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❑❡② Pr❡❞✐❝t✐♦♥ ❙❡❝✉r✐t② ♦❢ ❑❡②❡❞ ❙♣♦♥❣❡s ❇❛rt ▼❡♥♥✐♥❦ ❘❛❞❜♦✉❞ ❯♥✐✈❡rs✐t② ✭❚❤❡ ◆❡t❤❡r❧❛♥❞s✮ ❋❛st ❙♦❢t✇❛r❡ ❊♥❝r②♣t✐♦♥ ✷✵✶✾ ▼❛r❝❤ ✷✻✱ ✷✵✶✾ ✶ ✴ ✶✽

❙♣♦♥❣❡s ❬❇❉P❱✵✼❪ M pad trunc Z m pad trunc z r 0 0 r f f f f f f outer π π π π π π inner c 0 c 0 absorbing squeezing • ❈r②♣t♦❣r❛♣❤✐❝ ❤❛s❤ ❢✉♥❝t✐♦♥ • ❙❍❆✲✸✱ ❳❖❋s✱ ❧✐❣❤t✇❡✐❣❤t ❤❛s❤✐♥❣✱ ✳ ✳ ✳ • ❇❡❤❛✈❡s ❛s ❘❖ ✉♣ t♦ q✉❡r② ❝♦♠♣❧❡①✐t② ≈ 2 c/ 2 ❬❇❉P❱✵✽❪ ✷ ✴ ✶✽

■♥♥❡r✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❈❉❍❑◆✶✷✱❆❉▼❱✶✺✱◆❨✶✻❪ ❋✉❧❧✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❇❉P❱✶✷✱●P❚✶✺✱▼❘❱✶✺❪ ❑❡②❡❞ ❙♣♦♥❣❡s M pad trunc Z K � m pad trunc z r 0 0 r f f f f f f outer π π π π π π inner c 0 c 0 absorbing squeezing • ❖✉t❡r✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❇❉P❱✶✶✱❆❉▼❱✶✺✱◆❨✶✻❪ ✸ ✴ ✶✽

❋✉❧❧✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❇❉P❱✶✷✱●P❚✶✺✱▼❘❱✶✺❪ ❑❡②❡❞ ❙♣♦♥❣❡s M pad trunc Z m pad trunc z r 0 0 r f f f f f f outer π π π π π π inner c 0 c K absorbing squeezing • ❖✉t❡r✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❇❉P❱✶✶✱❆❉▼❱✶✺✱◆❨✶✻❪ • ■♥♥❡r✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❈❉❍❑◆✶✷✱❆❉▼❱✶✺✱◆❨✶✻❪ ✸ ✴ ✶✽

❑❡②❡❞ ❙♣♦♥❣❡s M pad trunc Z m pad trunc z r 0 0 r f f f f f f π π π π π π c 0 c K absorbing squeezing • ❖✉t❡r✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❇❉P❱✶✶✱❆❉▼❱✶✺✱◆❨✶✻❪ • ■♥♥❡r✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❈❉❍❑◆✶✷✱❆❉▼❱✶✺✱◆❨✶✻❪ • ❋✉❧❧✲❑❡②❡❞ ❙♣♦♥❣❡ ❬❇❉P❱✶✷✱●P❚✶✺✱▼❘❱✶✺❪ ✸ ✴ ✶✽

❙✐♠♣❧✐✜❡❞ ❙❡❝✉r✐t② ❇♦✉♥❞ ✿ ❞❛t❛ ✭❝♦♥str✉❝t✐♦♥✮ ❝♦♠♣❧❡①✐t② ✲ ✿ t✐♠❡ ✭♣r✐♠✐t✐✈❡✮ ❝♦♠♣❧❡①✐t② ♣r♦❜❛❜✐❧✐t② t❤❛t ❛❞✈❡rs❛r② ♣r❡❞✐❝ts ❦❡② ❙❡❝✉r✐t② ♦❢ ❑❡②❡❞ ❙♣♦♥❣❡ m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c → − c ❢♦r F = OKS ❛♥❞ 0 ❢♦r F = FKS • F ∈ { OKS , FKS } ✹ ✴ ✶✽

♣r♦❜❛❜✐❧✐t② t❤❛t ❛❞✈❡rs❛r② ♣r❡❞✐❝ts ❦❡② ❙❡❝✉r✐t② ♦❢ ❑❡②❡❞ ❙♣♦♥❣❡ m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c → − c ❢♦r F = OKS ❛♥❞ 0 ❢♦r F = FKS • F ∈ { OKS , FKS } ❙✐♠♣❧✐✜❡❞ ❙❡❝✉r✐t② ❇♦✉♥❞ • M ✿ ❞❛t❛ ✭❝♦♥str✉❝t✐♦♥✮ ❝♦♠♣❧❡①✐t② M 2 2 c + MN + Adv key ✲ pre ( N ) • N ✿ t✐♠❡ ✭♣r✐♠✐t✐✈❡✮ ❝♦♠♣❧❡①✐t② F 2 c ✹ ✴ ✶✽

❙❡❝✉r✐t② ♦❢ ❑❡②❡❞ ❙♣♦♥❣❡ m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c → − c ❢♦r F = OKS ❛♥❞ 0 ❢♦r F = FKS • F ∈ { OKS , FKS } ❙✐♠♣❧✐✜❡❞ ❙❡❝✉r✐t② ❇♦✉♥❞ • M ✿ ❞❛t❛ ✭❝♦♥str✉❝t✐♦♥✮ ❝♦♠♣❧❡①✐t② M 2 2 c + MN + Adv key ✲ pre ( N ) • N ✿ t✐♠❡ ✭♣r✐♠✐t✐✈❡✮ ❝♦♠♣❧❡①✐t② F 2 c ♣r♦❜❛❜✐❧✐t② t❤❛t ❛❞✈❡rs❛r② ♣r❡❞✐❝ts ❦❡② ✹ ✴ ✶✽

❑❡② Pr❡❞✐❝t✐♦♥ ❙❡❝✉r✐t② m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c Adv key ✲ pre ( N ) F • ❆❞✈❡rs❛r② ♠❛❦❡s N q✉❡r✐❡s t♦ π • ❑❡② K r❛♥❞♦♠❧② ❞r❛✇♥ • ❆❞✈❡rs❛r② ✇✐♥s ✐❢ q✉❡r② ❤✐st♦r② ✏❝♦✈❡rs K ✑ ✺ ✴ ✶✽

▼♦r❡ ❚❤❛♥ ❖♥❡ ❑❡② ❇❧♦❝❦ ✲ ❇② ●❛➸✐ ❡t ❛❧✳ ❬●P❚✶✺❪ ❯s❡❞ ✐♥ ♠❛♥② s♣♦♥❣❡ ♣r♦♦❢s ❑❡② Pr❡❞✐❝t✐♦♥ ❙❡❝✉r✐t②✿ ❊①✐st✐♥❣ ❇♦✉♥❞s m 1 m 2 m µ z 1 z 2 K 1 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c ❖♥❡ ❑❡② ❇❧♦❝❦ ( N ) ≤ N Adv key ✲ pre • ❆❞✈❡rs❛r② ♠❛❦❡s N q✉❡r✐❡s F 2 k • ◗✉❡r② ❤✐st♦r② ❝♦✈❡rs ❛t ♠♦st N ❦❡②s ✻ ✴ ✶✽

❑❡② Pr❡❞✐❝t✐♦♥ ❙❡❝✉r✐t②✿ ❊①✐st✐♥❣ ❇♦✉♥❞s m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c ❖♥❡ ❑❡② ❇❧♦❝❦ ( N ) ≤ N Adv key ✲ pre • ❆❞✈❡rs❛r② ♠❛❦❡s N q✉❡r✐❡s F 2 k • ◗✉❡r② ❤✐st♦r② ❝♦✈❡rs ❛t ♠♦st N ❦❡②s ▼♦r❡ ❚❤❛♥ ❖♥❡ ❑❡② ❇❧♦❝❦ ( N ) � b λ N Adv key ✲ pre • ❇② ●❛➸✐ ❡t ❛❧✳ ❬●P❚✶✺❪ F 2 k/ 2 • ❯s❡❞ ✐♥ ♠❛♥② s♣♦♥❣❡ ♣r♦♦❢s ✻ ✴ ✶✽

❑❡② Pr❡❞✐❝t✐♦♥ ❙❡❝✉r✐t②✿ ■♠♣❧✐❝❛t✐♦♥ ❢♦r ❖❑❙ m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c c c c c c ❈❛s❡ ♦❢ ( b, c, r, k ) = (320 , 256 , 64 , 64) M 2 M 2 2 c + MN + N 2 256 + MN 2 256 + N = 2 c 2 k 2 64 ❈❛s❡ ♦❢ ( b, c, r, k ) = (320 , 256 , 64 , 128) M 2 M 2 2 c + MN + N 2 256 + MN 2 256 + N = 2 c 2 k/ 2 2 64 ✼ ✴ ✶✽

◆❡✇ ❆♥❛❧②s✐s m µ K 1 K λ m 1 z 1 z 2 0 • • π π π π π 0 c ′ c ′ c ′ c ′ c c ( N ) � c λ − 1 N Adv key ✲ pre F 2 k • ▲♦ss c ❞✉❡ t♦ ❧✉❝❦② ♠✉❧t✐✲❝♦❧❧✐s✐♦♥s ✭✐♥ ♦❧❞ ❜♦✉♥❞✿ b ✮ • 2 k ✐♥ ❞❡♥♦♠✐♥❛t♦r ✭✐♥ ♦❧❞ ❜♦✉♥❞✿ 2 k/ 2 ✮ • ❇❡st ❛tt❛❝❦✿ ❛r♦✉♥❞ 2 k q✉❡r✐❡s ✽ ✴ ✶✽

❣♦❛❧✿ ❜♦✉♥❞ ★ ♣❛t❤s ❢r♦♠ t♦ Pr♦♦❢ ■❞❡❛ • ❚r❡❡✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✭❛s ✐♥ ❬●P❚✶✺❪ ✮ V 0 V 1 V 2 V 3 0 b • ✾ ✴ ✶✽

❣♦❛❧✿ ❜♦✉♥❞ ★ ♣❛t❤s ❢r♦♠ t♦ Pr♦♦❢ ■❞❡❛ • ❚r❡❡✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✭❛s ✐♥ ❬●P❚✶✺❪ ✮ V 0 V 1 V 2 V 3 u • L 0 b • ✾ ✴ ✶✽

❣♦❛❧✿ ❜♦✉♥❞ ★ ♣❛t❤s ❢r♦♠ t♦ Pr♦♦❢ ■❞❡❛ • ❚r❡❡✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✭❛s ✐♥ ❬●P❚✶✺❪ ✮ V 0 V 1 V 2 V 3 u • L • 0 b • • • ✾ ✴ ✶✽

❣♦❛❧✿ ❜♦✉♥❞ ★ ♣❛t❤s ❢r♦♠ t♦ Pr♦♦❢ ■❞❡❛ • ❚r❡❡✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✭❛s ✐♥ ❬●P❚✶✺❪ ✮ V 0 V 1 V 2 V 3 • u • • L • 0 b • • • • • • ✾ ✴ ✶✽

❣♦❛❧✿ ❜♦✉♥❞ ★ ♣❛t❤s ❢r♦♠ t♦ Pr♦♦❢ ■❞❡❛ • ❚r❡❡✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✭❛s ✐♥ ❬●P❚✶✺❪ ✮ V 0 V 1 V 2 V 3 • u • • L • • 0 b • • • • • • • • ✾ ✴ ✶✽

Pr♦♦❢ ■❞❡❛ • ❚r❡❡✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✭❛s ✐♥ ❬●P❚✶✺❪ ✮ V 0 V 1 V 2 V 3 • u • • L • • 0 b • • • • • • • • � �� � ❣♦❛❧✿ ❜♦✉♥❞ ★ ♣❛t❤s ❢r♦♠ V 0 t♦ V 3 ✾ ✴ ✶✽

❆rr♦✇s ✐♥❞✐❝❛t❡ q✉❡r② ❞✐r❡❝t✐♦♥✱ ❝✐r❝❧❡s ✐♥❞✐❝❛t❡ ✐♥♥❡r ❝♦❧❧✐s✐♦♥s ❚❤✐s q✉❡r② ✜①❡s ✐♥♥❡r ♣❛rt ♦❢ s❡❝♦♥❞✲❧❛st ❧❛②❡r ❝♦♥✜❣✳ ✿ ❝♦♥✜❣✳ ✿ ❝♦♥✜❣✳ ✿ ❝♦♥✜❣✳ ✿ ❈♦♥s✐❞❡r ❝♦♥✜❣✉r❛t✐♦♥s ❢♦r t❤❡s❡ ❧❛②❡rs ■♥❞✉❝t✐✈❡ r❡❛s♦♥✐♥❣ ♦♥ ♥♦♥✲♦❝❝✉rr❡♥❝❡ ♦❢ ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥s Pr♦♦❢ ■❞❡❛ • ❋✐① ❛♥② q✉❡r② ❢r♦♠ V 2 t♦ V 3 ✿ N ♦♣t✐♦♥s ✶✵ ✴ ✶✽