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[ᱥᱟᱯᱲᱟᱣ | ᱯᱷᱮᱰᱟᱛ ᱥᱟᱯᱲᱟᱣ]

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[ᱥᱟᱯᱲᱟᱣ | ᱯᱷᱮᱰᱟᱛ ᱥᱟᱯᱲᱟᱣ]

↑ 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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