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❙✐♠✐❧❛r✐t✐❡s ❜❡t✇❡❡♥ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥✿ ❤♦✇ ❢❛r ❝❛♥ ✇❡ ❣♦❄

❆♥♥❡ ❈❛♥t❡❛✉t ■♥r✐❛✱ ❋r❛♥❝❡ ❛♥❞ ❉❚❯✱ ❉❡♥♠❛r❦ ❆♥♥❡✳❈❛♥t❡❛✉t❅✐♥r✐❛✳❢r ❤tt♣✿✴✴✇✇✇✲r♦❝q✳✐♥r✐❛✳❢r✴s❡❝r❡t✴❆♥♥❡✳❈❛♥t❡❛✉t✴ ❙❆❈ ✷✵✶✸ ❜❛s❡❞ ♦♥ ❛ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▲❛rs ❑♥✉❞s❡♥ ❛♥❞ ●r❡❣♦r ▲❡❛♥❞❡r

❖✉t❧✐♥❡

• ▲♦✇✲❧❛t❡♥❝② ❛♥❞ ❧✐❣❤t✇❡✐❣❤t ❝✐♣❤❡rs
• ▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ ✐♥✈♦❧✉t✐♦♥❛❧ ❝✐♣❤❡rs ❛♥❞

✐♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s

• ▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ r❡✢❡❝t✐♦♥ ❝✐♣❤❡rs
• P❘■◆❈❊

✶

■t❡r❛t❡❞ ❜❧♦❝❦ ❝✐♣❤❡rs

F (1)

✲ F (2) ✲ ❡ ❡ ❡

✳✳✳

F (r)

✲ ✲ ✲

❦❡② s❝❤❡❞✉❧❡

❄ ❄ ❄ ❄

k1 k2 kr

♣❧❛✐♥t❡①t x

y ❝✐♣❤❡rt❡①t K ♠❛st❡r ❦❡②

✇❤❡r❡ ❡❛❝❤ F (i) ✐s ❛ ❦❡②❡❞ ♣❡r♠✉t❛t✐♦♥ ♦❢ Fn

2 ✳

✷

▲✐❣❤t✇❡✐❣❤t ❜❧♦❝❦ ❝✐♣❤❡rs ❆❊❙ ❬❉❛❡♠❡♥✲❘✐❥♠❡♥ ✾✽❪❬❋■P❙ P❯❇ ✶✾✼❪

• ❜❧♦❝❦s✐③❡✿ 128 ❜✐ts
• ❙❜♦① ♦♣❡r❛t❡s ♦♥ 8 ❜✐ts
• ❧✐♥❡❛r ❞✐✛✉s✐♦♥ ❧❛②❡r ✐s ❛ ❧✐♥❡❛r ♣❡r♠✉t❛t✐♦♥ ♦❢
• F28

4

❚♦ ♠❛❦❡ ✐t s♠❛❧❧❡r ✐♥ ❤❛r❞✇❛r❡✿

• ❜❧♦❝❦s✐③❡✿ 64 ❜✐ts
• s♠❛❧❧❡r ❙❜♦①✱ ♦♥ 3 ♦r 4 ❜✐ts
• ❧✐♥❡❛r ❞✐✛✉s✐♦♥ ❧❛②❡r ♦✈❡r ❛ s♠❛❧❧❡r ❛❧♣❤❛❜❡t
• s✐♠♣❧✐✜❡❞ ❦❡②✲s❝❤❡❞✉❧❡

✸

❚❤❡ ✉s✉❛❧ ❞❡s✐❣♥ str❛t❡❣②✿ P❘❊❙❊◆❚ ❬❇♦❣❞❛♥♦✈ ❡t ❛❧✳ ✵✼❪

S S S S S S S S S S S S S S S S

ski 64 bits

31 r♦✉♥❞s ✭✰ ❛ ❦❡② ❛❞❞✐t✐♦♥✮

✹

▲✐❣❤t✇❡✐❣❤t ❜✉t s❡❝✉r❡✳✳✳ ■♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ r♦✉♥❞s✦

• P❘❊❙❊◆❚ ❬❇♦❣❞❛♥♦✈ ❡t ❛❧✳ ✵✼❪✳ ✸✶ r♦✉♥❞s
• ▲❊❉ ❬●✉♦ ❡t ❛❧✳ ✶✶❪✿

▲❊❉✲✻✹✿ ✸✷ r♦✉♥❞s✱ ▲❊❉✲✶✷✽✿ ✹✽ r♦✉♥❞s

• ❙P❊❈❑ ❬❇❡❛✉❧✐❡✉ ❡t ❛❧✳ ✶✸❪✿

❙P❊❈❑✻✹✴✶✷✽✿ ✷✼ r♦✉♥❞s✱ ❙P❊❈❑✶✷✽✴✷✺✻✿ ✸✹ r♦✉♥❞s

• ❙■▼❖◆ ❬❇❡❛✉❧✐❡✉ ❡t ❛❧✳ ✶✸❪✿

❙■▼❖◆✻✹✴✶✷✽✿ ✹✹ r♦✉♥❞s✱ ❙■▼❖◆✶✷✽✴✷✺✻✿ ✼✷ r♦✉♥❞s

✺

❉♦❡s ❧✐❣❤t✇❡✐❣❤t ♠❡❛♥ ✏❧✐❣❤t ✰ ✇❛✐t✑❄ ❬❑♥❡➸❡✈✐➣ ❡t ❛❧✳ ✶✷❪

✻

❉♦❡s ❧✐❣❤t✇❡✐❣❤t ♠❡❛♥ ✏❧✐❣❤t ✰ ✇❛✐t✑❄ ❬❑♥❡➸❡✈✐➣ ❡t ❛❧✳ ✶✷❪

✼ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✴ ♦ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ✇ ✛ ✲

❙❊❈❯❘■❚❨ ❙P❊❊❉ ❆❘❊❆ ▲♦✇✲❧❛t❡♥❝② ❡♥❝r②♣t✐♦♥✳

• ▼❡♠♦r② ❡♥❝r②♣t✐♦♥
• ❱❆◆❊❚ ✭❱❡❤✐❝✉❧❛r ❛❞✲❤♦❝ ♥❡t✇♦r❦✮
• ❡♥❝r②♣t✐♦♥ ❢♦r ❤✐❣❤✲s♣❡❡❞ ♥❡t✇♦r❦✐♥❣✳✳✳

✼

❍♦✇ ❝❛♥ ✇❡ ❞❡s✐❣♥ ❛ ❢❛st ❛♥❞ ❧✐❣❤t✇❡✐❣❤t ❝✐♣❤❡r❄ ❯♥r♦❧❧❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳

• s♠❛❧❧ ♥✉♠❜❡r ♦❢ r♦✉♥❞s❀
• ❡❛❝❤ r♦✉♥❞ ♦❢ ❡♥❝r②♣t✐♦♥ ❛♥❞ ❞❡❝r②♣t✐♦♥ s❤♦✉❧❞ ❤❛✈❡

❛ ❧♦✇ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝♦st❀

• t❤❡ r♦✉♥❞s ❞♦ ♥♦t ♥❡❡❞ t♦ ❜❡ s✐♠✐❧❛r✳

❘❡❧❛t❡❞ ♦♣❡♥ ♣r♦❜❧❡♠✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♣r♦✈✐❞❡ s❡❝✉r✐t② ❛r❣✉♠❡♥ts ❢♦r ❛ ❝✐♣❤❡r ✐t❡r❛t✐♥❣ ✈❡r② ❞✐✛❡r❡♥t r♦✉♥❞s❄

✽

▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ ✐♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s

✾

❲❤❡♥ ❧✐❣❤t✇❡✐❣❤t ❡♥❝r②♣t✐♦♥ ✇❛s r❡❛❧❧② ❛♥ ✐ss✉❡✳✳✳

❤tt♣✿✴✴✇✇✇✳♥s❛✳❣♦✈✴♠✉s❡✉♠✴❡♥✐❣♠❛✳❤t♠❧ ✶✵

❙❝❤❡r❜✐✉s✬ s♦❧✉t✐♦♥✿ ❛❞❞ ❛ r❡✢❡❝t♦r

❆ ❇ ❈ ❉ ❊ ❋

✡ ☛ ✠ ✟

❆ ❇ ❈ ❉ ❊ ❋

✡ ☛ ✠ ✟ ✡ ☛ ✠ ✟

❇ ❉ ❆ ❇ ❈ ❉ ❊ ❋ ❆ ❈ ❊ ❋ ❇ ❉ ❆ ❇ ❈ ❉ ❊ ❋ ❆ ❈ ❊ ❋ ❇ ❉ ❆ ❇ ❈ ❉ ❊ ❋ ❆ ❈ ❊ ❋ ❦❡②❜♦❛r❞ r❡✢❡❝t♦r ❧❛♠♣❜♦❛r❞

EK = F −1

K

• M ◦ FK ✇❤❡r❡ M = M−1

✶✶

❈❛♥ EK ❜❡ ❛♥ ✐♥✈♦❧✉t✐♦♥❄ ❋✐①❡❞ ♣♦✐♥ts✳❬❨♦✉ss❡❢✲❚❛✈❛r❡s✲❍❡②s ✾✻❪

• ❆ r❛♥❞♦♠ ♣❡r♠✉t❛t✐♦♥ ♦❢ Fn

2 ❤❛s 1 ✜①❡❞ ♣♦✐♥t ♦♥ ❛✈❡r❛❣❡❀

• ❆ r❛♥❞♦♠ ✐♥✈♦❧✉t✐♦♥ ♦❢ Fn

2 ❤❛s 2

n 2 + O(1) ✜①❡❞ ♣♦✐♥ts✳

■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r EK = F −1

K

• M ◦ FK

EK ❤❛s t❤❡ s❛♠❡ ❝②❝❧❡ str✉❝t✉r❡ ✭❛♥❞ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ✜①❡❞

♣♦✐♥ts✮ ❛s M✳

• ❊♥✐❣♠❛✿ t❤❡ r❡✢❡❝t♦r ❤❛s ♥♦ ✜①❡❞ ♣♦✐♥ts❀
• ❉❊❙ ✇✐t❤ ❛ ✇❡❛❦ ❦❡②✿ M ✐s t❤❡ s✇❛♣♣✐♥❣ ♦❢ t❤❡ ✷ ❤❛❧✈❡s

→ ■t ❤❛s 232 ✜①❡❞ ♣♦✐♥ts ❬❈♦♣♣❡rs♠✐t❤ ✽✺❪✳

✶✷

❆❞❞ s♦♠❡ ✇❤✐t❡♥✐♥❣ ❦❡②s ❬❘✐✈❡st ✽✹❪

F X ❝♦♥str✉❝t✐♦♥

✍✌ ✎☞

✰

✲ ✲ ❄

k0

✍✌ ✎☞

✰

✲ ✲ ❄

k2 F

❄

k1 c m

❙❧✐❞❡ ❛tt❛❝❦ ✇✐t❤ ❝♦♠♣❧❡①✐t② 2

n+1 2

❬❨♦✉ss❡❢✲❚❛✈❛r❡s✲❍❡②s ✾✻❪❬❉✉♥❦❡❧♠❛♥ ❡t ❛❧✳ ✶✷❪

✍✌ ✎☞

✰

✲ ✲ ❄ ✍✌ ✎☞

✰

✲ ✲ ❄

c F

❄

k1 k0 y x k2 m

✍✌ ✎☞

✰

✲ ✲ ❄ ✍✌ ✎☞

✰

✲ ✲ ❄

F

❄

k1 k0 y x k2 m′ c′ ■❢ (m, c) ❛♥❞ (m′, c′) s❛t✐s❢② m ⊕ c = m′ ⊕ c′✱ t❤❡♥ ❝❤❡❝❦ ✇❤❡t❤❡r k0 ⊕ k2 = m′ ⊕ c ✳

✶✸

❯s✐♥❣ ✐♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s ❊①❛♠♣❧❡s✿

• ❋❡✐st❡❧ ❝✐♣❤❡rs
• ✐♥✈♦❧✉t✐♦♥❛❧ ❙P◆s ❬❨♦✉ss❡❢✲❚❛✈❛r❡s✲❍❡②s ✾✻❪
• ❑❤❛③❛❞ ❬❇❛rr❡t♦✲❘✐❥♠❡♥ ✵✵❪
• ❆◆❯❇■❙ ❬❇❛rr❡t♦✲❘✐❥♠❡♥ ✵✵❪
• ◆❖❊❑❊❖◆ ❬❉❛❡♠❡♥ ❡t ❛❧✳ ✵✵❪
• ■❈❊❇❊❘● ❬❙t❛♥❞❛❡rt ❡t ❛❧✳ ✵✹❪✳✳✳

✶✹

❆❊❙ s✉♣❡r❜♦①

S S S S

❦

✰

❄ ❦

✰

❄ ❦

✰

❄ ❦

✰

❄

S S S S

❄ ❄ ❄ ❄

L

❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ✲ ✲ ✲ ✲

K S ✐s ❛ ♣❡r♠✉t❛t✐♦♥ ♦✈❡r Fm

2

❚❤❡ ❞✐✛✉s✐♦♥ ❧❛②❡r ✐s ❧✐♥❡❛r ♦✈❡r F2m ❛♥❞ ❤❛s ♠❛①✐♠❛❧ ❜r❛♥❝❤ ♥✉♠❜❡r✳

✶✺

■♥✈♦❧✉t✐♦♥❛❧ ❙❜♦①❡s ✇✐t❤ ❛♥ ❙P◆ ▼❛①✐♠❛❧ ❡①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t② ❢♦r ❛ t✇♦✲r♦✉♥❞ ❞✐✛❡r❡♥t✐❛❧✿

MEDP2 = max

a=0,b Prx,K[∆EK(x) = b|∆x = a]

❋♦r t❤❡ ❆❊❙ ❙❜♦① S(x) = ℓ(x254)✿

MEDP2 = 53 × 2−34 ❬❑❡❧✐❤❡r✲❙✉✐ ✵✼❪

❋♦r t❤❡ ♥❛✐✈❡ ❙❜♦① S(x) = x254✿

MEDP2 = 79 × 2−34 ❬❉❛❡♠❡♥✲❘✐❥♠❡♥ ✵✻❪

→ ❍✐❣❤❡st ♣♦ss✐❜❧❡ ✈❛❧✉❡ ❢♦r ❛ ❢✉♥❝t✐♦♥ ❤❛✈✐♥❣ s✐♠✐❧❛r ✈❛❧✉❡s ✐♥ ✐ts ❞✐✛❡r❡♥❝❡ t❛❜❧❡ ❬P❛r❦ ❡t ❛❧✳ ✵✸❪

✶✻

❆ ♥❡✇ ❜♦✉♥❞ ✭♣❛rt✐❝✉❧❛r ❝❛s❡✮ ❬❈✳✲❘♦✉é ✶✸❪ ❈♦♥s✐❞❡r ❛♥ ❙P◆ ✇✐t❤ ❛ ♥♦♥❧✐♥❡❛r ❧❛②❡r ❝♦♠♣♦s❡❞ ♦❢ t ♣❛r❛❧❧❡❧ ❛♣♣❧✐✲ ❝❛t✐♦♥s ♦❢ ❛ ❢✉♥❝t✐♦♥ S ♦✈❡r F2m ❛♥❞ ✇✐t❤ ❛♥ ▼❉❙ ❧✐♥❡❛r ❞✐✛✉s✐♦♥ ❧❛②❡r ♦✈❡r F2m✱ ✐❢ S(x) = ℓ(xs) ♦r S(x) = (ℓ(x))s ✇❤❡r❡ ℓ ✐s ❛♥ ❛✣♥❡ ♣❡r♠✉t❛t✐♦♥ ♦❢ Fm

2 ✱ ✇❡ ❤❛✈❡

MEDP2 ≤ 2−m(t+1) max

1≤u≤t

max

α,β=0

• γ∈F∗

2m

δ(α, γ)uδ(γ, β)t+1−u

✇❤❡r❡ δ(a, b) = #{x ∈ Fm

2 ,

S(x + a) + S(x) = b}✳

▼♦r❡♦✈❡r✱ t❤❡ ❜♦✉♥❞ ✐s t✐❣❤t ❢♦r ❛❧❧ ▼❉❙ ❧✐♥❡❛r ❧❛②❡rs ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞s✿

• S(x) = xs❀
• S(x) = ℓ(xs) ❛♥❞ t❤❡ ♠❛①✐♠✉♠ ✐s ❛tt❛✐♥❡❞ ❢♦r u = 1✳

✶✼

❉✐✛❡r❡♥❝❡ t❛❜❧❡ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦✈❡r F16 1 ζ ζ2 ζ3 ζ4 ζ5 ζ6 ζ7 ζ8 ζ9 ζ10 ζ11 ζ12 ζ13 ζ14 1 ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ζ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ζ2 ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ζ3 ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ζ4 ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ζ5 ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ζ6 ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ζ7 ✷ ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ζ8 ✵ ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ζ9 ✵ ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ζ10 ✷ ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ζ11 ✷ ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ζ12 ✵ ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ζ13 ✷ ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ζ14 ✷ ✹ ✵ ✵ ✵ ✵ ✷ ✵ ✷ ✵ ✵ ✷ ✷ ✵ ✷

✶✽

MEDP2 ❢♦r ❆❊❙ ❛♥❞ ✈❛r✐❛♥ts 2−m(t+1) max

1≤u≤t max α,β=0

• γ∈F∗

2m

δ(α, γ)uδ(γ, β)t+1−u

❆❊❙ ❙❜♦① S(x) = ℓ(x254)✳

→ MEDP2=53 × 2−34

◆❛✐✈❡ ❙❜♦① S(x) = x254✳

δ(a, b) = δ(b, a) max

α,β=0

• γ∈F∗

2m

δ(α, γ)uδ(γ, β)t+1−u = max

α,β=0

• γ∈F∗

2m

δ(α, γ)uδ(β, γ)t+1−u = max

α=0

• γ∈F∗

2m

δ(α, γ)t+1 → MEDP2=79 × 2−34

✶✾

▼✐♥✐♠✐③✐♥❣ t❤❡ ♦✈❡r❤❡❛❞ ♦❢ ❞❡❝r②♣t✐♦♥✿ r❡✢❡❝t✐♦♥ ❝✐♣❤❡rs

✷✵

❘❡✢❡❝t✐♦♥ ❝✐♣❤❡rs ❉❡✜♥✐t✐♦♥✳ ❆ ❜❧♦❝❦ ❝✐♣❤❡r E ✐s ❛ r❡✢❡❝t✐♦♥ ❝✐♣❤❡r ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣❡r♠✉t❛t✐♦♥ P ♦❢ t❤❡ ❦❡② s♣❛❝❡ s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ K✱

(EK)−1 = EP (K)

❊①❛♠♣❧❡s✳

• ❋❡✐st❡❧ ❝✐♣❤❡r ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t r♦✉♥❞ ❦❡②s✿

P (k1, . . . , kr) = (kr, . . . , k1)

• ❘❙❆✿

P ❂ ✐♥✈❡rs✐♦♥ ♠♦❞✉❧♦ (p − 1)(q − 1)✳

✷✶

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ ♣❡r♠✉t❛t✐♦♥

(EK)−1 = EP (K)

✐♠♣❧✐❡s

EK = EP 2(K)

❈❤♦✐❝❡ ♦❢ P ✳

P s❤♦✉❧❞ ❜❡ ❛♥ ✐♥✈♦❧✉t✐♦♥✳

❊①❛♠♣❧❡✿

P (K) = K ⊕ α

✷✷

■t❡r❛t❡❞ r❡✢❡❝t✐♦♥ ❝✐♣❤❡r ✇✐t❤ P (K) = K ⊕ α ❊♥❝r②♣t✐♦♥✿ F1

✲ ❄

K F2

✲ ❄

K Fr

✲ ❄

K F −1

2

✲ ❄

K⊕α F −1

1

✲ ❄

K⊕α

✲ ✲

M

✲

F −1

r

✲ ✲ ❄

K⊕α m c ❉❡❝r②♣t✐♦♥✿ F1

✲ ❄

K⊕α F2

✲ ❄

K⊕α Fr

✲ ❄

K⊕α F −1

2

✲ ❄

K F −1

1

✲ ❄

K

✲ ✲

M

✲

F −1

r

✲ ✲ ❄

K c m ✇❤❡r❡ M ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥✳

✷✸

❊①❛♠♣❧❡ ♦❢ ❛ r❡✢❡❝t✐♦♥ ❝✐♣❤❡r ✇✐t❤ P (k1, k2) = (k2 ⊕ α, k1 ⊕ α) F1

✲ ❄

k1 F2

✲ ❄

k2 Fr

✲ ❄

k1 F −1

2

✲ ❄

k1⊕α F −1

1

✲ ❄

k2⊕α

✲ ✲

M

✲

F −1

r

✲ ✲ ❄

k2⊕α m c

• E(k1,k2)

−1 = E(k2⊕α,k1⊕α)

❋♦r ❛❧❧ ❦❡②s ✇✐t❤ k2 = k1 ⊕ α✱ t❤❡ ❝✐♣❤❡r ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥✱ ❛♥❞ ✐t ❤❛s t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ✜①❡❞ ♣♦✐♥ts ❛s M✳ → ▲❛r❣❡ ❝❧❛ss ♦❢ ✇❡❛❦ ❦❡②s✳

✷✹

❋✐①❡❞ ♣♦✐♥ts ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ ♣❡r♠✉t❛t✐♦♥ ❋✐①❡❞ ♣♦✐♥ts ♦❢ P ✳ ❚❤❡ ❦❡②s ❢♦r ✇❤✐❝❤ t❤❡ ❡♥❝r②♣t✐♦♥ ❢✉♥❝t✐♦♥ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❞❡t❡❝t❡❞ ✇✐t❤ O(2

n 2) ♣❧❛✐♥t❡①t✲❝✐♣❤❡rt❡①t ♣❛✐rs✳

❈❤♦✐❝❡ ♦❢ P ✳

P s❤♦✉❧❞ ❜❡ ❛♥ ✐♥✈♦❧✉t✐♦♥ ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts✳

❊①❛♠♣❧❡✿

P (K) = K ⊕ α

✷✺

❖♥ r❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡rs ❢♦r r❡✢❡❝t✐♦♥ ❝✐♣❤❡rs ❚r✐✈✐❛❧ r❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡rs✿ ❛r❡ ♥♦t ❝♦♥s✐❞❡r❡❞✳ ✭t❤❡② ♠❛② ❜❡ ✐♠♣♦rt❛♥t ✐♥ s♦♠❡ s❝❡♥❛r✐♦s✱ ❡✳❣✳✱ ❬■✇❛t❛✲❑✉r♦s❛✇❛ ✵✸❪✮ ❘❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡rs✿ ♠❛② ❤❛✈❡ ❛♥ ✐♠♣❛❝t ✐♥ ❛ s✐♥❣❧❡✲❦❡② ♠♦❞❡❧✳ ❆ r❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡r ❢♦r EK ✐♥✈♦❧✈✐♥❣ t✇♦ ❦❡②s K ❛♥❞ K′ r❡❧❛t❡❞ ❜② K′ = P (K) ✐s ❛ ❞✐st✐♥❣✉✐s❤❡r ✐♥ t❤❡ s✐♥❣❧❡✲❦❡② ♠♦❞❡❧✳ → ❘❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡rs ♠❛② ❜❡ r❡❧❡✈❛♥t✦

✷✻

❖♥ ❞✐✛❡r❡♥t✐❛❧ r❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡rs ❉✐st✐♥❣✉✐s❤❡rs ✐♥✈♦❧✈✐♥❣ K ❛♥❞ K′ = P (K) s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞✳ ❚✇♦ str❛t❡❣✐❡s✿

• ❈❤♦♦s❡ P s✉❝❤ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉❝❤ ❞✐st✐♥❣✉✐s❤❡rs ✐s ✈❡r②

✉♥❧✐❦❡❧②✱ ❡✳❣✳✱ s✉❝❤ t❤❛t K ⊕ P (K) ❤❛s ❛❧✇❛②s ❛ ❤✐❣❤ ✇❡✐❣❤t❀

• ❈❤♦♦s❡ P s✉❝❤ t❤❛t s✉❝❤ r❡❧❛t❡❞✲❦❡② ❞✐st✐♥❣✉✐s❤❡rs ❝❛♥

❜❡ ❡①♣❧♦✐t❡❞ ❢♦r ❛ ❢❡✇ K ♦♥❧②✳ ❚r❛❞❡✲♦✛ ❜❡t✇❡❡♥

min

K wt(K ⊕ P (K)) ❛♥❞ max δ

#{K : K ⊕ P (K) = δ}

❋♦r P (K) = K ⊕ α ✇❤❡r❡ wt(α) ✐s ❤✐❣❤✱ ✇❡ ♠❛①✐♠✐③❡ t❤❡ ✜rst q✉❛♥t✐t②✳

✷✼

P❘■◆❈❊

✷✽

❘❡✢❡❝t✐♦♥ ❝✐♣❤❡r ✇✐t❤ P (K) = K ⊕ α

♥

+

✲ ✲ ✲ ♥

+

✲ ✲ ✲ ♥

+

✲ ♥

+

✲ ✲ ♥

+

✲ ✲ ♥

+

✲ ✲

M

✲

F −1

r

✲

F2 Fr F1 F −1

2

✲

F −1

1

❄

k

❄

k

❄

k

❄

k⊕α

❄

k⊕α

✲ ❄

k⊕α m c

✷✾

■♥❝r❡❛s✐♥❣ t❤❡ ❦❡② ❧❡♥❣t❤ FX ❝♦♥str✉❝t✐♦♥ ❬❘✐✈❡st ✽✹❪

k = (k0||k1)

✚✙ ✛✘

✰

✲ ✲ ❄

k0

✚✙ ✛✘

✰

✲ ✲ ❄

π(k0)

P❘■◆❈❊core

❄

k1 c m

✇✐t❤ π(x) = (x ≫ 1) ⊕ (x ≫ 63) → (k0 ⊕ k1, π(k0) ⊕ k1) t❛❦❡s ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ✇❤❡♥ (k0, k1) ✈❛r✐❡s✳

✸✵

❙❡❝✉r✐t② ♦❢ t❤❡ FX ❝♦♥str✉❝t✐♦♥ ❬❑✐❧✐❛♥✲❘♦❣❛✇❛② ✾✻❪

F Xk0,k1,k2(m) = Fk1(m ⊕ k0) ⊕ k2

FXk F, F −1

✻ ❄ ✻ ❄

π F, F −1

✻ ❄ ✻ ❄

k0, k1, k2

✲ ✻

A m (x, k) ❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❛♥② ❛❞✈❡rs❛r② ✇❤♦ ♠❛❦❡s D q✉❡r✐❡s t♦ E = F X ❛♥❞ T q✉❡r✐❡s t♦ (F, F −1) ✐s ❛t ♠♦st

DT 2−(κ1+n−1)

✸✶

■♠♣❛❝t ♦❢ t❤❡ r❡✢❡❝t✐♦♥ ♣r♦♣❡rt② ♦♥ t❤❡ F X ❝♦♥str✉❝t✐♦♥ ■❞❡❛❧ r❡✢❡❝t✐♦♥ ❝✐♣❤❡r ✇✐t❤ ❝♦✉♣❧✐♥❣ ♣❡r♠✉t❛t✐♦♥ P ✳ ■❢ P ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts✱ t❤❡ ❦❡② s♣❛❝❡ ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ❛s

Fκ1

2

= H ∪ P (H)

✇❤❡r❡ H ❝♦♥t❛✐♥s ❤❛❧❢ ♦❢ t❤❡ ❦❡②s✳ ▲❡t F ❜❡ ❛♥ ✐❞❡❛❧ ❜❧♦❝❦ ❝✐♣❤❡r ✇✐t❤ ❦❡② s♣❛❝❡ H✳ ❲❡ ❡①t❡♥❞ ✐t ❜②

˜ Fk(x) =

  

Fk(x)

✐❢ k ∈ H

F −1

P (k)(x)

✐❢ k ∈ P (H) ❙❡❝✉r✐t② ♦❢ t❤❡ ˜

F X ❝♦♥str✉❝t✐♦♥✳

❚❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❛♥② ❛❞✈❡rs❛r② ✇❤♦ ♠❛❦❡s D q✉❡r✐❡s t♦ E = ˜

F X

❛♥❞ T q✉❡r✐❡s t♦ (F, F −1) ✐s ❛t ♠♦st

DT 2−(κ1+n−2) .

✸✷

P❛r❛♠❡t❡rs

• ❇❧♦❝❦ s✐③❡✿ ✻✹ ❜✐ts
• ❑❡② s✐③❡✿ ✶✷✽ ❜✐ts
• ◆❜ ♦❢ ❙❜♦① ❧❛②❡rs✿ ✶✷

❙❡❝✉r✐t② ❝❧❛✐♠ ✐♥ t❤❡ s✐♥❣❧❡✲❦❡② ♠♦❞❡❧✿ 126✲❜✐t s❡❝✉r✐t② ❚❤❡r❡ ✐s ♥♦ ❛tt❛❝❦ ✇✐t❤ t✐♠❡ ❛♥❞ ❞❛t❛ ❝♦♠♣❧❡①✐t✐❡s ❛r❡ s✉❝❤ t❤❛t

DT ≪ 2126

❇❡st ❛tt❛❝❦✳ ▼✐t▼ ❛tt❛❝❦ ♦♥ 8 r♦✉♥❞s ✇✐t❤ DT = 2124 ❬❈✳ ◆❛②❛✲P❧❛s❡♥❝✐❛ ❱❛②ss✐èr❡ ✶✸❪✳

✸✸

❈♦♥❝❧✉s✐♦♥s ❛♥❞ ♦♣❡♥ ✐ss✉❡s

• ■♥✈♦❧✉t✐♦♥❛❧ ❜✉✐❧❞✐♥❣✲❜❧♦❝❦s ♠❛② ✐♥tr♦❞✉❝❡ s♦♠❡ ✇❡❛❦♥❡ss❡s ✐♥

s♦♠❡ ❝❛s❡s✳ ❍♦✇ ❝❛♥ ✇❡ ✉s❡ t❤❡♠ ✐♥ s❡❝✉r❡ ✇❛②❄

• ❘❡✢❡❝t✐♦♥ ❝✐♣❤❡rs ❝♦♥s✐❞❡r❛❜❧② r❡❞✉❝❡ t❤❡ ♦✈❡r❤❡❛❞ ♦♥ ❞❡❝r②♣t✐♦♥

♦♥ t♦♣ ♦❢ ❡♥❝r②♣t✐♦♥ ❢♦r ✉♥r♦❧❧❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s✳

• ❋✐♥❞ s♦♠❡ ♦t❤❡r ❦❡② s❝❤❡❞✉❧❡s ✭✇♦r❦ ✐♥ ♣r♦❣r❡ss✮✳

✸✹