t r GF ( m ) - PowerPoint PPT Presentation

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

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

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

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF ( ✷ m ) P❇ ▼✉❧t✐♣❧✐❡r ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ❆♣♣❧✐❝❛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✐♥❣✳ Digital Signal Cryptosystems Error Control Processing Arithmetic Operations (Finite Fields, Integer Arithmetic, � , etc) ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF ( ✷ m ) ▼✉❧t✐♣❧✐❡r ✹✴✷✽ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤

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

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

▼♦t✐✈❛t✐♦♥ ❛♥❞ ■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❋✐♥✐t❡ ❋✐❡❧❞s ❇✐t✲▲❡✈❡❧ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❙❝❤❡♠❡s P♦❧②♥♦♠✐❛❧ ❇❛s✐s ✭P❇✮ Pr♦♣♦s❡❞ ❇✐t✲▲❡✈❡❧ GF ( ✷ m ) P❇ ▼✉❧t✐♣❧✐❡r ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ 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 ( ✷ ) ✳ � � α m − ✶ ,..., α , ✶ ✐s ❦♥♦✇♥ ❛s t❤❡ P❇✱ ✇❤❡r❡ p ( α ) = ✵ . ❚❤❡♥✱ ❆♥② ✜❡❧❞ ❡❧❡♠❡♥t A = ( a m − ✶ , ··· , a ✶ , a ✵ ) ∈ GF ( ✷ m ) , a i ∈ { ✵ , ✶ } , ✐s r❡♣r❡s❡♥t❡❞ ✇✳r✳t✳ t❤❡ P❇ ❛s m − ✶ ∑ a i α i . A = i = ✵ ◆❡✇ ❇✐t✲▲❡✈❡❧ ❙❡r✐❛❧ GF ( ✷ m ) ▼✉❧t✐♣❧✐❡r ✼✴✷✽ ❍❛②ss❛♠ ❊❧✲❘❛③♦✉❦ ❛♥❞ ❆r❛s❤ ❘❡②❤❛♥✐✲▼❛s♦❧❡❤