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✴✷✽

slide-3
SLIDE 3

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✸✴✷✽

slide-4
SLIDE 4

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s

▼❛♥② ❛♣♣❧✐❝❛t✐♦♥s ✉s❡ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ♦✈❡r GF (✷m)

❈r②♣t♦❣r❛♣❤②✿ ❡❧❧✐♣t✐❝ ❝✉r✈❡✱ ❆❊❙ ❊rr♦r ❝♦♥tr♦❧ ❝♦❞✐♥❣✿ ❘❡❡❞✲❙♦❧♦♠♦♥ ❝♦❞❡ ❉✐❣✐t❛❧ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣✳

Cryptosystems Error Control Arithmetic Operations (Finite Fields, Integer Arithmetic,, etc) Digital Signal Processing

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✹✴✷✽

slide-5
SLIDE 5

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✺✴✷✽

slide-6
SLIDE 6

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✻✴✷✽

slide-7
SLIDE 7

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❘❡♣r❡s❡♥t❛t✐♦♥

❚❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t ❜❛s❡s t♦ r❡♣r❡s❡♥t ❛ ✜❡❧❞ ❡❧❡♠❡♥t✳

P♦❧②♥♦♠✐❛❧ ❜❛s✐s✱ ♥♦r♠❛❧ ❜❛s✐s✱ ❞✉❛❧ ❜❛s✐s✱ ❡t❝✳

P❇ ♦✛❡rs ❡✣❝✐❡♥t ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❝♦♠♣❛r❡❞ t♦ ♦t❤❡r ❜❛s❡s✳ ❚♦ ❝♦♥str✉❝t ❛ P❇✿

❋✐♥❞ p (x)✿ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ m ♦✈❡r GF (✷)✳ ❚❤❡♥✱

  • αm−✶,...,α,✶
  • ✐s ❦♥♦✇♥ ❛s t❤❡ P❇✱ ✇❤❡r❡ p(α) = ✵.

❆♥② ✜❡❧❞ ❡❧❡♠❡♥t A = (am−✶,··· ,a✶,a✵) ∈ GF(✷m), ai ∈ {✵,✶}, ✐s r❡♣r❡s❡♥t❡❞ ✇✳r✳t✳ t❤❡ P❇ ❛s A =

m−✶

i=✵

aiαi.

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✼✴✷✽

slide-8
SLIDE 8

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❘❡♣r❡s❡♥t❛t✐♦♥

❚❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t ❜❛s❡s t♦ r❡♣r❡s❡♥t ❛ ✜❡❧❞ ❡❧❡♠❡♥t✳

P♦❧②♥♦♠✐❛❧ ❜❛s✐s✱ ♥♦r♠❛❧ ❜❛s✐s✱ ❞✉❛❧ ❜❛s✐s✱ ❡t❝✳

P❇ ♦✛❡rs ❡✣❝✐❡♥t ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❝♦♠♣❛r❡❞ t♦ ♦t❤❡r ❜❛s❡s✳ ❚♦ ❝♦♥str✉❝t ❛ P❇✿

❋✐♥❞ p (x)✿ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ m ♦✈❡r GF (✷)✳ ❚❤❡♥✱

  • αm−✶,...,α,✶
  • ✐s ❦♥♦✇♥ ❛s t❤❡ P❇✱ ✇❤❡r❡ p(α) = ✵.

❆♥② ✜❡❧❞ ❡❧❡♠❡♥t A = (am−✶,··· ,a✶,a✵) ∈ GF(✷m), ai ∈ {✵,✶}, ✐s r❡♣r❡s❡♥t❡❞ ✇✳r✳t✳ t❤❡ P❇ ❛s A =

m−✶

i=✵

aiαi.

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✼✴✷✽

slide-9
SLIDE 9

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❘❡♣r❡s❡♥t❛t✐♦♥

❚❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t ❜❛s❡s t♦ r❡♣r❡s❡♥t ❛ ✜❡❧❞ ❡❧❡♠❡♥t✳

P♦❧②♥♦♠✐❛❧ ❜❛s✐s✱ ♥♦r♠❛❧ ❜❛s✐s✱ ❞✉❛❧ ❜❛s✐s✱ ❡t❝✳

P❇ ♦✛❡rs ❡✣❝✐❡♥t ♠✉❧t✐♣❧✐❝❛t✐♦♥s ❝♦♠♣❛r❡❞ t♦ ♦t❤❡r ❜❛s❡s✳ ❚♦ ❝♦♥str✉❝t ❛ P❇✿

❋✐♥❞ p (x)✿ ❛♥ ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ m ♦✈❡r GF (✷)✳ ❚❤❡♥✱

  • αm−✶,...,α,✶
  • ✐s ❦♥♦✇♥ ❛s t❤❡ P❇✱ ✇❤❡r❡ p(α) = ✵.

❆♥② ✜❡❧❞ ❡❧❡♠❡♥t A = (am−✶,··· ,a✶,a✵) ∈ GF(✷m), ai ∈ {✵,✶}, ✐s r❡♣r❡s❡♥t❡❞ ✇✳r✳t✳ t❤❡ P❇ ❛s A =

m−✶

i=✵

aiαi.

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✼✴✷✽

slide-10
SLIDE 10

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✽✴✷✽

slide-11
SLIDE 11

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ♦✈❡r GF (✷m)

▲❡t A ❛♥❞ B ❜❡ t✇♦ ✜❡❧❞ ❡❧❡♠❡♥ts r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ P❇✳

❆❞❞✐t✐♦♥✱ A+B, ✐s ❜✐t✲✇✐s❡ ❳❖❘ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳

m−✶

i=✵

aiαi

  • A

+

m−✶

j=✵

bjαj

  • B

=

m−✶

k=✵

(ak ⊕bk)αk

  • A+B

. ❋✐♥✐t❡ ✜❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥✱ C = A·B ♠♦❞ p (α), ✐s ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ ❛❞❞✐t✐♦♥✳ ✶✳ P♦❧②♥♦♠✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ A ❛♥❞ B✳      

m−✶

i=✵

aiαi

  • A

      ·      

m−✶

j=✵

bjαj

  • B

      =

m−✶

i=✵ m−✶

j=✵

aibjαi+j

  • A·B

. ✷✳ ▼♦❞✉❧♦ r❡❞✉❝t✐♦♥ ♦❢ αi+j ✐❢ i +j ≥ m✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✾✴✷✽

slide-12
SLIDE 12

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ♦✈❡r GF (✷m)

▲❡t A ❛♥❞ B ❜❡ t✇♦ ✜❡❧❞ ❡❧❡♠❡♥ts r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ P❇✳

❆❞❞✐t✐♦♥✱ A+B, ✐s ❜✐t✲✇✐s❡ ❳❖❘ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳

m−✶

i=✵

aiαi

  • A

+

m−✶

j=✵

bjαj

  • B

=

m−✶

k=✵

(ak ⊕bk)αk

  • A+B

. ❋✐♥✐t❡ ✜❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥✱ C = A·B ♠♦❞ p (α), ✐s ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ ❛❞❞✐t✐♦♥✳ ✶✳ P♦❧②♥♦♠✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ A ❛♥❞ B✳      

m−✶

i=✵

aiαi

  • A

      ·      

m−✶

j=✵

bjαj

  • B

      =

m−✶

i=✵ m−✶

j=✵

aibjαi+j

  • A·B

. ✷✳ ▼♦❞✉❧♦ r❡❞✉❝t✐♦♥ ♦❢ αi+j ✐❢ i +j ≥ m✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✾✴✷✽

slide-13
SLIDE 13

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ♦✈❡r GF (✷m)

▲❡t A ❛♥❞ B ❜❡ t✇♦ ✜❡❧❞ ❡❧❡♠❡♥ts r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ P❇✳

❆❞❞✐t✐♦♥✱ A+B, ✐s ❜✐t✲✇✐s❡ ❳❖❘ ♦♣❡r❛t✐♦♥s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳

m−✶

i=✵

aiαi

  • A

+

m−✶

j=✵

bjαj

  • B

=

m−✶

k=✵

(ak ⊕bk)αk

  • A+B

. ❋✐♥✐t❡ ✜❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥✱ C = A·B ♠♦❞ p (α), ✐s ♠♦r❡ ❝♦♠♣❧❡① t❤❛♥ ❛❞❞✐t✐♦♥✳ ✶✳ P♦❧②♥♦♠✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ A ❛♥❞ B✳      

m−✶

i=✵

aiαi

  • A

      ·      

m−✶

j=✵

bjαj

  • B

      =

m−✶

i=✵ m−✶

j=✵

aibjαi+j

  • A·B

. ✷✳ ▼♦❞✉❧♦ r❡❞✉❝t✐♦♥ ♦❢ αi+j ✐❢ i +j ≥ m✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✾✴✷✽

slide-14
SLIDE 14

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ♦✈❡r GF (✷m)✲ ❈♦♥t✐♥✉❡❞

❊①♣♦♥❡♥t✐❛t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❜② r❡♣❡❛t✐♥❣ sq✉❛r✐♥❣ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ Ae =

m−✶

i=✵

Aei✷i. ❊①♣♦♥❡♥t ✐♥ r❛❞✐①✲✷ ✐s e = ∑m−✶

i=✵ ei✷i✳

■♥✈❡rs❡ ♦❜t❛✐♥❡❞ ✐❢ e = ✷m −✷ ✭❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠✮✳ ❚❤❡r❡❢♦r❡✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✵✴✷✽

slide-15
SLIDE 15

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ♦✈❡r GF (✷m)✲ ❈♦♥t✐♥✉❡❞

❊①♣♦♥❡♥t✐❛t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❜② r❡♣❡❛t✐♥❣ sq✉❛r✐♥❣ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ Ae =

m−✶

i=✵

Aei✷i. ❊①♣♦♥❡♥t ✐♥ r❛❞✐①✲✷ ✐s e = ∑m−✶

i=✵ ei✷i✳

■♥✈❡rs❡ ♦❜t❛✐♥❡❞ ✐❢ e = ✷m −✷ ✭❋❡r♠❛t✬s ▲✐tt❧❡ ❚❤❡♦r❡♠✮✳ ❚❤❡r❡❢♦r❡✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s t❤❡ ♠❛✐♥ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✵✴✷✽

slide-16
SLIDE 16

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❈❛t❡❣♦r✐❡s ♦❢ GF (✷m) ▼✉❧t✐♣❧✐❡rs

❚❤❡r❡ ❛r❡ t✇♦ ❣❡♥❡r❛❧ ❝❛t❡❣♦r✐❡s ❢♦r GF (✷m) ♠✉❧t✐♣❧✐❡rs✿ P❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥s✳

❍✐❣❤ ❛r❡❛ ❛♥❞ t❤r♦✉❣❤♣✉t ❘❡s✉❧t ✐♥ ❛ s✐♥❣❧❡ ❝❧♦❝❦ ❝②❝❧❡✳ ❆ttr❛❝t✐✈❡ ❢♦r ❤✐❣❤ ♣❡r❢♦r♠❛♥❝❡ ❛♣♣❧✐❝❛t✐♦♥s✳

❇✐t✲❧❡✈❡❧ ✭❇▲✮ ❝♦♠♣✉t❛t✐♦♥s✳

▲♦✇❡st ❛r❡❛ ❘❡s✉❧t ✐♥ m ❝❧♦❝❦ ❝②❝❧❡s ❆ttr❛❝t✐✈❡ ❢♦r r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳

A B AB

m m m

GF(2m) Multiplication 1 clock cycle m clock cycles

Parallel Bit-Level

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✶✴✷✽

slide-17
SLIDE 17

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❈❛t❡❣♦r✐❡s ♦❢ GF (✷m) ▼✉❧t✐♣❧✐❡rs

❚❤❡r❡ ❛r❡ t✇♦ ❣❡♥❡r❛❧ ❝❛t❡❣♦r✐❡s ❢♦r GF (✷m) ♠✉❧t✐♣❧✐❡rs✿ P❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥s✳

❍✐❣❤ ❛r❡❛ ❛♥❞ t❤r♦✉❣❤♣✉t ❘❡s✉❧t ✐♥ ❛ s✐♥❣❧❡ ❝❧♦❝❦ ❝②❝❧❡✳ ❆ttr❛❝t✐✈❡ ❢♦r ❤✐❣❤ ♣❡r❢♦r♠❛♥❝❡ ❛♣♣❧✐❝❛t✐♦♥s✳

❇✐t✲❧❡✈❡❧ ✭❇▲✮ ❝♦♠♣✉t❛t✐♦♥s✳

▲♦✇❡st ❛r❡❛ ❘❡s✉❧t ✐♥ m ❝❧♦❝❦ ❝②❝❧❡s ❆ttr❛❝t✐✈❡ ❢♦r r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳

A B AB

m m m

GF(2m) Multiplication 1 clock cycle m clock cycles

Parallel Bit-Level

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✶✴✷✽

slide-18
SLIDE 18

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❈❛t❡❣♦r✐❡s ♦❢ GF (✷m) ▼✉❧t✐♣❧✐❡rs

❚❤❡r❡ ❛r❡ t✇♦ ❣❡♥❡r❛❧ ❝❛t❡❣♦r✐❡s ❢♦r GF (✷m) ♠✉❧t✐♣❧✐❡rs✿ P❛r❛❧❧❡❧ ❝♦♠♣✉t❛t✐♦♥s✳

❍✐❣❤ ❛r❡❛ ❛♥❞ t❤r♦✉❣❤♣✉t ❘❡s✉❧t ✐♥ ❛ s✐♥❣❧❡ ❝❧♦❝❦ ❝②❝❧❡✳ ❆ttr❛❝t✐✈❡ ❢♦r ❤✐❣❤ ♣❡r❢♦r♠❛♥❝❡ ❛♣♣❧✐❝❛t✐♦♥s✳

❇✐t✲❧❡✈❡❧ ✭❇▲✮ ❝♦♠♣✉t❛t✐♦♥s✳

▲♦✇❡st ❛r❡❛ ❘❡s✉❧t ✐♥ m ❝❧♦❝❦ ❝②❝❧❡s ❆ttr❛❝t✐✈❡ ❢♦r r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳

A B AB

m m m

GF(2m) Multiplication 1 clock cycle m clock cycles

Parallel Bit-Level

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✶✴✷✽

slide-19
SLIDE 19

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

✶✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❙■P❖✮ ❬✶❪✿ ❖♥❡ ✐♥♣✉t✱ A ∈ GF(✷m), ✐s ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥✳ ❆♥♦t❤❡r ✐♥♣✉t✱ B, ❡♥t❡rs ❜✐t✲❜②✲❜✐t ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

AB

m

A

m 1

B

loading generate m iterations

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✷✴✷✽

slide-20
SLIDE 20

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

✶✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❙■P❖✮ ❬✶❪✿ ❖♥❡ ✐♥♣✉t✱ A ∈ GF(✷m), ✐s ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥✳ ❆♥♦t❤❡r ✐♥♣✉t✱ B, ❡♥t❡rs ❜✐t✲❜②✲❜✐t ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

AB

m

A

m 1

B

loading generate m iterations

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✷✴✷✽

slide-21
SLIDE 21

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

✶✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❙■P❖✮ ❬✶❪✿ ❖♥❡ ✐♥♣✉t✱ A ∈ GF(✷m), ✐s ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥✳ ❆♥♦t❤❡r ✐♥♣✉t✱ B, ❡♥t❡rs ❜✐t✲❜②✲❜✐t ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

AB

m

A

m 1

B

loading generate m iterations

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✷✴✷✽

slide-22
SLIDE 22

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

✶✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❙■P❖✮ ❬✶❪✿ ❖♥❡ ✐♥♣✉t✱ A ∈ GF(✷m), ✐s ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥✳ ❆♥♦t❤❡r ✐♥♣✉t✱ B, ❡♥t❡rs ❜✐t✲❜②✲❜✐t ❞✉r✐♥❣ ❝♦♠♣✉t❛t✐♦♥✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

AB

m

A

m 1

B

loading generate m iterations

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✷✴✷✽

slide-23
SLIDE 23

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✷✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲❙■❙❖✮ ❬✷✱ ✸❪✿ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ ❜✐t✲❜②✲❜✐t ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ❜✐t✲❜②✲❜✐t ■t t❛❦❡s ✷m ❝❧♦❝❦ ❝②❝❧❡s t♦ ❣❡♥❡r❛t❡ t❤❡ r❡s✉❧t✳

AB

1

A

1 1

B

2m iterations

❬✷❪ ▼✳ ❍❛s❛♥ ❛♥❞ ❱✳ ❇❤❛r❣❛✈❛✱ ✏❉✐✈✐s✐♦♥ ❛♥❞ ❇✐t✲❙❡r✐❛❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ●❋✭qm✮✱✑ ❈♦♠♣✉t❡rs ❛♥❞ ❉✐❣✐t❛❧ ❚❡❝❤♥✐q✉❡s✱ ■❊❊ Pr♦❝❡❡❞✐♥❣s ❊✱ ✈♦❧✳ ✶✸✾✱ ♣♣✳ ✷✸✵✕✷✸✻✱ ▼❛② ✶✾✾✷ ❬✸❪ ❆✳ ❆❧✲❑❤♦r❛✐❞❧② ❛♥❞ ▼✳ ❑✳ ■❜r❛❤✐♠✱ ✏❋✐♥✐t❡ ✜❡❧❞ s❡r✐❛❧✲s❡r✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✴r❡❞✉❝t✐♦♥ str✉❝t✉r❡ ❛♥❞ ♠❡t❤♦❞✱✑ ❆♣r✳ ✷✵✵✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✸✴✷✽

slide-24
SLIDE 24

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✷✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲❙■❙❖✮ ❬✷✱ ✸❪✿ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ ❜✐t✲❜②✲❜✐t ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ❜✐t✲❜②✲❜✐t ■t t❛❦❡s ✷m ❝❧♦❝❦ ❝②❝❧❡s t♦ ❣❡♥❡r❛t❡ t❤❡ r❡s✉❧t✳

AB

1

A

1 1

B

2m iterations

❬✷❪ ▼✳ ❍❛s❛♥ ❛♥❞ ❱✳ ❇❤❛r❣❛✈❛✱ ✏❉✐✈✐s✐♦♥ ❛♥❞ ❇✐t✲❙❡r✐❛❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ●❋✭qm✮✱✑ ❈♦♠♣✉t❡rs ❛♥❞ ❉✐❣✐t❛❧ ❚❡❝❤♥✐q✉❡s✱ ■❊❊ Pr♦❝❡❡❞✐♥❣s ❊✱ ✈♦❧✳ ✶✸✾✱ ♣♣✳ ✷✸✵✕✷✸✻✱ ▼❛② ✶✾✾✷ ❬✸❪ ❆✳ ❆❧✲❑❤♦r❛✐❞❧② ❛♥❞ ▼✳ ❑✳ ■❜r❛❤✐♠✱ ✏❋✐♥✐t❡ ✜❡❧❞ s❡r✐❛❧✲s❡r✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✴r❡❞✉❝t✐♦♥ str✉❝t✉r❡ ❛♥❞ ♠❡t❤♦❞✱✑ ❆♣r✳ ✷✵✵✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✸✴✷✽

slide-25
SLIDE 25

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✷✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲❙■❙❖✮ ❬✷✱ ✸❪✿ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ ❜✐t✲❜②✲❜✐t ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ❜✐t✲❜②✲❜✐t ■t t❛❦❡s ✷m ❝❧♦❝❦ ❝②❝❧❡s t♦ ❣❡♥❡r❛t❡ t❤❡ r❡s✉❧t✳

AB

1

A

1 1

B

2m iterations

❬✷❪ ▼✳ ❍❛s❛♥ ❛♥❞ ❱✳ ❇❤❛r❣❛✈❛✱ ✏❉✐✈✐s✐♦♥ ❛♥❞ ❇✐t✲❙❡r✐❛❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ●❋✭qm✮✱✑ ❈♦♠♣✉t❡rs ❛♥❞ ❉✐❣✐t❛❧ ❚❡❝❤♥✐q✉❡s✱ ■❊❊ Pr♦❝❡❡❞✐♥❣s ❊✱ ✈♦❧✳ ✶✸✾✱ ♣♣✳ ✷✸✵✕✷✸✻✱ ▼❛② ✶✾✾✷ ❬✸❪ ❆✳ ❆❧✲❑❤♦r❛✐❞❧② ❛♥❞ ▼✳ ❑✳ ■❜r❛❤✐♠✱ ✏❋✐♥✐t❡ ✜❡❧❞ s❡r✐❛❧✲s❡r✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✴r❡❞✉❝t✐♦♥ str✉❝t✉r❡ ❛♥❞ ♠❡t❤♦❞✱✑ ❆♣r✳ ✷✵✵✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✸✴✷✽

slide-26
SLIDE 26

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✷✳ ❇✐t✲❧❡✈❡❧ ❙❡r✐❛❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲❙■❙❖✮ ❬✷✱ ✸❪✿ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ ❜✐t✲❜②✲❜✐t ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ❜✐t✲❜②✲❜✐t ■t t❛❦❡s ✷m ❝❧♦❝❦ ❝②❝❧❡s t♦ ❣❡♥❡r❛t❡ t❤❡ r❡s✉❧t✳

AB

1

A

1 1

B

2m iterations

❬✷❪ ▼✳ ❍❛s❛♥ ❛♥❞ ❱✳ ❇❤❛r❣❛✈❛✱ ✏❉✐✈✐s✐♦♥ ❛♥❞ ❇✐t✲❙❡r✐❛❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ♦✈❡r ●❋✭qm✮✱✑ ❈♦♠♣✉t❡rs ❛♥❞ ❉✐❣✐t❛❧ ❚❡❝❤♥✐q✉❡s✱ ■❊❊ Pr♦❝❡❡❞✐♥❣s ❊✱ ✈♦❧✳ ✶✸✾✱ ♣♣✳ ✷✸✵✕✷✸✻✱ ▼❛② ✶✾✾✷ ❬✸❪ ❆✳ ❆❧✲❑❤♦r❛✐❞❧② ❛♥❞ ▼✳ ❑✳ ■❜r❛❤✐♠✱ ✏❋✐♥✐t❡ ✜❡❧❞ s❡r✐❛❧✲s❡r✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✴r❡❞✉❝t✐♦♥ str✉❝t✉r❡ ❛♥❞ ♠❡t❤♦❞✱✑ ❆♣r✳ ✷✵✵✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✸✴✷✽

slide-27
SLIDE 27

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✸✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲P■❙❖✮ ❬✹❪✳ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖♥❡ ♦✉t♣✉t ❜✐t ✐s ❣❡♥❡r❛t❡❞ ♣❡r ❛ ❝❧♦❝❦ ❝②❝❧❡✳ ❆❧❧ ♦✉t♣✉t ❜✐ts ❛r❡ ❣❡♥❡r❛t❡❞ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

A AB

m m

B

1

m iterations loading

❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✹✴✷✽

slide-28
SLIDE 28

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✸✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲P■❙❖✮ ❬✹❪✳ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖♥❡ ♦✉t♣✉t ❜✐t ✐s ❣❡♥❡r❛t❡❞ ♣❡r ❛ ❝❧♦❝❦ ❝②❝❧❡✳ ❆❧❧ ♦✉t♣✉t ❜✐ts ❛r❡ ❣❡♥❡r❛t❡❞ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

A AB

m m

B

1

m iterations loading

❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✹✴✷✽

slide-29
SLIDE 29

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✸✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲P■❙❖✮ ❬✹❪✳ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖♥❡ ♦✉t♣✉t ❜✐t ✐s ❣❡♥❡r❛t❡❞ ♣❡r ❛ ❝❧♦❝❦ ❝②❝❧❡✳ ❆❧❧ ♦✉t♣✉t ❜✐ts ❛r❡ ❣❡♥❡r❛t❡❞ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

A AB

m m

B

1

m iterations loading

❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✹✴✷✽

slide-30
SLIDE 30

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✸✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ ❙❡r✐❛❧✲♦✉t ✭❇▲✲P■❙❖✮ ❬✹❪✳ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖♥❡ ♦✉t♣✉t ❜✐t ✐s ❣❡♥❡r❛t❡❞ ♣❡r ❛ ❝❧♦❝❦ ❝②❝❧❡✳ ❆❧❧ ♦✉t♣✉t ❜✐ts ❛r❡ ❣❡♥❡r❛t❡❞ ❛❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s✳

A AB

m m

B

1

m iterations loading

❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✹✴✷✽

slide-31
SLIDE 31

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✹✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲P■P❖✮ ❬✺❪✿ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r ✷tω−✷ +✶ ❝❧♦❝❦ ❝②❝❧❡s✳

❋✐❡❧❞ ♣♦❧②♥♦♠✐❛❧ ✐s p (x) = xm +∑ω−✷

i=✶ xti +✶✳

AB

m

A

m m

B

loading generate 2tw-2+1 iterations

❬✺❪ ❏✳ ■♠❛ñ❛✱ ✏▲♦✇ ▲❛t❡♥❝② ●❋✭✷m✮ P♦❧②♥♦♠✐❛❧ ❇❛s✐s ▼✉❧t✐♣❧✐❡r✱✑ ■❊❊❊ ❚r❛♥s✳ ❈✐r❝✉✐ts ❙②st✳ ■✱ ❘❡❣✳ P❛♣❡rs✱ ✈♦❧✳ ✺✽✱ ♣♣✳ ✾✸✺✕✾✹✻✱ ▼❛② ✷✵✶✶ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✺✴✷✽

slide-32
SLIDE 32

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✹✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲P■P❖✮ ❬✺❪✿ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r ✷tω−✷ +✶ ❝❧♦❝❦ ❝②❝❧❡s✳

❋✐❡❧❞ ♣♦❧②♥♦♠✐❛❧ ✐s p (x) = xm +∑ω−✷

i=✶ xti +✶✳

AB

m

A

m m

B

loading generate 2tw-2+1 iterations

❬✺❪ ❏✳ ■♠❛ñ❛✱ ✏▲♦✇ ▲❛t❡♥❝② ●❋✭✷m✮ P♦❧②♥♦♠✐❛❧ ❇❛s✐s ▼✉❧t✐♣❧✐❡r✱✑ ■❊❊❊ ❚r❛♥s✳ ❈✐r❝✉✐ts ❙②st✳ ■✱ ❘❡❣✳ P❛♣❡rs✱ ✈♦❧✳ ✺✽✱ ♣♣✳ ✾✸✺✕✾✹✻✱ ▼❛② ✷✵✶✶ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✺✴✷✽

slide-33
SLIDE 33

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✹✳ ❇✐t✲❧❡✈❡❧ P❛r❛❧❧❡❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲P■P❖✮ ❬✺❪✿ ❚✇♦ ✐♥♣✉ts ❛r❡ ♣r❡✲❧♦❛❞❡❞ ❜❡❢♦r❡ ❝♦♠♣✉t❛t✐♦♥s✳ ❖✉t♣✉t ✐s ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r ✷tω−✷ +✶ ❝❧♦❝❦ ❝②❝❧❡s✳

❋✐❡❧❞ ♣♦❧②♥♦♠✐❛❧ ✐s p (x) = xm +∑ω−✷

i=✶ xti +✶✳

AB

m

A

m m

B

loading generate 2tw-2+1 iterations

❬✺❪ ❏✳ ■♠❛ñ❛✱ ✏▲♦✇ ▲❛t❡♥❝② ●❋✭✷m✮ P♦❧②♥♦♠✐❛❧ ❇❛s✐s ▼✉❧t✐♣❧✐❡r✱✑ ■❊❊❊ ❚r❛♥s✳ ❈✐r❝✉✐ts ❙②st✳ ■✱ ❘❡❣✳ P❛♣❡rs✱ ✈♦❧✳ ✺✽✱ ♣♣✳ ✾✸✺✕✾✹✻✱ ▼❛② ✷✵✶✶ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✺✴✷✽

slide-34
SLIDE 34

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-35
SLIDE 35

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-36
SLIDE 36

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-37
SLIDE 37

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-38
SLIDE 38

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-39
SLIDE 39

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-40
SLIDE 40

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s ✲ ❈♦♥t✐♥✉❡❞

✺✳ ❇✐t✲❧❡✈❡❧ ❋✉❧❧② ❙❡r✐❛❧✲✐♥ P❛r❛❧❧❡❧✲♦✉t ✭❇▲✲❋❙■P❖✮✿ ◆♦ ♣r❡✲❧♦❛❞✐♥❣ ✐s r❡q✉✐r❡❞❀ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ✇❤✐❧❡ ✐♥♣✉ts ❡♥t❡r✳ ❘❡q✉✐r❡s ❧♦✇ ✐♥♣✉t ❞❛t❛✲♣❛t❤ r❡s♦✉r❝❡s✳ ❖✉t♣✉t ❣❡♥❡r❛t❡❞ ✐♥ ♣❛r❛❧❧❡❧ ❛❢t❡r m ❝②❝❧❡s✳ ■♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ ✐♥♣✉t ❞❛t❛✲♣❛t❤ ❝♦♥str❛✐♥❡❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ❊①✐sts ♦♥❧② ❢♦r ◆♦r♠❛❧ ❇❛s✐s ❬✻❪✳

AB

m

A

1 1

B

m iterations generate

■♥ t❤✐s ♣❛♣❡r✱ ❛ ♥❡✇ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ✉s✐♥❣ P❇ ✐s ♣r♦♣♦s❡❞✳

❬✻❪ ●✳✲▲✳ ❋❡♥❣✱ ✏❆ ❱▲❙■ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋❛st ■♥✈❡rs✐♦♥ ✐♥ ●❋✭✷m✮✱✑ ■❊❊❊ ❚r❛♥s✳ ❈♦♠♣✉t✳✱ ✈♦❧✳ ✸✽✱ ♥♦✳ ✶✵✱ ♣♣✳ ✶✸✽✸✕✶✸✽✻✱ ✶✾✽✾ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✻✴✷✽

slide-41
SLIDE 41

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✼✴✷✽

slide-42
SLIDE 42

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

▼❙❇✲✜rst ❈♦♥str✉❝t✐♦♥ ♦❢ ❋✐❡❧❞ ❊❧❡♠❡♥ts

▲❡t A = (am−✶,··· ,a✶,a✵) = ∑m−✶

i=✵ aiαi ∈ GF(✷m)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❝♦♥str✉❝ts A ❜✐t✲❜②✲❜✐t st❛rt✐♥❣ ❛t ▼❙❇✲✜rst✳ A(i) =am−✶−i +A(i−✶)α. ■t❡r❛t❡ A(i) ∈ GF(✷m) ❢r♦♠ i = ✵ t♦ m −✶✳

A(−✶) = ✵ = (✵,··· ,✵,✵)✳ A(✵) = am−✶ = (✵,···✵,am−✶) A(✶) = am−✶α +am−✷ = (✵,··· ,am−✶,am−✷) A = A(m−✶) = (am−✶,··· ,a✶,a✵)✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✽✴✷✽

slide-43
SLIDE 43

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

▼❙❇✲✜rst ❈♦♥str✉❝t✐♦♥ ♦❢ ❋✐❡❧❞ ❊❧❡♠❡♥ts

▲❡t A = (am−✶,··· ,a✶,a✵) = ∑m−✶

i=✵ aiαi ∈ GF(✷m)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❝♦♥str✉❝ts A ❜✐t✲❜②✲❜✐t st❛rt✐♥❣ ❛t ▼❙❇✲✜rst✳ A(i) =am−✶−i +A(i−✶)α. ■t❡r❛t❡ A(i) ∈ GF(✷m) ❢r♦♠ i = ✵ t♦ m −✶✳

A(−✶) = ✵ = (✵,··· ,✵,✵)✳ A(✵) = am−✶ = (✵,···✵,am−✶) A(✶) = am−✶α +am−✷ = (✵,··· ,am−✶,am−✷) A = A(m−✶) = (am−✶,··· ,a✶,a✵)✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✽✴✷✽

slide-44
SLIDE 44

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

▼❙❇✲✜rst ❈♦♥str✉❝t✐♦♥ ♦❢ ❋✐❡❧❞ ❊❧❡♠❡♥ts

▲❡t A = (am−✶,··· ,a✶,a✵) = ∑m−✶

i=✵ aiαi ∈ GF(✷m)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❝♦♥str✉❝ts A ❜✐t✲❜②✲❜✐t st❛rt✐♥❣ ❛t ▼❙❇✲✜rst✳ A(i) =am−✶−i +A(i−✶)α. ■t❡r❛t❡ A(i) ∈ GF(✷m) ❢r♦♠ i = ✵ t♦ m −✶✳

A(−✶) = ✵ = (✵,··· ,✵,✵)✳ A(✵) = am−✶ = (✵,···✵,am−✶) A(✶) = am−✶α +am−✷ = (✵,··· ,am−✶,am−✷) A = A(m−✶) = (am−✶,··· ,a✶,a✵)✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✽✴✷✽

slide-45
SLIDE 45

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

▼❙❇✲✜rst ❈♦♥str✉❝t✐♦♥ ♦❢ ❋✐❡❧❞ ❊❧❡♠❡♥ts

▲❡t A = (am−✶,··· ,a✶,a✵) = ∑m−✶

i=✵ aiαi ∈ GF(✷m)

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❝♦♥str✉❝ts A ❜✐t✲❜②✲❜✐t st❛rt✐♥❣ ❛t ▼❙❇✲✜rst✳ A(i) =am−✶−i +A(i−✶)α. ■t❡r❛t❡ A(i) ∈ GF(✷m) ❢r♦♠ i = ✵ t♦ m −✶✳

A(−✶) = ✵ = (✵,··· ,✵,✵)✳ A(✵) = am−✶ = (✵,···✵,am−✶) A(✶) = am−✶α +am−✷ = (✵,··· ,am−✶,am−✷) A = A(m−✶) = (am−✶,··· ,a✶,a✵)✳

A(-1) A(0) A(m-1) am-1 a1 am-1 a0 A(1) am-1am-2

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✽✴✷✽

slide-46
SLIDE 46

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❑❡② ❋♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ Pr♦♣♦s❡❞ ▼✉❧t✐♣❧✐❡r

Pr♦♣♦s✐t✐♦♥ ✶✿ ▲❡t A ❛♥❞ B ❜❡ t✇♦ ❛r❜✐tr❛r② GF (✷m) ❡❧❡♠❡♥ts r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ P❇ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❞❡❣r❡❡ m ✐rr❡❞✉❝✐❜❧❡ ♣♦❧②♥♦♠✐❛❧ p (x) = xm +∑ω−✷

i=✶ xti +✶ ✇✐t❤ ω

♥♦♥③❡r♦ t❡r♠s✳ ▲❡t ✉s ❞❡✜♥❡ Ci = A(i)B(i) ♠♦❞ p (α) ✱ ✇❤❡r❡ A(i) ❛♥❞ B(i) ❛r❡ ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s A(i) =am−✶−i +A(i−✶)α B(i) =bm−✶−i +B(i−✶)α ✇❤❡r❡ i = ✵,...,m −✶✱ A(−✶) = B(−✶) = ✵✱ ❛♥❞ α ✐s t❤❡ r♦♦t ♦❢ p (x)✳ ❚❤❡♥✱ ❜❛s❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rr❡♥❝❡ ♦♥ Ci✱ ♦♥❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ A ❛♥❞ B✱ ❛s AB = Cm−✶✿ Ci =am−✶−ibm−✶−i +

  • am−✶−iB(i−✶) +bm−✶−iA(i−✶)

α +Ci−✶α✷ ♠♦❞ p (α). i = ✵,...,m −✶✱ ✇❤❡r❡ C−✶ = A(−✶)B(−✶) ♠♦❞ p (α) = ✵✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✶✾✴✷✽

slide-47
SLIDE 47

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✵✴✷✽

slide-48
SLIDE 48

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

Pr♦♣♦s❡❞ ❆r❝❤✐t❡❝t✉r❡

Ci = am−✶−ibm−✶−i +

  • am−✶−iB(i−✶) +bm−✶−iA(i−✶)

α +Ci−✶α✷ ♠♦❞ p (α).

▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② α ✐s ❥✉st ❛ ❧❡❢t s❤✐❢t✳ ❖♥❧② r❡❞✉❝t✐♦♥ ✐s ❞✉❡ t♦ ♠✉❧t✐♣❧② ❜② α✷✳ ❚❤r❡❡ r❡❣✐st❡rs X, Y , ❛♥❞ Z ❛r❡ ✐♥✐t✐❛❧❧② ❝❧❡❛r❡❞✳ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ t♦ X ❛♥❞ Y s❡r✐❛❧❧② ❢r♦♠ ▼❙❇✳ ❆❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s Z ❝♦♥t❛✐♥s AB ♠♦❞ p (α).

  • ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤

◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✶✴✷✽

slide-49
SLIDE 49

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

Pr♦♣♦s❡❞ ❆r❝❤✐t❡❝t✉r❡

Ci = am−✶−ibm−✶−i +

  • am−✶−iB(i−✶) +bm−✶−iA(i−✶)

α +Ci−✶α✷ ♠♦❞ p (α).

▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② α ✐s ❥✉st ❛ ❧❡❢t s❤✐❢t✳ ❖♥❧② r❡❞✉❝t✐♦♥ ✐s ❞✉❡ t♦ ♠✉❧t✐♣❧② ❜② α✷✳ ❚❤r❡❡ r❡❣✐st❡rs X, Y , ❛♥❞ Z ❛r❡ ✐♥✐t✐❛❧❧② ❝❧❡❛r❡❞✳ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ t♦ X ❛♥❞ Y s❡r✐❛❧❧② ❢r♦♠ ▼❙❇✳ ❆❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s Z ❝♦♥t❛✐♥s AB ♠♦❞ p (α).

  • ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤

◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✶✴✷✽

slide-50
SLIDE 50

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

Pr♦♣♦s❡❞ ❆r❝❤✐t❡❝t✉r❡

Ci = am−✶−ibm−✶−i +

  • am−✶−iB(i−✶) +bm−✶−iA(i−✶)

α +Ci−✶α✷ ♠♦❞ p (α).

▼✉❧t✐♣❧✐❝❛t✐♦♥ ❜② α ✐s ❥✉st ❛ ❧❡❢t s❤✐❢t✳ ❖♥❧② r❡❞✉❝t✐♦♥ ✐s ❞✉❡ t♦ ♠✉❧t✐♣❧② ❜② α✷✳ ❚❤r❡❡ r❡❣✐st❡rs X, Y , ❛♥❞ Z ❛r❡ ✐♥✐t✐❛❧❧② ❝❧❡❛r❡❞✳ ■♥♣✉ts ❛r❡ ❡♥t❡r❡❞ t♦ X ❛♥❞ Y s❡r✐❛❧❧② ❢r♦♠ ▼❙❇✳ ❆❢t❡r m ❝❧♦❝❦ ❝②❝❧❡s Z ❝♦♥t❛✐♥s AB ♠♦❞ p (α).

1 1

<Z>

  • m

m m m-1 m-1 m-1

m 1

  • b

i

  • m 1
  • b

b

m 1

  • a

i

  • m 1
  • a

a m-1 m-1 1 m B(i-1) A(i-1) am-1-iB(i-1) bm-1-iA(i-1) Ci-1 am-1-ibm-1-i am-1-ibm-1-i + (bm-1-iA(i-1)+am-1-iB(i-1)) Ci-12 mod p() Ci

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✶✴✷✽

slide-51
SLIDE 51

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s

❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s

Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✷✴✷✽

slide-52
SLIDE 52

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♠♣❧❡①✐t✐❡s

Pr♦♣♦s✐t✐♦♥ ✷✿ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❣❛t❡s ❛♥❞ ♣r♦♣❛❣❛t✐♦♥ ❞❡❧❛② ✭P❉✮ ✐♥ t❤❡ ♣r♦♣♦s❡❞ P❇ ♠✉❧t✐♣❧✐❡r ♦✈❡r GF(✷m) ❛r❡ ❛s ❢♦❧❧♦✇s✿

  • #ANDs = ✷m −✶,

#FFs = ✸m −✷, #XORs = ✷m +Nα✷ −✶, ❛♥❞ PD =♠❛①{Tα✷ +TX ,TA +✷TX }. ❋♦r t❤❡ ✜✈❡ ◆■❙❚ r❡❝♦♠♠❡♥❞❡❞ ✜❡❧❞s ❢♦r ❊❈❉❙❆✱ t❤❡ ❝♦♠♣❧❡①✐t✐❡s ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② α✷ ❛r❡ ❛s ❢♦❧❧♦✇s✿ m ✶✻✸ ✷✸✸ ✷✽✸ ✹✵✾ ✺✼✶ Nα✷ ✻ ✷ ✻ ✷ ✻ Tα✷ ✷TX TX TX TX TX

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✸✴✷✽

slide-53
SLIDE 53

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❙♣❛❝❡ ❈♦♠♣❧❡①✐t② ❈♦♠♣❛r✐s♦♥

▼✉❧t✐♣❧✐❡r ❋❋ ❆◆❉ ❳❖❘ ✷ : ✶ ✶✲❜✐t ▼❯❳ ✷ : ✶ ✶✲❜✐t ▼❯❳ ❋✉♥❝t✐♦♥❛❧ P❛r❛❧❧❡❧ ▲♦❛❞✐♥❣ ▼❙❇ ❇▲✲❙■P❖ ❬✶❪ ✷m m m +ω −✷ ✵ m LSB BL-SIPO [1] ✷m m m +ω −✷ ✵ m ❇▲✲P■❙❖ ❬✹❪ ✸m +tω−✷ −✶ ✷m −✶ (ω −✶)(m −✶)+ω −✸+∑ω−✷

i=✶ ti

✵ ✷m P■P❖ ❬✺❪ ✺m −✶

m✷+m ✷ m✷+m ✷

✹m ✷m ▼❙❇ ❇▲✲❋❙■P❖ ✭t❤✐s ✇♦r❦✮ ✸m −✷ ✷m −✶ ✷m +Nα✷ −✶ ✵ ✵ p (x) = xm +∑ω−✷

i=✶ xti +✶ ✐s t❤❡ ✜❡❧❞ ♣♦❧②♥♦♠✐❛❧✳

❚❤❡ ❛r❡❛ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❜✐t✲❧❡✈❡❧ ♠✉❧t✐♣❧✐❡r ✐s ❧❛r❣❡r t❤❛♥ t❤❡ ♦♥❡ ♣r♦♣♦s❡❞ ✐♥ ❇▲✲❙■P❖ ❬✶❪✳ ❝❧♦s❡ t♦ t❤❛t ♦❢ ❇▲✲P■❙❖ ❬✹❪✳ ❜❡tt❡r t❤❛♥ t❤❛t ♦❢ ❇▲✲P■P❖ ❬✺❪✳

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❬✺❪ ❏✳ ■♠❛ñ❛✱ ✏▲♦✇ ▲❛t❡♥❝② ●❋✭✷m✮ P♦❧②♥♦♠✐❛❧ ❇❛s✐s ▼✉❧t✐♣❧✐❡r✱✑ ■❊❊❊ ❚r❛♥s✳ ❈✐r❝✉✐ts ❙②st✳ ■✱ ❘❡❣✳ P❛♣❡rs✱ ✈♦❧✳ ✺✽✱ ♣♣✳ ✾✸✺✕✾✹✻✱ ▼❛② ✷✵✶✶ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✹✴✷✽

slide-54
SLIDE 54

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❚✐♠❡ ❈♦♠♣❧❡①✐t② ❈♦♠♣❛r✐s♦♥

▼✉❧t✐♣❧✐❡r Pr♦♣❛❣❛t✐♦♥ ❙❡r✐❛❧ Pr❡✲❧♦❛❞✐♥❣ ❈♦♠♣✉t❛t✐♦♥ ❉❡❧❛② ▲❛t❡♥❝② ▲❛t❡♥❝② ▼❙❇ ❇▲✲❙■P❖ ❬✶❪ TA +TX m m LSB BL-SIPO [1] TA +TX m m ❇▲✲P■❙❖ ❬✹❪ TA +

  • ✶+⌈❧♦❣✷ (ω −✶)⌉+⌈❧♦❣✷ (m)⌉
  • TX

m m P■P❖ ❬✺❪ TA +⌈❧♦❣✷ m⌉TX +✷TM m ✷tω−✷ +✶ ▼❙❇ ❇▲✲❋❙■P❖ ✭t❤✐s ✇♦r❦✮ ♠❛①{Tα✷ +TX,TA +✷TX} ✵ m TA✱ TX✱ ❛♥❞ TM ❞❡♥♦t❡ t❤❡ ❞❡❧❛② ✐♥ ✷✲✐♥♣✉ts ❆◆❉✱ ❳❖❘✱ ❛♥❞ ✷✲t♦✲✶ ▼❯❳✱ r❡s♣❡❝t✐✈❡❧②✳ p (x) = xm +∑ω−✷

i=✶ xti +✶ ✐s t❤❡ ✜❡❧❞ ♣♦❧②♥♦♠✐❛❧✳

❙♣❡❡❞ ♦❢ ❇▲✲❋❙■P❖ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ m✱ s✐♠✐❧❛r t♦ ❇▲✲❙■P❖✳ ❇▲✲❋❙■P❖ ❤❛s s❧✐❣❤t❧② ❧♦✇❡r s♣❡❡❞ ❝♦♠♣❛r❡❞ t♦ ❇▲✲❙■P❖✳ ❇▲✲❋❙■P❖ ❤❛s ❤✐❣❤❡r s♣❡❡❞ ❝♦♠♣❛r❡❞ t♦ ❇▲✲P■❙❖ ❛♥❞ ❇▲✲P■P❖✳ ❚❤❡ ♣r♦♣♦s❡❞ ❇▲✲❋❙■P❖ ♠✉❧t✐♣❧✐❡r ♦✛❡rs t❤❡ ❤✐❣❤❡st t❤r♦✉❣❤♣✉t ✐❢ ✐♥♣✉ts ❛r❡ s❡r✐❛❧❧② ❧♦❛❞❡❞ ♦♥❡ ❜✐t ❛t ❛ t✐♠❡✳

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❬✺❪ ❏✳ ■♠❛ñ❛✱ ✏▲♦✇ ▲❛t❡♥❝② ●❋✭✷m✮ P♦❧②♥♦♠✐❛❧ ❇❛s✐s ▼✉❧t✐♣❧✐❡r✱✑ ■❊❊❊ ❚r❛♥s✳ ❈✐r❝✉✐ts ❙②st✳ ■✱ ❘❡❣✳ P❛♣❡rs✱ ✈♦❧✳ ✺✽✱ ♣♣✳ ✾✸✺✕✾✹✻✱ ▼❛② ✷✵✶✶ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✺✴✷✽

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SLIDE 55

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❋♦r♠✉❧❛t✐♦♥s ❆r❝❤✐t❡❝t✉r❡ ❈♦♠♣❧❡①✐t✐❡s ❛♥❞ ❈♦♠♣❛r✐s♦♥

❈♦♠♣❛r✐s♦♥s✿ p (x) = x✷✸✸ +x✼✹ +✶

▼✉❧t✐♣❧✐❡r ▼P❉

▲❛t❡♥❝② ❚P✴● ❅ ✶ ●❍③ ns P■▲ ❙■▲ P■▲ ❙■▲ P■▲ ❙■▲ ▲❙❇ ❇▲✲❙■P❖ ❬✶❪ ✵.✵✼ ✷✾✼✸ ✷✺✵✼ ✷✸✸ ✹✻✻ ✸✸✻ ✶✾✾ ▼❙❇ ❇▲✲❙■P❖ ❬✶❪ ✵.✵✼ ✷✾✼✸ ✷✺✵✼ ✷✸✸ ✹✻✻ ✸✸✻ ✶✾✾ ❇▲✲P■❙❖ ❬✹❪ ✵.✹✸ ✺✹✽✹ ✹✺✺✷ ✷✸✸ ✹✻✻ ✶✽✷ ✶✶✵ P■P❖ ❬✺❪ ✵.✹✶ ✾✺✼✺✾ ✾✹✽✷✼ ✶✹✾ ✸✽✷ ✶✻ ✻ ▼❙❇ ❇▲✲❋❙■P❖ ✵.✶✶ ✹✶✷✾ ✹✶✷✾ ✷✸✸ ✷✸✸ ✷✹✷ ✷✹✷

▼P❉ ❂ ♠❛①✐♠✉♠ ♣r♦♣❛❣❛t✐♦♥ ❞❡❧❛②✳ ●❊ ❂ ★ ◆❆◆❉ ❡q✉✐✈❛❧❡♥❝❡✳ P■▲ ❂ ♣❛r❛❧❧❡❧ ✐♥♣✉t ❧♦❛❞✐♥❣✳ ❙■▲ ❂ s❡r✐❛❧ ✐♥♣✉t ❧♦❛❞✐♥❣✳ ❚P✴● ❂ ♥♦r♠❛❧✐③❡❞ t❤r♦✉❣❤♣✉t✳ ❊st✐♠❛t❡s ❜❛s❡❞ ♦♥ ✻✺♥♠ ❙❚▼✐❝r♦❡❧❡❝tr♦♥✐❝s st❛♥❞❛r❞ ❈▼❖❙ ❧✐❜r❛r②✳

❬✶❪ ❚✳ ❇❡t❤ ❛♥❞ ❉✳ ●♦❧❧♠❛♥✱ ✏❆❧❣♦r✐t❤♠ ❊♥❣✐♥❡❡r✐♥❣ ❢♦r P✉❜❧✐❝ ❑❡② ❆❧❣♦r✐t❤♠s✱✑ ■❊❊❊ ❏✳ ❙❡❧✳ ❆r❡❛s ❈♦♠♠✉♥✳✱ ✈♦❧✳ ✼✱ ♥♦✳ ✹✱ ♣♣✳ ✹✺✽✕✹✻✻✱ ✶✾✽✾ ❬✹❪ ❆✳ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤✱ ✏❆ ◆❡✇ ❇✐t✲❙❡r✐❛❧ ❆r❝❤✐t❡❝t✉r❡ ❢♦r ❋✐❡❧❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❯s✐♥❣ P♦❧②♥♦♠✐❛❧ ❇❛s❡s✱✑ ✐♥ ❈r②♣t♦❣r❛♣❤✐❝ ❍❛r❞✇❛r❡ ❛♥❞ ❊♠❜❡❞❞❡❞ ❙②st❡♠s ✲ ❈❍❊❙ ✷✵✵✽ ✭❊✳ ❖s✇❛❧❞ ❛♥❞ P✳ ❘♦❤❛t❣✐✱ ❡❞s✳✮✱ ♥♦✳ ✺✶✺✹ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣♣✳ ✸✵✵✕✸✶✹✱ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥ ✷✵✵✽ ❬✺❪ ❏✳ ■♠❛ñ❛✱ ✏▲♦✇ ▲❛t❡♥❝② ●❋✭✷m✮ P♦❧②♥♦♠✐❛❧ ❇❛s✐s ▼✉❧t✐♣❧✐❡r✱✑ ■❊❊❊ ❚r❛♥s✳ ❈✐r❝✉✐ts ❙②st✳ ■✱ ❘❡❣✳ P❛♣❡rs✱ ✈♦❧✳ ✺✽✱ ♣♣✳ ✾✸✺✕✾✹✻✱ ▼❛② ✷✵✶✶ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✻✴✷✽

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SLIDE 56

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❈♦♥❝❧✉s✐♦♥✿ ❲❡ ❤❛✈❡ ♣r♦♣♦s❡❞ ❛ ♥❡✇ ▼❙❇✲✜rst ❜✐t✲❧❡✈❡❧ ❢✉❧❧② s❡r✐❛❧✲✐♥ ♣❛r❛❧❧❡❧✲♦✉t P❇ ♠✉❧t✐♣❧✐❡r✳ ❚❤❡ ♣r♦♣♦s❡❞ ♠✉❧t✐♣❧✐❡r ✐♠♣r♦✈❡s t❤r♦✉❣❤♣✉t ✐♥ r❡s♦✉r❝❡ ❝♦♥str❛✐♥❡❞ ❡♥✈✐r♦♥♠❡♥ts ✇✐t❤ ❧♦✇ ❞❛t❛✲♣❛t❤ ✐♥♣✉t✳ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s ✐s t❤❡ ✜rst t✐♠❡ t❤❛t s✉❝❤ ❛ ❇▲✲❋❙■P❖ P❇ ♠✉❧t✐♣❧✐❡r ✐s ♣r♦♣♦s❡❞✳ ❋✉t✉r❡ ❲♦r❦✿ ❍❛r❞✇❛r❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛r❝❤✐t❡❝t✉r❡ ❉✐❣✐t ❧❡✈❡❧ ❡①t❡♥s✐♦♥ ♦❢ ❛♥ ▼❙❉✲✜rst ❉▲✲❋❙■P❖ P❇ ♠✉❧t✐♣❧✐❡r✳ ❈♦♥str✉❝t ❛♥ ❛r❝❤✐t❡❝t✉r❡ ❢♦r ▲❙❉✲✜rst ❉▲✲❋❙■P❖ P❇ ♠✉❧t✐♣❧✐❡r✳

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✼✴✷✽

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SLIDE 57

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF (✷m) P❇ ▼✉❧t✐♣❧✐❡r ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦

❚❤❛♥❦ ❨♦✉✦

❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF (✷m) ▼✉❧t✐♣❧✐❡r ✷✽✴✷✽