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P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ ✼ ✴ ✷✸

P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ ✼ ✴ ✷✸

P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) ✼ ✴ ✷✸

P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s ✼ ✴ ✷✸

P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P ✼ ✴ ✷✸

P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s ✼ ✴ ✷✸

P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ ✼ ✴ ✷✸

P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ ✼ ✴ ✷✸

❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❳♦P ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ✼ ✴ ✷✸

❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❳♦P d ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ✼ ✴ ✷✸

■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❳♦P d ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ✼ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② P❛t❛r✐♥✬s ❘❡s✉❧t • ❊①tr❡♠❡❧② ♣♦✇❡r❢✉❧ ❧♦✇❡r ❜♦✉♥❞ • ❍❛s r❡♠❛✐♥❡❞ r❛t❤❡r ✉♥❦♥♦✇♥ s✐♥❝❡ ✐♥tr♦❞✉❝t✐♦♥ ✭✷✵✵✸✮ ❆✉t❤♦rs P✉❜❧✐❝❛t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ▼✐rr♦r ❇♦✉♥❞ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✸ ❋❡✐st❡❧ ❙✉❜♦♣t✐♠❛❧ P❛t❛r✐♥ ❈❘❨P❚❖ ✷✵✵✹ ❋❡✐st❡❧ P❛t❛r✐♥ ■❈■❙❈ ✷✵✵✺ ❋❡✐st❡❧ ❖♣t✐♠❛❧ ✐♥ O ( · ) P❛t❛r✐♥✱ ▼♦♥tr❡✉✐❧ ■❈■❙❈ ✷✵✵✺ ❇❡♥❡s P❛t❛r✐♥ ■❈■❚❙ ✷✵✵✽ ❳♦P P❛t❛r✐♥ ❆❋❘■❈❆❈❘❨P❚ ✷✵✵✽ ❇❡♥❡s P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✽✼ ❳♦P ❈♦♥❝r❡t❡ ❜♦✉♥❞ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✵✴✷✾✸ ❋❡✐st❡❧ P❛t❛r✐♥ ❡Pr✐♥t ✷✵✶✸✴✸✻✽ ❳♦P ❳♦P d ❈♦❣❧✐❛t✐✱ ▲❛♠♣❡✱ P❛t❛r✐♥ ❋❙❊ ✷✵✶✹ ❱♦❧t❡✱ ◆❛❝❤❡❢✱ ▼❛rr✐èr❡ ❡Pr✐♥t ✷✵✶✻✴✶✸✻ ❋❡✐st❡❧ ■✇❛t❛✱ ▼❡♥♥✐♥❦✱ ❱✐③ár ❡Pr✐♥t ✷✵✶✻✴✶✵✽✼ ❈❊◆❈ ✼ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② ❙②st❡♠ ♦❢ ❊q✉❛t✐♦♥s • r ❞✐st✐♥❝t ✉♥❦♥♦✇♥s P = { P 1 , . . . , P r } • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s P a i ⊕ P b i = λ i • ❙✉r❥❡❝t✐♦♥ ϕ : { a 1 , b 1 , . . . , a q , b q } → { 1 , . . . , r } ●r❛♣❤ ❇❛s❡❞ ❱✐❡✇ P b 1 λ 1 P b 3 P a 8 P a 9 P a 1 = P a 2 P a 6 λ 8 λ 3 λ 9 λ 2 P b 8 = P b 9 = P b 10 = P a 11 λ 6 λ 4 P b 2 = P a 3 = P b 4 P a 4 = P a 5 λ 11 λ 10 P b 6 P a 10 λ 5 P b 5 P b 11 λ 7 P b 7 P a 7 ✽ ✴ ✷✸

■❢ ♦r ♦r ❈♦♥tr❛❞✐❝t✐♦♥✿ ♦r ♦r ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ ❛♥❞ ❝❤♦✐❝❡s ❢♦r ❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✶ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 2 P c ✾ ✴ ✷✸

■❢ ❛♥❞ ❝❤♦✐❝❡s ❢♦r ❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✶ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 2 P c ■❢ λ 1 = 0 ♦r λ 2 = 0 ♦r λ 1 = λ 2 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c ♦r P a = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ✾ ✴ ✷✸

❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✶ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 2 P c ■❢ λ 1 = 0 ♦r λ 2 = 0 ♦r λ 1 = λ 2 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c ♦r P a = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ λ 1 , λ 2 � = 0 ❛♥❞ λ 1 � = λ 2 • 2 n ❝❤♦✐❝❡s ❢♦r P a ✾ ✴ ✷✸

❋✐①❡s ✭✇❤✐❝❤ ✐s ❛s ❞❡s✐r❡❞✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✶ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 2 P c ■❢ λ 1 = 0 ♦r λ 2 = 0 ♦r λ 1 = λ 2 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c ♦r P a = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ λ 1 , λ 2 � = 0 ❛♥❞ λ 1 � = λ 2 • 2 n ❝❤♦✐❝❡s ❢♦r P a • ❋✐①❡s P b = λ 1 ⊕ P a ✭✇❤✐❝❤ ✐s � = P a ❛s ❞❡s✐r❡❞✮ ✾ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✶ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 2 P c ■❢ λ 1 = 0 ♦r λ 2 = 0 ♦r λ 1 = λ 2 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c ♦r P a = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ λ 1 , λ 2 � = 0 ❛♥❞ λ 1 � = λ 2 • 2 n ❝❤♦✐❝❡s ❢♦r P a • ❋✐①❡s P b = λ 1 ⊕ P a ✭✇❤✐❝❤ ✐s � = P a ❛s ❞❡s✐r❡❞✮ • ❋✐①❡s P c = λ 2 ⊕ P b ✭✇❤✐❝❤ ✐s � = P a , P b ❛s ❞❡s✐r❡❞✮ ✾ ✴ ✷✸

■❢ ♦r ❈♦♥tr❛❞✐❝t✐♦♥✿ ♦r ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ ❝❤♦✐❝❡s ❢♦r ✭✇❤✐❝❤ ✜①❡s ✮ ❋♦r ❛♥❞ ✇❡ r❡q✉✐r❡ ❆t ❧❡❛st ❝❤♦✐❝❡s ❢♦r ✭✇❤✐❝❤ ✜①❡s ✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✷ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 λ 2 P c P d P c ⊕ P d = λ 2 ✶✵ ✴ ✷✸

■❢ ❝❤♦✐❝❡s ❢♦r ✭✇❤✐❝❤ ✜①❡s ✮ ❋♦r ❛♥❞ ✇❡ r❡q✉✐r❡ ❆t ❧❡❛st ❝❤♦✐❝❡s ❢♦r ✭✇❤✐❝❤ ✜①❡s ✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✷ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 λ 2 P c P d P c ⊕ P d = λ 2 ■❢ λ 1 = 0 ♦r λ 2 = 0 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ✶✵ ✴ ✷✸

❋♦r ❛♥❞ ✇❡ r❡q✉✐r❡ ❆t ❧❡❛st ❝❤♦✐❝❡s ❢♦r ✭✇❤✐❝❤ ✜①❡s ✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✷ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 λ 2 P c P d P c ⊕ P d = λ 2 ■❢ λ 1 = 0 ♦r λ 2 = 0 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ λ 1 , λ 2 � = 0 • 2 n ❝❤♦✐❝❡s ❢♦r P a ✭✇❤✐❝❤ ✜①❡s P b ✮ ✶✵ ✴ ✷✸

❆t ❧❡❛st ❝❤♦✐❝❡s ❢♦r ✭✇❤✐❝❤ ✜①❡s ✮ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✷ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 λ 2 P c P d P c ⊕ P d = λ 2 ■❢ λ 1 = 0 ♦r λ 2 = 0 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ λ 1 , λ 2 � = 0 • 2 n ❝❤♦✐❝❡s ❢♦r P a ✭✇❤✐❝❤ ✜①❡s P b ✮ • ❋♦r P c ❛♥❞ P d ✇❡ r❡q✉✐r❡ • P c � = P a , P b • P d = λ 2 ⊕ P c � = P a , P b ✶✵ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✷ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 λ 2 P c P d P c ⊕ P d = λ 2 ■❢ λ 1 = 0 ♦r λ 2 = 0 • ❈♦♥tr❛❞✐❝t✐♦♥✿ P a = P b ♦r P b = P c • ❙❝❤❡♠❡ ✐s ❞❡❣❡♥❡r❛t❡ ■❢ λ 1 , λ 2 � = 0 • 2 n ❝❤♦✐❝❡s ❢♦r P a ✭✇❤✐❝❤ ✜①❡s P b ✮ • ❋♦r P c ❛♥❞ P d ✇❡ r❡q✉✐r❡ • P c � = P a , P b • P d = λ 2 ⊕ P c � = P a , P b • ❆t ❧❡❛st 2 n − 4 ❝❤♦✐❝❡s ❢♦r P c ✭✇❤✐❝❤ ✜①❡s P d ✮ ✶✵ ✴ ✷✸

■❢ ❈♦♥tr❛❞✐❝t✐♦♥✿ ❡q✉❛t✐♦♥s s✉♠ t♦ ❙❝❤❡♠❡ ❝♦♥t❛✐♥s ❛ ❝✐r❝❧❡ ■❢ ❖♥❡ r❡❞✉♥❞❛♥t ❡q✉❛t✐♦♥✱ ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✸ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 3 λ 2 P c ⊕ P a = λ 3 P c • ❆ss✉♠❡ λ i � = 0 ❛♥❞ λ i � = λ j ✶✶ ✴ ✷✸

■❢ ❖♥❡ r❡❞✉♥❞❛♥t ❡q✉❛t✐♦♥✱ ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✸ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 3 λ 2 P c ⊕ P a = λ 3 P c • ❆ss✉♠❡ λ i � = 0 ❛♥❞ λ i � = λ j ■❢ λ 1 ⊕ λ 2 ⊕ λ 3 � = 0 • ❈♦♥tr❛❞✐❝t✐♦♥✿ ❡q✉❛t✐♦♥s s✉♠ t♦ 0 = λ 1 ⊕ λ 2 ⊕ λ 3 • ❙❝❤❡♠❡ ❝♦♥t❛✐♥s ❛ ❝✐r❝❧❡ ✶✶ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r②✿ ❚♦② ❊①❛♠♣❧❡ ✸ λ 1 • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✿ P a P b P a ⊕ P b = λ 1 P b ⊕ P c = λ 2 λ 3 λ 2 P c ⊕ P a = λ 3 P c • ❆ss✉♠❡ λ i � = 0 ❛♥❞ λ i � = λ j ■❢ λ 1 ⊕ λ 2 ⊕ λ 3 � = 0 • ❈♦♥tr❛❞✐❝t✐♦♥✿ ❡q✉❛t✐♦♥s s✉♠ t♦ 0 = λ 1 ⊕ λ 2 ⊕ λ 3 • ❙❝❤❡♠❡ ❝♦♥t❛✐♥s ❛ ❝✐r❝❧❡ ■❢ λ 1 ⊕ λ 2 ⊕ λ 3 = 0 • ❖♥❡ r❡❞✉♥❞❛♥t ❡q✉❛t✐♦♥✱ ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ✶✶ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r②✿ ❚✇♦ Pr♦❜❧❡♠❛t✐❝ ❈❛s❡s ❈✐r❝❧❡ ❉❡❣❡♥❡r❛❝② λ 1 P b 1 P b 1 = P a 2 P a 1 = P a 2 P a 8 λ 2 λ 1 P b 2 = P a 3 λ 2 λ 1 ⊕ λ 2 ⊕ · · · ⊕ λ 7 λ 3 P b 2 = P b 3 P a 3 = P a 4 P a 1 = P b 5 λ 3 P b 7 = P b 8 P b 3 = P a 4 λ 7 λ 5 λ 4 λ 4 P b 4 = P a 5 P b 4 = P a 5 λ 5 λ 6 P b 6 = P b 7 P b 5 = P a 6 ✶✷ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r②✿ ▼❛✐♥ ❘❡s✉❧t ❙②st❡♠ ♦❢ ❊q✉❛t✐♦♥s • r ❞✐st✐♥❝t ✉♥❦♥♦✇♥s P = { P 1 , . . . , P r } • ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s P a i ⊕ P b i = λ i • ❙✉r❥❡❝t✐♦♥ ϕ : { a 1 , b 1 , . . . , a q , b q } → { 1 , . . . , r } ▼❛✐♥ ❘❡s✉❧t ■❢ t❤❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✐s ❝✐r❝❧❡✲❢r❡❡ ❛♥❞ ♥♦♥✲❞❡❣❡♥❡r❛t❡✱ t❤❡ ♥✉♠❜❡r ♦❢ s♦❧✉t✐♦♥s t♦ P s✉❝❤ t❤❛t P a � = P b ❢♦r ❛❧❧ ❞✐st✐♥❝t a, b ∈ { 1 , . . . , r } ✐s ❛t ❧❡❛st (2 n ) r 2 nq ♣r♦✈✐❞❡❞ t❤❡ ♠❛①✐♠✉♠ tr❡❡ s✐③❡ ξ s❛t✐s✜❡s ( ξ − 1) 2 · r ≤ 2 n / 67 ✶✸ ✴ ✷✸

❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ■♥♣✉ts t♦ ❛r❡ ❛❧❧ ❞✐st✐♥❝t✿ ✉♥❦♥♦✇♥s ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P p 0 �· p y x 1 �· ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } ✶✹ ✴ ✷✸

❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ■♥♣✉ts t♦ ❛r❡ ❛❧❧ ❞✐st✐♥❝t✿ ✉♥❦♥♦✇♥s ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P p 0 �· p y x 1 �· ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p (0 � x i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p (1 � x i ) =: P b i ✶✹ ✴ ✷✸

■♥♣✉ts t♦ ❛r❡ ❛❧❧ ❞✐st✐♥❝t✿ ✉♥❦♥♦✇♥s ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P p 0 �· p y x 1 �· ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p (0 � x i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p (1 � x i ) =: P b i • ❙②st❡♠ ♦❢ q ❡q✉❛t✐♦♥s P a i ⊕ P b i = y i ✶✹ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P p 0 �· p y x 1 �· ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p (0 � x i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p (1 � x i ) =: P b i • ❙②st❡♠ ♦❢ q ❡q✉❛t✐♦♥s P a i ⊕ P b i = y i • ■♥♣✉ts t♦ p ❛r❡ ❛❧❧ ❞✐st✐♥❝t✿ 2 q ✉♥❦♥♦✇♥s ✶✹ ✴ ✷✸

■❢ ✿ ❛t ❧❡❛st s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s ❆♣♣❧②✐♥❣ ▼✐rr♦r ❚❤❡♦r② ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ♣r♦✈✐❞❡❞ t❤❛t ❢♦r ❛❧❧ ▼❛①✐♠✉♠ tr❡❡ s✐③❡ ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P P a 1 P a 2 P a q y q y 1 y 2 · · · P b q P b 1 P b 2 ✶✺ ✴ ✷✸

■❢ ✿ ❛t ❧❡❛st s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P P a 1 P a 2 P a q y q y 1 y 2 · · · P b q P b 1 P b 2 ❆♣♣❧②✐♥❣ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ♣r♦✈✐❞❡❞ t❤❛t y i � = 0 ❢♦r ❛❧❧ i • ▼❛①✐♠✉♠ tr❡❡ s✐③❡ 2 ✶✺ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P P a 1 P a 2 P a q y q y 1 y 2 · · · P b q P b 1 P b 2 ❆♣♣❧②✐♥❣ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ♣r♦✈✐❞❡❞ t❤❛t y i � = 0 ❢♦r ❛❧❧ i • ▼❛①✐♠✉♠ tr❡❡ s✐③❡ 2 • ■❢ 2 q ≤ 2 n / 67 ✿ ❛t ❧❡❛st (2 n ) 2 q s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s 2 nq ✶✺ ✴ ✷✸

❣✐✈❡s ❋♦r ❛♥② ❣♦♦❞ tr❛♥s❝r✐♣t✿ ❣✐✈❡s ❇❛❞ tr❛♥s❝r✐♣t✿ ✐❢ ❢♦r s♦♠❡ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P ❍✲❈♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡ ❬P❛t✾✶✱P❛t✵✽✱❈❙✶✹❪ ▲❡t ε ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❣♦♦❞ tr❛♥s❝r✐♣ts τ ✿ Pr [ XoP ❣✐✈❡s τ ] ≥ 1 − ε Pr [ f ❣✐✈❡s τ ] ❚❤❡♥✱ Adv prf XoP ( q ) ≤ ε + Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] ✶✻ ✴ ✷✸

❣✐✈❡s ❋♦r ❛♥② ❣♦♦❞ tr❛♥s❝r✐♣t✿ ❣✐✈❡s ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P ❍✲❈♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡ ❬P❛t✾✶✱P❛t✵✽✱❈❙✶✹❪ ▲❡t ε ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❣♦♦❞ tr❛♥s❝r✐♣ts τ ✿ Pr [ XoP ❣✐✈❡s τ ] ≥ 1 − ε Pr [ f ❣✐✈❡s τ ] ❚❤❡♥✱ Adv prf XoP ( q ) ≤ ε + Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] • ❇❛❞ tr❛♥s❝r✐♣t✿ ✐❢ y i = 0 ❢♦r s♦♠❡ i • Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] = q/ 2 n ✶✻ ✴ ✷✸

❣✐✈❡s ▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P ❍✲❈♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡ ❬P❛t✾✶✱P❛t✵✽✱❈❙✶✹❪ ▲❡t ε ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❣♦♦❞ tr❛♥s❝r✐♣ts τ ✿ Pr [ XoP ❣✐✈❡s τ ] ≥ 1 − ε Pr [ f ❣✐✈❡s τ ] ❚❤❡♥✱ Adv prf XoP ( q ) ≤ ε + Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] • ❇❛❞ tr❛♥s❝r✐♣t✿ ✐❢ y i = 0 ❢♦r s♦♠❡ i • Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] = q/ 2 n • ❋♦r ❛♥② ❣♦♦❞ tr❛♥s❝r✐♣t✿ • Pr [ XoP ❣✐✈❡s τ ] ≥ (2 n ) 2 q 1 · 2 nq (2 n ) 2 q ✶✻ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P ❍✲❈♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡ ❬P❛t✾✶✱P❛t✵✽✱❈❙✶✹❪ ▲❡t ε ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❣♦♦❞ tr❛♥s❝r✐♣ts τ ✿ Pr [ XoP ❣✐✈❡s τ ] ≥ 1 − ε Pr [ f ❣✐✈❡s τ ] ❚❤❡♥✱ Adv prf XoP ( q ) ≤ ε + Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] • ❇❛❞ tr❛♥s❝r✐♣t✿ ✐❢ y i = 0 ❢♦r s♦♠❡ i • Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] = q/ 2 n • ❋♦r ❛♥② ❣♦♦❞ tr❛♥s❝r✐♣t✿ • Pr [ XoP ❣✐✈❡s τ ] ≥ (2 n ) 2 q 1 · 2 nq (2 n ) 2 q 1 • Pr [ f ❣✐✈❡s τ ] = 2 nq ✶✻ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P ❍✲❈♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡ ❬P❛t✾✶✱P❛t✵✽✱❈❙✶✹❪ ▲❡t ε ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❣♦♦❞ tr❛♥s❝r✐♣ts τ ✿ Pr [ XoP ❣✐✈❡s τ ] ≥ 1 − ε Pr [ f ❣✐✈❡s τ ] ❚❤❡♥✱ Adv prf XoP ( q ) ≤ ε + Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] • ❇❛❞ tr❛♥s❝r✐♣t✿ ✐❢ y i = 0 ❢♦r s♦♠❡ i • Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] = q/ 2 n • ❋♦r ❛♥② ❣♦♦❞ tr❛♥s❝r✐♣t✿ � • Pr [ XoP ❣✐✈❡s τ ] ≥ (2 n ) 2 q 1 · ε = 0 2 nq (2 n ) 2 q 1 • Pr [ f ❣✐✈❡s τ ] = 2 nq ✶✻ ✴ ✷✸

▼✐rr♦r ❚❤❡♦r② ❆♣♣❧✐❡❞ t♦ ❳♦P ❍✲❈♦❡✣❝✐❡♥t ❚❡❝❤♥✐q✉❡ ❬P❛t✾✶✱P❛t✵✽✱❈❙✶✹❪ ▲❡t ε ≥ 0 ❜❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❣♦♦❞ tr❛♥s❝r✐♣ts τ ✿ Pr [ XoP ❣✐✈❡s τ ] ≥ 1 − ε Pr [ f ❣✐✈❡s τ ] ❚❤❡♥✱ Adv prf XoP ( q ) ≤ ε + Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] • ❇❛❞ tr❛♥s❝r✐♣t✿ ✐❢ y i = 0 ❢♦r s♦♠❡ i • Pr [ ❜❛❞ tr❛♥s❝r✐♣t ❢♦r f ] = q/ 2 n • ❋♦r ❛♥② ❣♦♦❞ tr❛♥s❝r✐♣t✿ � • Pr [ XoP ❣✐✈❡s τ ] ≥ (2 n ) 2 q 1 · ε = 0 2 nq (2 n ) 2 q 1 • Pr [ f ❣✐✈❡s τ ] = 2 nq Adv prf XoP ( q ) ≤ q/ 2 n ✶✻ ✴ ✷✸

❳♦r ♦❢ ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♠✐❞❞❧❡ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✬s ❛❧❧ ✉♥✐q✉❡✱ ✬s ♠❛② ❝♦❧❧✐❞❡ ❊❉▼ x p 1 p 2 y x ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } ✶✼ ✴ ✷✸

❳♦r ♦❢ ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♠✐❞❞❧❡ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✬s ❛❧❧ ✉♥✐q✉❡✱ ✬s ♠❛② ❝♦❧❧✐❞❡ ❊❉▼ x p 1 p 2 y x ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } ✶✼ ✴ ✷✸

❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✬s ❛❧❧ ✉♥✐q✉❡✱ ✬s ♠❛② ❝♦❧❧✐❞❡ ❊❉▼ x p 1 p 2 y x ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❳♦r ♦❢ ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♠✐❞❞❧❡ ✶✼ ✴ ✷✸

❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✬s ❛❧❧ ✉♥✐q✉❡✱ ✬s ♠❛② ❝♦❧❧✐❞❡ ❊❉▼ x p 1 p 2 y x ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❳♦r ♦❢ ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♠✐❞❞❧❡ • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p 1 ( x i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ y i �→ p − 1 2 ( y i ) =: P b i ✶✼ ✴ ✷✸

✬s ❛❧❧ ✉♥✐q✉❡✱ ✬s ♠❛② ❝♦❧❧✐❞❡ ❊❉▼ x p 1 p 2 y x ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❳♦r ♦❢ ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♠✐❞❞❧❡ • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p 1 ( x i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ y i �→ p − 1 2 ( y i ) =: P b i • ❙②st❡♠ ♦❢ q ❡q✉❛t✐♦♥s P a i ⊕ P b i = x i ✶✼ ✴ ✷✸

❊❉▼ x p 1 p 2 y x ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( x 1 , y 1 ) , . . . , ( x q , y q ) } • ❳♦r ♦❢ ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♠✐❞❞❧❡ • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ x i �→ p 1 ( x i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ y i �→ p − 1 2 ( y i ) =: P b i • ❙②st❡♠ ♦❢ q ❡q✉❛t✐♦♥s P a i ⊕ P b i = x i • x i ✬s ❛❧❧ ✉♥✐q✉❡✱ y i ✬s ♠❛② ❝♦❧❧✐❞❡ ✶✼ ✴ ✷✸

❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ❛s ❢♦r ❛❧❧ ▼❛① tr❡❡ s✐③❡ ✿ ♣r♦✈✐❞❡❞ ♥♦ ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ ■❢ ✿ ❛t ❧❡❛st s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s ❍✲❝♦❡✣❝✐❡♥t t❡❝❤♥✐q✉❡✿ ❝♦✈❡rs ✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② ❊❉▼ P a ξ 1+ P a q ✕ ξs + P a 1 1 1 x q ✕ ξ s + x ξ 1 + x 1 1 1 P a 2 P a ξ 1+ P a q ✕ ξs + x q ✕ ξ s + x ξ 1 + x 2 2 2 2 2 P b 1 P b 2 · · · P b s ξ 2 x ξ 1 + x q x ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ✶✽ ✴ ✷✸

❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ❛s ❢♦r ❛❧❧ ▼❛① tr❡❡ s✐③❡ ✿ ♣r♦✈✐❞❡❞ ♥♦ ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ ■❢ ✿ ❛t ❧❡❛st s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s ❍✲❝♦❡✣❝✐❡♥t t❡❝❤♥✐q✉❡✿ ❊❉▼ P a ξ 1+ P a q ✕ ξs + P a 1 1 1 x q ✕ ξ s + x ξ 1 + x 1 1 1 P a 2 P a ξ 1+ P a q ✕ ξs + x q ✕ ξ s + x ξ 1 + x 2 2 2 2 2 P b 1 P b 2 · · · P b s ξ 2 x ξ 1 + x q x ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ❝♦✈❡rs ✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② ✶✽ ✴ ✷✸

■❢ ✿ ❛t ❧❡❛st s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s ❍✲❝♦❡✣❝✐❡♥t t❡❝❤♥✐q✉❡✿ ❊❉▼ P a ξ 1+ P a q ✕ ξs + P a 1 1 1 x q ✕ ξ s + x ξ 1 + x 1 1 1 P a 2 P a ξ 1+ P a q ✕ ξs + x q ✕ ξ s + x ξ 1 + x 2 2 2 2 2 P b 1 P b 2 · · · P b s ξ 2 x ξ 1 + x q x ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ❝♦✈❡rs ✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p 1 • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ❛s x i � = x j ❢♦r ❛❧❧ i � = j • ▼❛① tr❡❡ s✐③❡ ξ + 1 ✿ ♣r♦✈✐❞❡❞ ♥♦ ( ξ + 1) ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ ✶✽ ✴ ✷✸

❍✲❝♦❡✣❝✐❡♥t t❡❝❤♥✐q✉❡✿ ❊❉▼ P a ξ 1+ P a q ✕ ξs + P a 1 1 1 x q ✕ ξ s + x ξ 1 + x 1 1 1 P a 2 P a ξ 1+ P a q ✕ ξs + x q ✕ ξ s + x ξ 1 + x 2 2 2 2 2 P b 1 P b 2 · · · P b s ξ 2 x ξ 1 + x q x ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ❝♦✈❡rs ✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p 1 • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ❛s x i � = x j ❢♦r ❛❧❧ i � = j • ▼❛① tr❡❡ s✐③❡ ξ + 1 ✿ ♣r♦✈✐❞❡❞ ♥♦ ( ξ + 1) ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ • ■❢ ξ 2 q ≤ 2 n / 67 ✿ ❛t ❧❡❛st (2 n ) s · (2 n − 1) q s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s 2 nq ✶✽ ✴ ✷✸

❊❉▼ P a ξ 1+ P a q ✕ ξs + P a 1 1 1 x q ✕ ξ s + x ξ 1 + x 1 1 1 P a 2 P a ξ 1+ P a q ✕ ξs + x q ✕ ξ s + x ξ 1 + x 2 2 2 2 2 P b 1 P b 2 · · · P b s ξ 2 x ξ 1 + x q x ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ❝♦✈❡rs ✐♥❞❡♣❡♥❞❡♥t ♣❡r♠✉t❛t✐♦♥s ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p 1 • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ❛s x i � = x j ❢♦r ❛❧❧ i � = j • ▼❛① tr❡❡ s✐③❡ ξ + 1 ✿ ♣r♦✈✐❞❡❞ ♥♦ ( ξ + 1) ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ • ■❢ ξ 2 q ≤ 2 n / 67 ✿ ❛t ❧❡❛st (2 n ) s · (2 n − 1) q s♦❧✉t✐♦♥s t♦ ✉♥❦♥♦✇♥s 2 nq � q EDM ( q ) ≤ q/ 2 n + • ❍✲❝♦❡✣❝✐❡♥t t❡❝❤♥✐q✉❡✿ Adv prf � / 2 nξ ξ +1 ✶✽ ✴ ✷✸

❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ❊①tr❛ ✐ss✉❡✿ ♠❛② ❝♦❧❧✐❞❡ ❊❲❈❉▼ ν p 1 p 2 ν t h ( m ) ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( ν 1 , m 1 , t 1 ) , . . . , ( ν q , m q , t q ) } ✶✾ ✴ ✷✸

❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ❙②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ❊①tr❛ ✐ss✉❡✿ ♠❛② ❝♦❧❧✐❞❡ ❊❲❈❉▼ ν p 1 p 2 ν t h ( m ) ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( ν 1 , m 1 , t 1 ) , . . . , ( ν q , m q , t q ) } ✶✾ ✴ ✷✸

❊①tr❛ ✐ss✉❡✿ ♠❛② ❝♦❧❧✐❞❡ ❊❲❈❉▼ ν p 1 p 2 ν t h ( m ) ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( ν 1 , m 1 , t 1 ) , . . . , ( ν q , m q , t q ) } • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ν i �→ p 1 ( ν i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ t i �→ p − 1 2 ( t i ) =: P b i • ❙②st❡♠ ♦❢ q ❡q✉❛t✐♦♥s P a i ⊕ P b i = ν i ⊕ h ( m i ) ✶✾ ✴ ✷✸

❊❲❈❉▼ ν p 1 p 2 ν t h ( m ) ●❡♥❡r❛❧ ❙❡tt✐♥❣ • ❆❞✈❡rs❛r② ❣❡ts tr❛♥s❝r✐♣t τ = { ( ν 1 , m 1 , t 1 ) , . . . , ( ν q , m q , t q ) } • ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ ν i �→ p 1 ( ν i ) =: P a i ❛♥❞ ❊❛❝❤ t✉♣❧❡ ❝♦rr❡s♣♦♥❞s t♦ t i �→ p − 1 2 ( t i ) =: P b i • ❙②st❡♠ ♦❢ q ❡q✉❛t✐♦♥s P a i ⊕ P b i = ν i ⊕ h ( m i ) • ❊①tr❛ ✐ss✉❡✿ ν i ⊕ h ( m i ) ♠❛② ❝♦❧❧✐❞❡ ✶✾ ✴ ✷✸

■❢ ✿ ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ♣r♦✈✐❞❡❞ ✐♥ ❛❧❧ tr❡❡s ▼❛① tr❡❡ s✐③❡ ✿ ♣r♦✈✐❞❡❞ ♥♦ ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ ❊❲❈❉▼ P a ξ 1+ P a 1 P a q ✕ ξs + 1 ν ξ 1 + 1 ν ν 1 ⊕ h ( m 1 ) q ✕ ξ 1 ⊕ h ( m ξ 1 + + s 1 ⊕ h ( m ν q ✕ ξ s + q ✕ ξ ν ξ 1 + 1 ) ν 2 + P a ξ 1+ P a q ✕ ξs + 2 ⊕ h ( m s ) P a 2 ⊕ h 2 ⊕ h ( m ξ 1 + 1 ( m 2 ) 2 2 ) 2 q ✕ ξ s 2 ) + P b 1 P b 2 · · · P b s ) ν ξ 1 ⊕ h ( m ξ 1 ) ξ 2 + ) m ξ 1 m ( q h ( h ⊕ ⊕ ν ξ 2 q + ν ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ✷✵ ✴ ✷✸

■❢ ✿ ❊❲❈❉▼ P a ξ 1+ P a 1 P a q ✕ ξs + 1 ν ξ 1 + 1 ν ν 1 ⊕ h ( m 1 ) q ✕ ξ 1 ⊕ h ( m ξ 1 + + s 1 ⊕ h ( m ν q ✕ ξ s + q ✕ ξ ν ξ 1 + 1 ) ν 2 + P a ξ 1+ P a q ✕ ξs + 2 ⊕ h ( m s ) P a 2 ⊕ h 2 ⊕ h ( m ξ 1 + 1 ( m 2 ) 2 2 ) 2 q ✕ ξ s 2 ) + P b 1 P b 2 · · · P b s ) ν ξ 1 ⊕ h ( m ξ 1 ) ξ 2 + ) m ξ 1 m ( q h ( h ⊕ ⊕ ν ξ 2 q + ν ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p 1 • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ♣r♦✈✐❞❡❞ ν i ⊕ h ( m i ) � = ν j ⊕ h ( m j ) ✐♥ ❛❧❧ tr❡❡s • ▼❛① tr❡❡ s✐③❡ ξ + 1 ✿ ♣r♦✈✐❞❡❞ ♥♦ ( ξ + 1) ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ ✷✵ ✴ ✷✸

❊❲❈❉▼ P a ξ 1+ P a 1 P a q ✕ ξs + 1 ν ξ 1 + 1 ν ν 1 ⊕ h ( m 1 ) q ✕ ξ 1 ⊕ h ( m ξ 1 + + s 1 ⊕ h ( m ν q ✕ ξ s + q ✕ ξ ν ξ 1 + 1 ) ν 2 + P a ξ 1+ P a q ✕ ξs + 2 ⊕ h ( m s ) P a 2 ⊕ h 2 ⊕ h ( m ξ 1 + 1 ( m 2 ) 2 2 ) 2 q ✕ ξ s 2 ) + P b 1 P b 2 · · · P b s ) ν ξ 1 ⊕ h ( m ξ 1 ) ξ 2 + ) m ξ 1 m ( q h ( h ⊕ ⊕ ν ξ 2 q + ν ξ 1 P a ξ 1 P a ξ 1+ P a q ξ 2 ❆♣♣❧②✐♥❣ ❘❡❧❛①❡❞ ▼✐rr♦r ❚❤❡♦r② • ❈✐r❝❧❡✲❢r❡❡✿ ♥♦ ❝♦❧❧✐s✐♦♥s ✐♥ ✐♥♣✉ts t♦ p 1 • ◆♦♥✲❞❡❣❡♥❡r❛t❡✿ ♣r♦✈✐❞❡❞ ν i ⊕ h ( m i ) � = ν j ⊕ h ( m j ) ✐♥ ❛❧❧ tr❡❡s • ▼❛① tr❡❡ s✐③❡ ξ + 1 ✿ ♣r♦✈✐❞❡❞ ♥♦ ( ξ + 1) ✲❢♦❧❞ ❝♦❧❧✐s✐♦♥ � q EWCDM ( q ) ≤ q/ 2 n + ǫ/ 2 n + • ■❢ ξ 2 q ≤ 2 n / 67 ✿ Adv prf � q / 2 nξ � � 2 ξ +1 ✷✵ ✴ ✷✸

✐❞❡♥t✐❝❛❧ ❡q✉✐✈❛❧❡♥t ✐s ❛t ❧❡❛st ❛s s❡❝✉r❡ ❛s ■❢ ✿ ❊❉▼❉ p 1 p 2 y x ✷✶ ✴ ✷✸

❡q✉✐✈❛❧❡♥t ✐s ❛t ❧❡❛st ❛s s❡❝✉r❡ ❛s ■❢ ✿ ❊❉▼❉ p 1 p 1 p 2 y p 1 p 2 y x x ✐❞❡♥t✐❝❛❧ ✷✶ ✴ ✷✸

✐s ❛t ❧❡❛st ❛s s❡❝✉r❡ ❛s ■❢ ✿ ❊❉▼❉ p 1 p 1 p 1 p 2 y p 1 p 2 y p 3 y x x x ✐❞❡♥t✐❝❛❧ ❡q✉✐✈❛❧❡♥t ✷✶ ✴ ✷✸

❊❉▼❉ p 1 p 1 p 1 p 2 y p 1 p 2 y p 3 y x x x ✐❞❡♥t✐❝❛❧ ❡q✉✐✈❛❧❡♥t • EDMD ✐s ❛t ❧❡❛st ❛s s❡❝✉r❡ ❛s XoP • ■❢ q ≤ 2 n / 67 ✿ Adv prf EDMD ( D ) ≤ q/ 2 n ✷✶ ✴ ✷✸

❊❉▼❉ ✐♥❞❡♣❡♥❞❡♥t✿ ❝❛s❝❛❞✐♥❣ ❤❛s ❧✐♠✐t❡❞ ✐♥✢✉❡♥❝❡ ❙❧✐❞✐♥❣ ✐ss✉❡s ✐❢ ❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ s❡❝✉r✐t② ❙✐♥❣❧❡✲❑❡② ❱❛r✐❛♥ts❄ ❊✭❲❈✮❉▼ p 1 p 2 y x h ( m ) • ✏❳♦P ✐♥ t❤❡ ♠✐❞❞❧❡✑ r❡❧✐❡s ♦♥ ✐♥✈❡rt✐♥❣ p 2 • ❚r✐❝❦ ❢❛✐❧s ✐❢ p 1 = p 2 ✷✷ ✴ ✷✸

❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ s❡❝✉r✐t② ❙✐♥❣❧❡✲❑❡② ❱❛r✐❛♥ts❄ ❊✭❲❈✮❉▼ ❊❉▼❉ p 1 p 2 y p 1 p 2 y x x h ( m ) • ✏❳♦P ✐♥ t❤❡ ♠✐❞❞❧❡✑ • p 1 , p 2 ✐♥❞❡♣❡♥❞❡♥t✿ r❡❧✐❡s ♦♥ ✐♥✈❡rt✐♥❣ p 2 ❝❛s❝❛❞✐♥❣ ❤❛s ❧✐♠✐t❡❞ ✐♥✢✉❡♥❝❡ • ❚r✐❝❦ ❢❛✐❧s ✐❢ p 1 = p 2 • ❙❧✐❞✐♥❣ ✐ss✉❡s ✐❢ p 1 = p 2 ✷✷ ✴ ✷✸

❙✐♥❣❧❡✲❑❡② ❱❛r✐❛♥ts❄ ❊✭❲❈✮❉▼ ❊❉▼❉ p 1 p 2 y p 1 p 2 y x x h ( m ) • ✏❳♦P ✐♥ t❤❡ ♠✐❞❞❧❡✑ • p 1 , p 2 ✐♥❞❡♣❡♥❞❡♥t✿ r❡❧✐❡s ♦♥ ✐♥✈❡rt✐♥❣ p 2 ❝❛s❝❛❞✐♥❣ ❤❛s ❧✐♠✐t❡❞ ✐♥✢✉❡♥❝❡ • ❚r✐❝❦ ❢❛✐❧s ✐❢ p 1 = p 2 • ❙❧✐❞✐♥❣ ✐ss✉❡s ✐❢ p 1 = p 2 ❈♦♥❥❡❝t✉r❡✿ ♦♣t✐♠❛❧ 2 n s❡❝✉r✐t② ✷✷ ✴ ✷✸

❖♣❡♥ ◗✉❡st✐♦♥s ❙✐♥❣❧❡✲❦❡② ✈❛r✐❛♥ts❄ ❉✉❛❧ ♦❢ ❊❲❈❉▼❄ ❋✉rt❤❡r ❛♣♣❧✐❝❛t✐♦♥s ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦ ❈♦♥❝❧✉s✐♦♥ ▼✐rr♦r ❚❤❡♦r② • P♦✇❡r❢✉❧ ❜✉t ✉♥❞❡r❡st✐♠❛t❡❞ t❡❝❤♥✐q✉❡ • ■♠♣❧✐❡s ✭❛❧♠♦st✮ ♦♣t✐♠❛❧ s❡❝✉r✐t② ♦❢ ❊✭❲❈✮❉▼ • ■♠♣❧✐❡s ♦♣t✐♠❛❧ s❡❝✉r✐t② ♦❢ ❊❉▼❉ ✷✸ ✴ ✷✸

❈♦♥❝❧✉s✐♦♥ ▼✐rr♦r ❚❤❡♦r② • P♦✇❡r❢✉❧ ❜✉t ✉♥❞❡r❡st✐♠❛t❡❞ t❡❝❤♥✐q✉❡ • ■♠♣❧✐❡s ✭❛❧♠♦st✮ ♦♣t✐♠❛❧ s❡❝✉r✐t② ♦❢ ❊✭❲❈✮❉▼ • ■♠♣❧✐❡s ♦♣t✐♠❛❧ s❡❝✉r✐t② ♦❢ ❊❉▼❉ ❖♣❡♥ ◗✉❡st✐♦♥s • ❙✐♥❣❧❡✲❦❡② ✈❛r✐❛♥ts❄ • ❉✉❛❧ ♦❢ ❊❲❈❉▼❄ • ❋✉rt❤❡r ❛♣♣❧✐❝❛t✐♦♥s ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦ ✷✸ ✴ ✷✸

❙✉♣♣♦rt✐♥❣ ❙❧✐❞❡s ❙❯PP❖❘❚■◆● ❙▲■❉❊❙ ✷✹ ✴ ✷✸

❉✐st✐♥❣✉✐s❤❡r ❤❛s q✉❡r② ❛❝❝❡ss t♦ ❡✐t❤❡r ♦r tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦r❛❝❧❡ ✐t ❝♦♠♠✉♥✐❝❛t❡s ✇✐t❤ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ IC E k p blockcipher random permutation • ❚✇♦ ♦r❛❝❧❡s✿ E k ✭❢♦r s❡❝r❡t r❛♥❞♦♠ ❦❡② k ✮ ❛♥❞ p ✷✺ ✴ ✷✸

tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦r❛❝❧❡ ✐t ❝♦♠♠✉♥✐❝❛t❡s ✇✐t❤ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ IC E k p blockcipher random permutation distinguisher D • ❚✇♦ ♦r❛❝❧❡s✿ E k ✭❢♦r s❡❝r❡t r❛♥❞♦♠ ❦❡② k ✮ ❛♥❞ p • ❉✐st✐♥❣✉✐s❤❡r D ❤❛s q✉❡r② ❛❝❝❡ss t♦ ❡✐t❤❡r E k ♦r p ✷✺ ✴ ✷✸

Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ IC E k p blockcipher random permutation distinguisher D • ❚✇♦ ♦r❛❝❧❡s✿ E k ✭❢♦r s❡❝r❡t r❛♥❞♦♠ ❦❡② k ✮ ❛♥❞ p • ❉✐st✐♥❣✉✐s❤❡r D ❤❛s q✉❡r② ❛❝❝❡ss t♦ ❡✐t❤❡r E k ♦r p • D tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦r❛❝❧❡ ✐t ❝♦♠♠✉♥✐❝❛t❡s ✇✐t❤ ✷✺ ✴ ✷✸

Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ IC E k p blockcipher random permutation distinguisher D • ❚✇♦ ♦r❛❝❧❡s✿ E k ✭❢♦r s❡❝r❡t r❛♥❞♦♠ ❦❡② k ✮ ❛♥❞ p • ❉✐st✐♥❣✉✐s❤❡r D ❤❛s q✉❡r② ❛❝❝❡ss t♦ ❡✐t❤❡r E k ♦r p • D tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦r❛❝❧❡ ✐t ❝♦♠♠✉♥✐❝❛t❡s ✇✐t❤ − Pr [ D p = 1] Adv prp D E k = 1 � � � � E ( D ) = � Pr � ✷✺ ✴ ✷✸

Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ IC F k f one-way function random function distinguisher D • ❚✇♦ ♦r❛❝❧❡s✿ F k ✭❢♦r s❡❝r❡t r❛♥❞♦♠ ❦❡② k ✮ ❛♥❞ f • ❉✐st✐♥❣✉✐s❤❡r D ❤❛s q✉❡r② ❛❝❝❡ss t♦ ❡✐t❤❡r F k ♦r f • D tr✐❡s t♦ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ♦r❛❝❧❡ ✐t ❝♦♠♠✉♥✐❝❛t❡s ✇✐t❤ � � D f = 1 �� Adv prf D F k = 1 � � F ( D ) = − Pr � Pr � � � ✷✻ ✴ ✷✸

❙❡❝✉r✐t② ❜♦✉♥❞✿ ❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s✿ ✐s ❛ s❡❝✉r❡ P❘P ◆✉♠❜❡r ♦❢ ❡♥❝r②♣t❡❞ ❜❧♦❝❦s ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ ✷✼ ✴ ✷✸

❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s✿ ✐s ❛ s❡❝✉r❡ P❘P ◆✉♠❜❡r ♦❢ ❡♥❝r②♣t❡❞ ❜❧♦❝❦s ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ � σ � Adv cpa CTR [ E ] ( σ ) ≤ Adv prp / 2 n E ( σ ) + 2 ✷✼ ✴ ✷✸

❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ � σ � Adv cpa CTR [ E ] ( σ ) ≤ Adv prp / 2 n E ( σ ) + 2 • ❈❚❘ [ E ] ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s✿ • E k ✐s ❛ s❡❝✉r❡ P❘P • ◆✉♠❜❡r ♦❢ ❡♥❝r②♣t❡❞ ❜❧♦❝❦s σ ≪ 2 n/ 2 ✷✼ ✴ ✷✸

❉✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦ ✐♥ ❜❧♦❝❦s✿ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ • m i ⊕ c i ✐s ❞✐st✐♥❝t ❢♦r ❛❧❧ σ ❜❧♦❝❦s • ❯♥❧✐❦❡❧② t♦ ❤❛♣♣❡♥ ❢♦r r❛♥❞♦♠ str✐♥❣ ✷✽ ✴ ✷✸

❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ P❡r♠✉t❛t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · E k E k E k m 1 m 2 m ℓ c 1 c 2 c ℓ • m i ⊕ c i ✐s ❞✐st✐♥❝t ❢♦r ❛❧❧ σ ❜❧♦❝❦s • ❯♥❧✐❦❡❧② t♦ ❤❛♣♣❡♥ ❢♦r r❛♥❞♦♠ str✐♥❣ • ❉✐st✐♥❣✉✐s❤✐♥❣ ❛tt❛❝❦ ✐♥ σ ≈ 2 n/ 2 ❜❧♦❝❦s✿ � σ � / 2 n � Adv cpa CTR [ E ] ( σ ) 2 ✷✽ ✴ ✷✸

❙❡❝✉r✐t② ❜♦✉♥❞✿ ❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s ✐s ❛ s❡❝✉r❡ P❘❋ ❇✐rt❤❞❛② ❜♦✉♥❞ s❡❝✉r✐t② ❧♦ss ❞✐s❛♣♣❡❛r❡❞ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · F k F k F k m 1 m 2 m ℓ c 1 c 2 c ℓ ✷✾ ✴ ✷✸

❈❚❘ ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s ✐s ❛ s❡❝✉r❡ P❘❋ ❇✐rt❤❞❛② ❜♦✉♥❞ s❡❝✉r✐t② ❧♦ss ❞✐s❛♣♣❡❛r❡❞ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · F k F k F k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ F ] ( σ ) ≤ Adv prf F ( σ ) ✷✾ ✴ ✷✸

❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ Ps❡✉❞♦r❛♥❞♦♠ ❋✉♥❝t✐♦♥ n + 1 n + 2 n + ℓ · · · · · · F k F k F k m 1 m 2 m ℓ c 1 c 2 c ℓ • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ F ] ( σ ) ≤ Adv prf F ( σ ) • ❈❚❘ [ F ] ✐s s❡❝✉r❡ ❛s ❧♦♥❣ ❛s F k ✐s ❛ s❡❝✉r❡ P❘❋ • ❇✐rt❤❞❛② ❜♦✉♥❞ s❡❝✉r✐t② ❧♦ss ❞✐s❛♣♣❡❛r❡❞ ✷✾ ✴ ✷✸

❇❡②♦♥❞ ❜✐rt❤❞❛②✲❜♦✉♥❞ ❜✉t ✷① ❛s ❡①♣❡♥s✐✈❡ ❛s ❈❚❘ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ ❳♦P 0 � n +1 1 � n +1 0 � n +2 1 � n +2 0 � n + ℓ 1 � n + ℓ E k E k · · · · · · E k E k E k E k m 1 m 2 m ℓ c 2 c ℓ c 1 • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ XoP ] ( σ ) ≤ Adv prf XoP ( σ ) ✸✵ ✴ ✷✸

❇❡②♦♥❞ ❜✐rt❤❞❛②✲❜♦✉♥❞ ❜✉t ✷① ❛s ❡①♣❡♥s✐✈❡ ❛s ❈❚❘ ❈♦✉♥t❡r ▼♦❞❡ ❇❛s❡❞ ♦♥ ❳♦P 0 � n +1 1 � n +1 0 � n +2 1 � n +2 0 � n + ℓ 1 � n + ℓ E k E k · · · · · · E k E k E k E k m 1 m 2 m ℓ c 2 c ℓ c 1 • ❙❡❝✉r✐t② ❜♦✉♥❞✿ Adv cpa CTR [ XoP ] ( σ ) ≤ Adv prf XoP ( σ ) ≤ Adv prp E (2 σ ) + σ/ 2 n ✸✵ ✴ ✷✸