ᱦᱚᱨᱰᱼᱠᱳᱨ ᱯᱨᱤᱰᱤᱠᱮᱴ

ᱠᱨᱤᱯᱴᱳᱯᱷᱨᱤ ᱨᱮ, ᱢᱤᱫ ᱥᱮᱫ ᱠᱷᱚᱱ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱯᱷ ᱨᱮᱭᱟᱜ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱮᱰᱤᱠᱮᱴ ᱫᱚ ᱦᱩᱭᱩᱜ ᱠᱟᱱᱟ ᱯᱮᱰᱤᱠᱮᱴ ᱵᱤ (ᱟᱭᱮᱢᱟᱜᱟᱱ , ᱢᱤᱫᱴᱟᱝ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱡᱟᱦᱟᱸ ᱨᱮᱭᱟᱜ ᱚᱯᱴᱩᱯᱩᱴ ᱢᱤᱫᱴᱟᱝ ᱵᱤᱴ ᱠᱟᱱᱟ , ᱡᱟᱦᱟᱫᱚ ᱞᱮᱠᱷᱟ ᱞᱟᱹᱜᱤᱫ ᱟᱞᱜᱟ (x′ ᱨᱮᱭᱟᱜ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱞᱮᱠᱟᱛᱮ , ᱢᱮᱱᱠᱷᱟᱱ ᱯᱷ (x′) ᱮᱢ ᱠᱟᱛᱮ ᱞᱮᱠᱷᱟ ᱠᱚᱨᱟᱣ ᱫᱚ ᱟᱹᱰᱤ ᱟᱱᱟᱴ ᱾ ᱟᱹᱨᱤ ᱞᱮᱠᱟᱛᱮ ᱡᱟᱦᱟᱱ ᱯᱚᱨᱟᱢᱮᱥ ᱟᱱᱟᱜ ᱯᱳᱞᱤᱱᱳᱢᱤᱭᱟᱞᱼᱴᱟᱭᱚᱢ ᱵᱟᱝ ᱢᱮᱱᱟᱜᱼᱟ (PPT′ ᱮᱞᱟᱨᱜᱟᱢ ᱡᱟᱦᱟᱸ ᱫᱚ ᱵᱤ (x′ ᱠᱷᱚᱱ ᱯᱷ (x′) ᱨᱮ ᱟᱭᱩᱢᱟᱹᱱ ᱠᱟᱛᱮ , ᱟᱭᱩᱢᱟᱹᱱ ᱞᱮᱠᱟᱛᱮ ᱟᱭᱩᱢᱟᱹᱱ ᱢᱤᱫ ᱥᱮ ᱚᱱᱟ ᱠᱷᱚᱱ ᱰᱷᱮᱨ , ᱟᱨ ᱟᱭᱩᱢᱟᱹᱱ ᱟᱭᱩᱢᱟᱹᱱ x:34 ᱾ ᱮᱴᱟᱜ ᱠᱟᱛᱷᱟ ᱞᱮᱠᱟᱛᱮ , ᱡᱩᱫᱤ x ᱫᱚ ᱟᱭᱩᱢᱟᱹᱱ ᱟᱱᱟᱜ ᱞᱮᱠᱟᱛᱮ ᱟᱭᱩᱢᱟᱹᱞ ᱨᱮ ᱵᱮᱱᱟᱣ ᱦᱳᱭᱳᱜᱼᱟ , ᱮᱱᱠᱷᱟᱱ ᱮᱢ ᱦᱳᱭᱳᱜᱼᱟ f (x′ , ᱡᱟᱦᱟᱸ ᱮᱴᱟᱜ PPT ᱵᱤᱨᱩᱫᱤᱭᱟᱹ ᱥᱩᱢᱩᱝ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱵᱤᱴ ᱵᱤ (xʼ) ᱟᱨ ᱢᱤᱫᱴᱟᱝ ᱟᱭᱩᱢᱟᱹᱱ ᱞᱮᱠᱟᱱ ᱵᱤᱴ ᱨᱮ ᱟᱭᱩᱢᱟᱹᱱᱟᱱ ᱮᱥᱮᱨ ᱥᱟᱶᱛᱮ ᱟᱭᱩᱢᱟᱹᱱ ᱫᱟᱲᱮ ᱥᱟᱶᱛᱮ , x ᱨᱮᱭᱟᱜ ᱡᱮᱞᱮᱧ ᱪᱮᱛᱟᱱ ᱨᱮ ᱵᱷᱮᱜᱟᱨ ᱠᱚᱨᱟᱣ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ ᱾^[᱑]

ᱢᱤᱫᱴᱟᱝ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱦᱚᱸ ᱚᱱᱟ ᱞᱮᱠᱟ ᱞᱮᱠᱟᱛᱮ ᱵᱚᱨᱱᱚᱱ ᱠᱚᱨᱟᱣ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ ᱾ ᱚᱱᱟ ᱫᱚ ᱦᱩᱭᱩᱜ ᱠᱟᱱᱟ , ᱡᱩᱫᱤ ' x ' ᱫᱚ ᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱞᱮᱠᱟᱛᱮ ' ᱨᱮᱱᱰᱚᱢ ' ᱨᱮ ᱵᱟᱪᱷᱱᱟᱣ ᱦᱳᱭᱳᱜᱼᱟ , ᱮᱱᱠᱷᱟᱱ ' f ' ᱮᱢ ᱮᱢ ᱠᱟᱛᱮ ᱡᱟᱦᱟᱱ PPT ᱮᱞᱟᱨᱜᱟᱢ ᱥᱩᱢᱩᱝ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱷᱚᱱᱥᱚᱱ ᱨᱮᱭᱟᱜ ᱜᱚᱱᱚᱝ ' h ' (' x ') ᱟᱨ ' |h ' (' ′′ | ') ᱨᱮᱱᱟᱜ ᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱨᱮᱱᱰᱚᱢ ᱵᱤᱴ ᱠᱚ ᱵᱷᱮᱜᱟᱨ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ , ᱡᱟᱦᱟᱸ ᱫᱚ ' x ' ᱨᱮᱭᱟᱜ ᱡᱮᱞᱮᱧ ᱪᱮᱛᱟᱱ ᱨᱮ ᱟᱹᱰᱤ ᱠᱚᱢ ᱜᱮ ᱥᱤᱜᱤᱞᱮ ᱧᱟᱢ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ ᱾^[᱒]^[᱓]

ᱢᱤᱫᱴᱟᱝ ᱦᱚᱨᱰᱼᱠᱳᱨ ᱵᱨᱮᱰᱤᱠᱮᱴ " ᱚᱱᱛᱩᱛᱚᱱᱛᱚ ᱞᱮᱠᱟᱛᱮ " ᱤᱱᱵᱷᱟᱨᱴᱤᱝ ᱯᱷ " ᱨᱮᱭᱟᱜ ᱦᱚᱨᱰᱱᱮᱥ ᱠᱚ ᱦᱟᱛᱟᱣ ᱞᱮᱫᱟ ᱾

ᱮᱱᱦᱚᱸ ᱢᱤᱫ ᱥᱮᱫᱛᱮ ᱠᱟᱹᱢᱤ ᱫᱚ ᱵᱚᱫᱚᱞ ᱠᱚᱨᱟᱣ ᱟᱹᱰᱤ ᱟᱱᱟᱴ , ᱮᱱᱦᱚᱸ ᱪᱤᱛᱟᱹᱨ ᱯᱷ (x) ᱠᱷᱚᱱ ᱮᱛᱚᱦᱚᱵ ᱪᱤᱛᱟᱹᱨ ᱥᱤ ᱤᱫᱤ ᱠᱟᱛᱮ ᱦᱤᱸᱥ ᱛᱚᱛᱛ ᱞᱮᱠᱷᱟ ᱞᱮᱠᱷᱟ ᱨᱮᱭᱟᱜ ᱠᱟᱹᱢᱤ ᱫᱟᱲᱮᱱᱟᱜ ᱵᱟᱨᱮᱛᱮ ᱡᱟᱦᱟᱱ ᱵᱟᱛᱟᱣᱟᱜ ᱫᱚ ᱵᱟᱝ ᱠᱟᱱᱟ ᱾ ᱡᱮᱞᱮᱠᱟ, ᱡᱚᱠᱷᱚᱱ RSA ᱫᱚ ᱢᱤᱫᱴᱟᱹᱝ ᱦᱚᱨ ᱯᱷᱚᱱᱠᱥᱚᱱ ᱞᱮᱠᱟᱛᱮ ᱠᱚ ᱢᱟᱱᱟᱣ ᱵᱟᱛᱟᱣ ᱮᱫᱟ, ᱯᱨᱟᱭᱤᱢᱮᱡᱽ ᱨᱮᱱᱟᱜ ᱡᱮᱠᱳᱵᱤ ᱥᱤᱢᱵᱚᱞ ᱫᱚ ᱱᱚᱶᱟ ᱪᱤᱛᱟᱹᱨ ᱠᱷᱚᱱ ᱟᱞᱜᱟ ᱛᱮ ᱠᱚᱢᱯᱩᱴᱟᱨ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ: ᱑᱒᱑

ᱱᱚᱶᱟ ᱫᱚ ᱴᱷᱟᱹᱣᱠᱟᱹ ᱠᱟᱱᱟ ᱡᱮ ᱡᱩᱫᱤ ᱢᱤᱫᱼᱛᱮᱼ ᱢᱤᱫ ᱯᱷᱚᱱᱥᱚᱱ ᱨᱮ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱮᱰᱤᱠᱮᱴ ᱢᱮᱱᱟᱜᱼᱟ ᱮᱱᱠᱷᱟᱱ ᱱᱚᱶᱟ ᱫᱚ ᱢᱤᱫ ᱞᱮᱠᱟᱛᱮ ᱦᱳᱭᱳᱜᱼᱟ ᱾ ᱳᱰᱰ ᱜᱳᱞᱰᱨᱟᱭᱤᱠ ᱟᱨ ᱞᱤᱭᱳᱱᱤᱰ ᱞᱮᱵᱷᱤᱱ (1989) ᱩᱫᱩᱜ ᱥᱚᱫᱚᱨ ᱞᱮᱫᱟᱭ ᱡᱮ , ᱪᱮᱫ ᱞᱮᱠᱟᱛᱮ ᱥᱟᱱᱟᱢ ᱢᱤᱫᱼᱥᱟᱠᱷᱟ ᱠᱟᱹᱢᱤ ᱫᱚ ᱢᱤᱫᱴᱟᱝ ᱢᱤᱫᱼᱥᱟᱥᱟᱠᱷᱟ ᱠᱟᱹᱢᱤ ᱧᱟᱢ ᱞᱟᱹᱜᱤᱫ ᱟᱹᱰᱤ ᱠᱚᱢ ᱞᱮᱠᱟᱛᱮ ᱵᱚᱱᱚᱫᱚᱞ ᱠᱚᱨᱟᱣ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ , ᱡᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱢᱤᱫᱴᱟᱝ ᱜᱚᱴᱟᱵᱩᱴᱟᱹ ᱦᱚᱨᱰᱼᱠᱳᱨ ᱵᱨᱮᱰᱤᱠᱮᱴ ᱢᱮᱱᱟᱜᱼᱟ ᱾^[᱔] ᱟᱭᱩᱢᱟᱹᱱ ᱟᱭᱢᱟᱜᱟᱱ ᱯᱷᱚᱱᱥᱚᱱ ᱦᱩᱭᱩᱜ ᱢᱮ ᱾ g (x′r′ = (f (x′R′) ᱡᱟᱦᱟᱸ ᱨᱮ r ᱨᱮᱭᱟᱜ ᱡᱮᱞᱮᱧ x ᱨᱮᱭᱟᱜ ᱡᱮᱞᱮᱧ ᱞᱮᱠᱟ ᱾ ᱟᱭᱩᱢᱟᱹᱱ xj , x ᱨᱮᱭᱟᱜ j ᱟᱱᱟᱜ ᱵᱤᱴ ᱟᱨ rj , r ᱨᱮᱭᱟᱜ j ᱟᱱᱟᱜ ᱵᱮᱴ ᱮ ᱩᱫᱩᱜᱟ ᱾ ᱚᱱᱟ ᱛᱟᱭᱚᱢ ᱾

$b(x,r):=\langle x,r\rangle =\bigoplus _{j}x_{j}r_{j}$

ᱱᱚᱶᱟ ᱫᱚ ' ᱜ ' ᱨᱮᱭᱟᱜ ᱢᱤᱫᱴᱟᱝ ᱦᱚᱨᱰ ᱠᱳᱨ ᱯᱮᱰᱤᱠᱮᱴ ᱾ ᱧᱮᱞ ᱢᱮ ᱡᱮ b (x′ r′ = < x′ rʼ) ᱡᱟᱦᱟᱸᱨᱮ < ′ ′ ·′ ᱵᱷᱮᱠᱴᱟᱨ ᱥᱯᱮᱥ (Z2′ n) ᱨᱮ ᱥᱟᱵᱟᱫ ᱟᱱ ᱤᱱᱟᱨ ᱯᱮᱠᱴᱚᱠ ᱮ ᱩᱫᱩᱜᱟ ᱾ ᱱᱚᱶᱟ ᱵᱨᱮᱰᱤᱠᱮᱴ ᱫᱚ ᱠᱚᱢᱯᱤᱭᱩᱴᱮᱥᱚᱱᱮᱞ ᱟᱱᱟᱴ ᱚᱡᱮᱛᱮ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱡᱟᱦᱟᱸ ᱫᱚ ᱱᱚᱶᱟ ᱞᱮᱠᱷᱟ ᱵᱟᱝ ᱟᱱᱟᱴ ᱚᱡᱮ ᱫᱚ ᱜ (x) ᱛᱚᱛᱛᱟᱹᱨᱤ ᱞᱮᱠᱟᱛᱮ ᱛᱚᱛᱛᱩᱛᱚ ᱾ ᱢᱮᱱᱠᱷᱟᱱ ᱡᱩᱫᱤ ᱱᱚᱶᱟ ᱵᱨᱮᱰᱤᱠᱮᱴ ᱫᱚ ᱠᱟᱹᱢᱤ ᱞᱮᱠᱟᱛᱮ ᱞᱮᱠᱷᱟ ᱞᱟᱹᱜᱤᱫ ᱡᱟᱦᱟᱱ ᱮᱞᱟᱨᱜᱟᱢ ᱢᱮᱱᱟᱜ ᱟ , ᱮᱱᱠᱷᱟᱱ ᱟᱨ ᱢᱤᱫᱴᱟᱝ ᱮᱞᱟᱨᱜᱟᱢ ᱢᱮᱱᱟᱜᱼᱟ ᱡᱟᱦᱟᱸ ᱫᱚ ᱠᱟᱹᱢᱤ ᱞᱮᱠᱟᱛᱮ ᱯᱷ ᱫᱚ ᱵᱮᱶᱦᱟᱨ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ ᱾

ᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱵᱮᱱᱟᱣ ᱳ (log |x|′) ᱚᱯᱴᱩᱯᱩᱴ ᱵᱤᱴ ᱥᱟᱶᱛᱮ ᱢᱤᱫᱴᱟᱝ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱮ ᱵᱮᱱᱟᱣᱼᱟ ᱾ ᱢᱚᱱᱮ ᱦᱩᱭᱩᱜ ᱢᱮ , ᱯᱷ ᱢᱤᱫᱴᱟᱝ ᱫᱟᱲᱮ ᱟᱱ ᱢᱤᱫᱼᱥᱟᱠᱷᱟᱱᱟᱜ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱾ g (x′ r′ = (f (x′ ′ r′) ᱡᱟᱦᱟᱸ ᱨᱮ | r| = 2|x| ᱞᱟᱹᱭ ᱢᱮ ᱾ ᱢᱤᱫᱴᱟᱝ ᱡᱮᱞᱮᱧ ᱯᱷᱚᱱᱥᱚᱱ l (n′ = O (log n′ s. t. l (n′′ ≤ n) ᱵᱟᱪᱷᱱᱟᱣ ᱢᱮ ᱾ ᱢᱮ ᱾

$b_{i}(x,r)=\bigoplus _{j}x_{j}r_{i+j}.$

ᱚᱱᱟ ᱛᱟᱭᱚᱢ h (x′ r′:= b1 (x′r′ b2 (x′ R′... bl (|x|′ (x′R′)) ᱢᱤᱫᱴᱟᱝ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱡᱟᱦᱟᱸ ᱨᱮᱭᱟᱜ ᱚᱯᱴᱩᱯᱚᱴ ᱞᱟᱹᱞᱤᱥ l (|x1|′)᱾^[᱕]

ᱡᱟᱦᱟᱸ ᱚᱠᱛᱮ ᱨᱮ ᱤᱱᱯᱩᱛ x ᱨᱮᱭᱟᱜ ᱢᱤᱫᱴᱟᱝ ᱟᱥᱚᱞ ᱵᱤᱴ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱦᱳᱭᱳᱜᱼᱟ ᱾ ᱫᱟᱹᱭᱠᱟᱹ ᱞᱮᱠᱟᱛᱮ , ᱟᱨ ᱮᱥ ᱮ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱨᱮ ᱡᱚᱛᱚ ᱵᱤᱴ ᱤᱱᱯᱩᱴ ᱫᱚ ᱟᱨ ᱮᱥ ᱮ ᱨᱮᱭᱟᱜ ᱢᱤᱫᱴᱟᱝ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱮᱰᱤᱠᱮᱴ ᱟᱨ ᱳ (log |x. x) ᱼ ᱨᱮᱭᱟᱜ ᱵᱞᱳᱠᱠᱚ ᱯᱳᱞᱤᱱᱳᱢᱤᱭᱟᱞ ᱚᱠᱛᱚ ᱨᱮ ᱨᱮᱱᱰᱚᱢ ᱵᱤᱴ ᱥᱴᱨᱤᱝ ᱠᱷᱚᱱ ᱵᱷᱮᱜᱟᱨ ᱵᱟᱝ ᱦᱳᱭᱳᱜᱼᱟ ᱾^[᱖]

ᱦᱚᱨᱰᱼᱠᱳᱨ ᱵᱨᱮᱰᱤᱠᱮᱴ ᱠᱚᱫᱚ ᱡᱟᱦᱟᱱ ᱢᱤᱫᱼᱥᱟᱠᱷᱟᱱᱟᱜ ᱯᱚᱨᱢᱩᱡᱚᱱ ᱠᱷᱚᱱ ᱢᱤᱫᱴᱟᱝ ᱥᱩᱰᱳᱨᱟᱱᱰᱚᱢ ᱡᱮᱱᱟᱴᱟᱨ ᱵᱮᱱᱟᱣ ᱨᱮᱭᱟᱜ ᱰᱟᱦᱟᱨ ᱠᱚ ᱮᱢᱼᱟ ᱾ ᱡᱚᱫᱤ ᱵᱤ ᱢᱤᱫᱼᱥᱟᱛᱚᱠ ᱯᱚᱨᱢᱩᱴᱤᱥᱚᱱ ᱯᱷᱼ ᱟᱨ ᱮᱥ ᱢᱤᱫᱴᱟᱝ ᱨᱮᱱᱰᱚᱢ ᱥᱤᱰᱼ ᱨᱮᱭᱟᱜ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱯᱮᱰᱤᱠᱮᱴ ᱦᱩᱭᱩᱜ ᱠᱷᱟᱱ ᱾

$\{b(f^{n}(s))\}_{n}$

ᱫᱚ ᱢᱤᱫ ᱥᱩᱰᱳᱨᱟᱱᱰᱚᱢ ᱵᱤᱴ ᱥᱤᱠᱣᱮᱸᱥ ᱠᱟᱱᱟ, ᱡᱟᱦᱟᱸᱨᱮ fn ᱫᱚ s ᱨᱮ f ᱮᱯᱞᱤᱠᱮᱥᱚᱱ ᱨᱮᱱᱟᱜ n-th ᱤᱴᱮᱨᱮᱥᱚᱱ ᱦᱩᱭᱩᱜ ᱠᱟᱱᱟ, ᱟᱨ b ᱫᱚ ᱡᱚᱛᱚ ᱨᱟᱣᱩᱱᱰ n ᱛᱮ ᱛᱮᱭᱟᱨ ᱟᱠᱟᱱ ᱦᱟᱨᱰ-ᱠᱚᱨ ᱵᱤᱴ ᱠᱟᱱᱟ: 132

ᱴᱨᱮᱯᱰᱳᱨ ᱢᱤᱫᱼᱥᱟᱠᱷᱟᱱᱟᱜ ᱯᱚᱨᱢᱩᱴᱟᱵᱷᱚᱱ ᱨᱮᱭᱟᱜ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱵᱨᱮᱰᱤᱠᱮᱴ (ᱴᱨᱮᱯᱰᱩᱨ ᱵᱨᱮᱰᱤᱠᱤᱴ ᱧᱩᱛᱩᱢ ᱛᱮ ᱩᱯᱨᱩᱢ) ᱥᱮᱢᱮᱱᱴᱤᱠ ᱞᱮᱠᱟᱛᱮ ᱨᱩᱠᱷᱤᱭᱟᱹᱣᱟᱱ ᱦᱚᱲᱼᱠᱤ ᱮᱱᱠᱨᱤᱯᱥᱚᱱ ᱥᱤᱠᱤᱢ ᱵᱮᱱᱟᱣ ᱞᱟᱹᱜᱤᱫ ᱵᱮᱣᱦᱟᱨ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ ᱾

ᱱᱚᱶᱟ ᱦᱚᱸ ᱧᱮᱞ ᱢᱮ

ᱛᱟᱹᱞᱤᱠᱟᱼᱰᱤᱠᱳᱰᱤᱝ (ᱵᱤᱪᱷᱚᱱᱟ ᱛᱟᱹᱞᱤᱠᱟ ᱰᱤᱠᱳᱰᱤᱝ) ᱜᱳᱞᱰᱨᱤᱪᱼᱞᱤᱵᱷᱤᱱ ᱨᱮᱭᱟᱜ ᱢᱩᱲᱩᱫ ᱦᱤᱸᱥ ᱢᱤᱫ ᱥᱮᱫ ᱠᱷᱚᱱ ᱯᱷᱟᱝᱠᱥᱚᱱ ᱠᱷᱚᱱ ᱦᱟᱨᱰᱼᱠᱳᱨ ᱵᱨᱮᱰᱤᱠᱮᱴ ᱵᱮᱱᱟᱣ ᱞᱟᱹᱜᱤᱫ ᱦᱟᱰᱟᱢᱟᱨᱰ ᱠᱳᱰ ᱨᱮᱭᱟᱜ ᱛᱟᱹᱞᱤᱠᱟ ᱼᱰᱤᱠᱳᱰᱤᱝ ᱞᱟᱹᱜᱤᱫ ᱢᱤᱫᱴᱟᱝ ᱮᱞᱟᱨᱜᱟᱢ ᱞᱮᱠᱟᱛᱮ ᱧᱮᱞ ᱫᱟᱲᱮᱭᱟᱜᱼᱟ ᱾

ᱥᱟᱹᱠᱷᱭᱟᱹᱛ

↑ Definition 2.4 in Lindell, Yehuda. "Foundations of Cryptography 89-856" (PDF). Computer Science, Bar Ilan University. Bar Ilan University. Archived from the original (PDF) on 19 January 2022. Retrieved 11 January 2016.
↑ Goldreich's FoC, vol 1, def 2.5.5.
↑ Definition 3 in Holenstein, Thomas; et al. "Complete Classification of Bilinear Hard-Core Functions" (PDF). IACR eprint. IACR. Retrieved 11 January 2016.
↑ O. Goldreich and L.A. Levin, A Hard-Core Predicate for all One-Way Functions, STOC 1989, pp25–32.
↑ Goldreich's FoC, vol 1, Theorem 2.5.6.
↑ J. Håstad, M. Naslund, The Security of all RSA and Discrete Log Bits (2004): Journal of the ACM, 2004.

[1] Definition 2.4 in Lindell, Yehuda. "Foundations of Cryptography 89-856" (PDF). Computer Science, Bar Ilan University. Bar Ilan University. Archived from the original (PDF) on 19 January 2022. Retrieved 11 January 2016.

[2] Goldreich's FoC, vol 1, def 2.5.5.

[3] Definition 3 in Holenstein, Thomas; et al. "Complete Classification of Bilinear Hard-Core Functions" (PDF). IACR eprint. IACR. Retrieved 11 January 2016.

[GoldLevin-4] O. Goldreich and L.A. Levin, A Hard-Core Predicate for all One-Way Functions, STOC 1989, pp25–32.

[5] Goldreich's FoC, vol 1, Theorem 2.5.6.

[JohanNaslund-6] J. Håstad, M. Naslund, The Security of all RSA and Discrete Log Bits (2004): Journal of the ACM, 2004.

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