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ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɍɟɨɪɟɦɚ 4 (ɤɢɬɚɣɫɤɚɹ ɬɟɨɪɟɦɚ ɨɛ ɨɫɬɚɬɤɚɯ). ȿɫɥɢ ɧɚɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ
, …, nk ɩɨɩɚɪɧɨ ɜɡɚɢɦɧɨ ɩɪɨɫɬɵ ɢ n = n1·… ·nk – ɢɯ ɩɪɨɢɡɜɟɞɟɧɢɟ. Ɍɨɝɞɚ ɞɥɹ
n
1
ɥɸɛɵɯ ɱɢɫɟɥ 0 r
< ni , i = 1, …, k ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ
i
{x = ri mod ni, i = 1, …, k}.
ɜɫɟɝɞɚ ɧɟ ɩɭɫɬɨ ɢ ɥɟɠɢɬ ɜ Zn. ȼ ɢɧɨɣ (ɱɚɫɬɨ ɩɨɥɟɡɧɨɣ) ɮɨɪɦɭɥɢɪɨɜɤɟ: ɞɥɹ ɥɸɛɵɯ ɰɟɥɵɯ x ɢ y ɢɦɟɟɦ:
ɯ = y (mod n) ɯ = y (mod ni), i = 1, …, n,
ɝɞɟ ɯ = y (mod n) ɨɡɧɚɱɚɟɬ ɫɪɚɜɧɟɧɢɟ ɩɨ ɦɨɞɭɥɸ n, ɬɨ ɟɫɬɶ (x – y) ɞɟɥɢɬɫɹ ɧɚ n.
§ 3. Ⱥɥɝɨɪɢɬɦ ɲɢɮɪɨɜɚɧɢɹ RSA
Ɍɟɩɟɪɶ ɪɚɡɛɟɪɟɦ ɩɪɢɧɰɢɩ ɲɢɮɪɨɜɚɧɢɹ RSA, ɭɩɨɦɹɧɭɬɵɣ ɜɨ ɜɜɟɞɟɧɢɢ.
Ⱥɥɝɨɪɢɬɦ ɲɢɮɪɨɜɚɧɢɹ RSA ɛɵɥ ɩɪɟɞɥɨɠɟɧ ɢ ɨɩɭɛɥɢɤɨɜɚɧ ɜ 1977 ɝɨɞɭ
ɚɦɟɪɢɤɚɧɫɤɢɦɢ ɦɚɬɟɦɚɬɢɤɚɦɢ ɢ ɧɚɡɜɚɧ ɩɨ ɩɟɪɜɵɦ ɛɭɤɜɚɦ ɢɯ ɮɚɦɢɥɢɣ
(Rivest R. L., Shamir A., Adleman L.). ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɨɧ ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ
ɧɚɢɛɨɥɟɟ ɩɨɩɭɥɹɪɧɵɯ ɚɥɝɨɪɢɬɦɨɜ. ɇɟɤɨɬɨɪɵɟ ɚɜɬɨɪɵ ɫɱɢɬɚɸɬ, ɱɬɨ RSA –
ɧɚɢɛɨɥɟɟ ɱɚɫɬɨ ɭɩɨɬɪɟɛɥɹɟɦɵɣ ɚɥɝɨɪɢɬɦ ɲɢɮɪɨɜɚɧɢɹ.
ɉɪɢɜɟɞɟɦ ɨɫɧɨɜɧɵɟ ɦɨɦɟɧɬɵ ɩɨɫɬɪɨɟɧɢɹ ɚɥɝɨɪɢɬɦɚ.
ȼ ɤɚɱɟɫɬɜɟ ɨɫɧɨɜɧɨɣ ɲɢɮɪɭɸɳɟɣ ɮɭɧɤɰɢɢ ɜ ɫɯɟɦɟ RSA ɩɪɢɧɹɬɚ
e
mod m, (6)
f(x) = x
ɝɞɟ ɱɢɫɥɨ ɯ – ɲɢɮɪɭɟɦɵɣ ɢɫɯɨɞɧɵɣ ɬɟɤɫɬ, ɟ ɢ m – ɡɚɞɚɧɧɵɟ ɧɚɬɭɪɚɥɶɧɵɟ
ɱɢɫɥɚ ɢ ɯ < m.
Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɱɢɫɥɚ ɯ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɪɨɫɬɟɣɲɢɣ ɫɬɚɧɞɚɪɬɧɵɣ
ɩɪɨɰɟɫɫ, ɧɚɡɵɜɚɟɦɵɣ ɩɟɪɟɤɨɞɢɪɨɜɤɨɣ. Ɉɧ ɫɨɫɬɨɢɬ ɜ ɩɨɫɢɦɜɨɥɶɧɨɣ ɡɚɦɟɧɟ
ɛɭɤɜ ɢ ɞɪɭɝɢɯ ɫɢɦɜɨɥɨɜ, ɫɨɞɟɪɠɚɳɢɯɫɹ ɜ ɬɟɤɫɬɟ, ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɢɦ
ɱɢɫɥɚ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɬɟɤɫɬɚ ɧɚ ɚɧɝɥɢɣɫɤɨɦ ɹɡɵɤɟ ɨɛɵɱɧɨ
ɩɪɢɦɟɧɹɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɫɬɚɧɞɚɪɬɧɚɹ ɩɟɪɟɤɨɞɢɪɨɜɤɚ: ɚ = 01, b = 02, …, z = 26, ɩɪɨɛɟɥ = 00,
ɢ ɬ.ɞ. ɉɨɥɭɱɟɧɧɵɟ ɧɚɛɨɪɵ ɰɢɮɪ ɡɚɩɢɫɵɜɚɸɬɫɹ ɞɪɭɝ ɡɚ ɞɪɭɝɨɦ, ɢ ɪɟɡɭɥɶɬɚɬ
ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɨɞɧɨ ɰɟɥɨɟ ɱɢɫɥɨ, ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɢɫɯɨɞɧɨɦɭ ɬɟɤɫɬɭ. ɇɚɩɪɢɦɟɪ, ɬɟɤɫɬ “baza” ɛɭɞɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ ɱɢɫɥɨ ɯ = 02012601.
Ⱦɥɹ ɪɚɫɲɢɮɪɨɜɚɧɢɹ ɫɨɨɛɳɟɧɢɹ ɭ = f(x) ɧɟɨɛɯɨɞɢɦɨ, ɡɧɚɹ ɭ, ɧɚɣɬɢ ɯ ɢɡ
ɭɪɚɜɧɟɧɢɹ
xe = ɭ mod m.
ɉɚɪɚ ɱɢɫɟɥ ɟ ɢ m ɫɱɢɬɚɟɬɫɹ ɢɡɜɟɫɬɧɨɣ, ɨɧɚ ɫɨɫɬɚɜɥɹɟɬ ɨɬɤɪɵɬɵɣ ɤɥɸɱ
ɫɯɟɦɵ ɲɢɮɪɨɜɚɧɢɹ.
Ⱥɜɬɨɪɵ ɫɯɟɦɵ RSA ɩɨɤɚɡɚɥɢ, ɱɬɨ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ x = f
-1
(y) – ɨɛɪɚɬɧɨɣ
ɮɭɧɤɰɢɢ ɞɨɫɬɚɬɨɱɧɨ ɜɵɱɢɫɥɢɬɶ
x = yd mod m, (7)
ɝɞɟ d – ɧɟɤɨɬɨɪɨɟ ɱɢɫɥɨ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɸ:
de mod ij(m) = 1, (8)
31
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɝɞɟ ij – ɮɭɧɤɰɢɹ ɗɣɥɟɪɚ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɨɬɫɸɞɚ, ɢɡ (6) ɢ ɬɟɨɪɟɦɵ ɗɣɥɟɪɚ
ɢɦɟɟɦ
d
mod m = ɯde mod m = ɯ
y
qij(m)
x mod m = x mod m. (9)
Ɉɛɪɚɬɢɬɟ ɜɧɢɦɚɧɢɟ ɧɚ ɜɚɠɧɨɟ ɡɚɦɟɱɚɧɢɟ ɧɢɠɟ.
ɑɢɫɥɨ d ɹɜɥɹɟɬɫɹ ɫɟɤɪɟɬɧɵɦ ɤɥɸɱɨɦ ɫɯɟɦɵ, ɧɟɨɛɯɨɞɢɦɵɦ ɞɥɹ ɪɚɫɲɢɮɪɨɜɚɧɢɹ. ɍɱɚɫɬɧɢɤ ɫɯɟɦɵ, ɡɧɚɸɳɢɣ ɜɫɸ ɨɪɝɚɧɢɡɚɰɢɸ ɫɯɟɦɵ RSA, ɧɨ ɧɟ
ɡɧɚɸɳɢɣ ɫɟɤɪɟɬɧɨɝɨ ɱɢɫɥɚ d, ɞɨɥɠɟɧ ɜɵɱɢɫɥɢɬɶ d ɩɨ ɮɨɪɦɭɥɟ (8). ȼ ɷɬɭ ɮɨɪɦɭɥɭ ɜɯɨɞɢɬ ɜɟɥɢɱɢɧɚ ij(m), ɤɨɬɨɪɭɸ ɦɨɠɧɨ ɥɟɝɤɨ ɜɵɱɢɫɥɢɬɶ, ɡɧɚɹ
ɪɚɡɥɨɠɟɧɢɟ m ɧɚ ɩɪɨɫɬɵɟ ɦɧɨɠɢɬɟɥɢ (ɩɨ ɮɨɪɦɭɥɟ 4) ɫɜɨɣɫɬɜ ɮɭɧɤɰɢɢ ɗɣɥɟɪɚ. Ⱦɥɹ
ɩɨɥɶɡɨɜɚɬɟɥɹ, ɧɟ ɡɧɚɸɳɟɝɨ ɫɟɤɪɟɬɧɨɝɨ ɱɢɫɥɚ d, ɟɞɢɧɫɬɜɟɧɧɚɹ ɜɨɡɦɨɠɧɨɫɬɶ
ɜɵɱɢɫɥɢɬɶ ij(m) – ɪɚɡɥɨɠɢɬɶ m ɧɚ ɩɪɨɫɬɵɟ ɦɧɨɠɢɬɟɥɢ, ɚ ɷɬɚ ɡɚɞɚɱɚ, ɤɚɤ ɭɠɟ
ɭɤɚɡɵɜɚɥɨɫɶ, ɧɟ ɢɦɟɟɬ ɷɮɮɟɤɬɢɜɧɨɝɨ ɚɥɝɨɪɢɬɦɚ.
Ⱦɥɹ ɩɪɢɦɟɪɚ ɜɵɱɢɫɥɢɦ ɱɢɫɥɨ d ɩɨ ɡɚɞɚɧɧɵɦ ɢ ɢɡɜɟɫɬɧɵɦ m = 91,
e = 29. Ɂɞɟɫɶ ɜ
ɤɚɱɟɫɬɜɟ «ɫɟɤɪɟɬɧɨɝɨ ɫɜɨɣɫɬɜɚ» ɢɫɩɨɥɶɡɭɟɬɫɹ ɪɚɡɥɨɠɟɧɢɟ m ɧɚ
ɩɪɨɫɬɵɟ ɦɧɨɠɢɬɟɥɢ (m = pq = 7·13). Ɍɨɝɞɚ
ij(m) = (ɪ–1)(q–1) = 6·12 = 72.
ɑɢɫɥɨ ɟ ɧɟ ɢɦɟɟɬ ɨɛɳɢɯ ɞɟɥɢɬɟɥɟɣ ɧɢ ɫ m = 91, ɧɢ ɫ ij(m) = 72, ɡɧɚɱɢɬ,
ɝɨɞɢɬɫɹ ɜ ɤɚɱɟɫɬɜɟ ɲɢɮɪɨɜɚɧɢɹ. ɇɚɣɞɟɦ d ɩɨ ɚɥɝɨɪɢɬɦɭ ɗɜɤɥɢɞɚ (ɬɨ ɟɫɬɶ
ɇɈȾ(ɟ, ij(m)) = ɇɈȾ(29, 72)):
72 = 2·29 + 14; 29 = 2ǜ14 + 1.
Ɉɬɫɸɞɚ ɢɦɟɟɦ
1 = 29 – 2·14 = 29 – 2·(72 – 2ǜ29) = 5·29 – 2·72.
Ɂɧɚɱɢɬ, d = 5.
ɉɪɟɞɫɬɚɜɢɦ ɬɟɩɟɪɶ ɚɥɝɨɪɢɬɦ ɲɢɮɪɨɜɚɧɢɹ ɢ ɞɟɲɢɮɪɨɜɚɧɢɹ ɩɨ ɫɯɟɦɟ
RSA:
Ⱥɥɝɨɪɢɬɦ 13 (ɫɯɟɦɚ RSA).
function f(a,b){while(a){var z=a; a=b%a; b=z}return z}
p=53; q=37; m=p*q; fm=(p–1)*(q–1); e=18; while(f(e,fm)!=1)e++;
s=e–1; d=1; while(s){s=(s+e)%fm; d++};
x=+prompt('ȼɜɟɫɬɢ ɯ',123); f=x; for(i=1; i<e; i++)f=(f*x)%m;
t='Ɉɬɤɪɵɬɵɣ ɤɥɸɱ (e='+e+', m='+m+') Ɂɚɲɢɮɪɨɜɚɧɨ x='+x+' ɤɚɤ f='+f;
x=f; for(i=1; i<d; i++)x=(f*x)%m;
t+='\nɁɚɤɪɵɬɵɣ ɤɥɸɱ (p='+p+', q='+q+', d='+d+') Ɋɚɫɲɢɮɪɨɜɚɧɨ f='+f;
t+=' ɤɚɤ x='+x
32
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɞɟɫɶ ɦɵ ɡɚɞɚɟɦ ɩɪɨɫɬɵɟ ɱɢɫɥɚ p = 53 ɢ q = 37 ɢ ɧɚɯɨɞɢɦ ɱɢɫɥɨ ɟ ɢɡ ɭɫɥɨɜɢɹ: ɇɈȾ(ɟ, ij(m)) = 1 (ɞɥɹ ɭɞɨɛɫɬɜɚ), ɡɚɬɟɦ d ɢɡ ɮɨɪɦɭɥɵ (8).
Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ d ɜ ɷɬɨɦ ɚɥɝɨɪɢɬɦɟ ɦɨɠɧɨ ɛɵɥɨ ɩɪɢɦɟɧɢɬɶ ɚɥɝɨɪɢɬɦ 12, ɤɚɤ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ.
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 13 ɞɥɹ m = 1961, e = 19.
Ɉɬɤɪɵɬɵɣ ɤɥɸɱ (e = 19, m = 1961) Ɂɚɲɢɮɪɨɜɚɧɨ x = 123 ɤɚɤ f = 1122
ɤɥɸɱ (p = 53, q = 37, d = 1675) Ɋɚɫɲɢɮɪɨɜɚɧɨ f = 1122 ɤɚɤ
x = 123
Ɂɚɤɪɵɬɵɣ
Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ ɫɜɨɟɝɨ ɦɟɬɨɞɚ ɚɜɬɨɪɵ ɫɯɟɦɵ RSA ɜ 1978 ɝɨɞɭ ɡɚɲɢɮɪɨɜɚɥɢ ɮɪɚɡɭ “The magic words are squeamish ossifrage” ɢ ɨɩɭɛɥɢɤɨɜɚɥɢ ɩɨɥɭɱɟɧɧɭɸ f(x) ɫ ɭɤɚɡɚɧɢɟɦ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɨɜ e = 9007 ɢ 129-ɡɧɚɱɧɨɟ m:
m = 114381625757888867669235779976146612010218296721242362562561842
935706935245733897830597123563958705058989075147599290026879543541.
Ⱦɨɩɨɥɧɢɬɟɥɶɧɨ ɫɨɨɛɳɚɥɨɫɶ, ɱɬɨ m = pq, ɝɞɟ p ɢ q – ɧɟɤɨɬɨɪɵɟ ɩɪɨɫɬɵɟ
ɱɢɫɥɚ, ɡɚɩɢɫɵɜɚɟɦɵɟ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 64 ɢ 65 ɞɟɫɹɬɢɱɧɵɦɢ ɡɧɚɤɚɦɢ. ɉɟɪɜɨɦɭ,
ɤɬɨ ɞɟɲɢɮɪɭɟɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɫɨɨɛɳɟɧɢɟ, ɛɵɥɚ ɨɛɟɳɚɧɚ ɧɚɝɪɚɞɚ
ɜ 100 ɞɨɥɥɚɪɨɜ.
ɉɪɨɲɥɨ 16 ɥɟɬ, ɤɨɝɞɚ ɜ 1994 ɝɨɞɭ D. Atkins, M. Graff, A. K. Lenstra
ɢ P. S. Leyland ɫɨɨɛɳɢɥɢ ɨ ɪɚɫɲɢɮɪɨɜɤɟ ɮɪɚɡɵ, ɩɪɟɞɥɨɠɟɧɧɨɣ ɜɵɲɟ. ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɱɢɫɥɚ p ɢ q ɨɤɚɡɚɥɢɫɶ ɪɚɜɧɵɦɢ:
p = 3490529510847650949147849619903898133417764638493387843990820577
q = 32769132993266709549961988190834461413177642967992942539798288533
ȼɵɩɨɥɧɟɧɢɟ ɜɵɱɢɫɥɟɧɢɣ ɩɨɬɪɟɛɨɜɚɥɨ ɤɨɥɨɫɫɚɥɶɧɵɯ ɪɟɫɭɪɫɨɜ. ȼ ɪɚɛɨɬɟ,
ɜɨɡɝɥɚɜɥɹɟɦɨɣ ɱɟɬɵɪɶɦɹ ɚɜɬɨɪɚɦɢ ɢ ɩɪɨɞɨɥɠɚɜɲɟɣɫɹ ɩɨɫɥɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɣ
ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɩɨɞɝɨɬɨɜɤɢ ɩɪɢɦɟɪɧɨ 220 ɞɧɟɣ, ɧɚ ɞɨɛɪɨɜɨɥɶɧɵɯ ɧɚɱɚɥɚɯ ɭɱɚɫɬɜɨɜɚɥɨ ɨɤɨɥɨ 600 ɱɟɥɨɜɟɤ ɢ ɩɪɢɦɟɪɧɨ 1600 ɤɨɦɩɶɸɬɟɪɨɜ, ɨɛɴɟɞɢɧɟɧɧɵɯ
ɜ ɫɟɬɶ Internet. ɑɟɫɬɧɵɦ ɬɪɭɞɨɦ ɡɚɪɚɛɨɬɚɧɧɵɟ 100 ɞɨɥɥɚɪɨɜ ɭɱɚɫɬɧɢɤɢ ɷɬɨɝɨ
ɝɪɚɧɞɢɨɡɧɨɝɨ ɩɪɨɟɤɬɚ ɩɟɪɟɞɚɥɢ ɜ ɮɨɧɞ ɪɚɡɜɢɬɢɹ ɩɪɨɝɪɚɦɦɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ.
ȼɚɠɧɨɟ ɡɚɦɟɱɚɧɢɟ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɩɨɥɭɱɟɧɢɢ
ɪɚɜɟɧɫɬɜɚ (9) ɦɵ ɩɪɟɞ-
ɩɨɥɨɠɢɥɢ, ɱɬɨ x ɢ m ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɟ ɱɢɫɥɚ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɟɫɥɢ m = pq –
ɩɪɨɢɡɜɟɞɟɧɢɟ ɩɪɨɫɬɵɯ ɱɢɫɟɥ p ɢ q, ɬɨ x ɧɟ ɞɨɥɠɧɨ ɞɟɥɢɬɶɫɹ ɧɚ p ɢ q. Ɉɞɧɚɤɨ,
ɷɬɨ ɧɟ ɟɫɬɶ ɨɛɹɡɚɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ. Ⱦɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɜɨɩɪɨɫɚ ɩɪɢɦɟɧɢɦ ɫɥɟɞɭɸɳɢɣ ɚɥɝɨɪɢɬɦ:
Ⱥɥɝɨɪɢɬɦ 14 (ɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɣ).
u=prompt('ȼɜɟɞɢ ɚ ɢ b','97,15').split(','); a=+u [0]; b=+u [1];
m=a*b; fm=(a–1)*(b–1)+1
s=a; for(i=1; i<b; i++)s=(s*a)%m; s1=b; for(i=1; i<a; i++)s1=(s1*b)%m;
x=b; s2=x; for(i=1; i<fm; i++)s2=(s2*x)%m;
c=''; for(j=1; j<m; j++){s3=j; for(i=1; i<fm; i++)s3=(s3*j)%m; if(s3!=j)c+=j+' '; }
'a='+a+', b='+b+'\na^b='+s+', b^a='+s1+'\n'+x+'^(fm+1)='+s2+'\nc='+c
33
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ȼ ɷɬɨɦ ɚɥɝɨɪɢɬɦɟ ɞɥɹ ɱɢɫɟɥ a, b, m = ab ɢ fm = (a – 1)(b – 1) + 1, ɤɨɬɨ-
ɪɵɯ ɜɵɱɢɫɥɹɸɬɫɹ ɱɢɫɥɚ a
ɜɟɬɫɬɜɟɧɧɨ. Ɍɚɤɠɟ b
ɱɢɫɥɚ j ɨɬ 1 ɞɨ m – 1 ɧɚ ɩɪɟɞɦɟɬ ɜɵɩɨɥɧɟɧɢɹ ɪɚɜɟɧɫɬɜɚ: j = j
ɷɬɨɦ, ɟɫɥɢ ɷɬɨ ɪɚɜɟɧɫɬɜɨ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɬɚɤɢɟ j ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɫ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɚ = 21 ɢ b = 10 x
b
mod m ɢ ba mod m ɢ ɫɪɚɜɧɢɜɚɸɬɫɹ ɫ a ɢ b – ɫɨɨɬ-
fm
ɫɪɚɜɧɢɜɚɟɬɫɹ ɫ b. ɂ, ɧɚɤɨɧɟɰ, ɩɪɨɫɦɚɬɪɢɜɚɸɬɫɹ ɜɫɟ
181
mod 210 = x ɞɥɹ ɥɸɛɨɝɨ ɯ ɨɬ 1 ɞɨ 209. Ɍɨ ɟɫɬɶ ɞɥɹ
fm
mod m, ɩɪɢ
ɲɢɮɪɨɜɚɧɢɹ RSA ɧɟɨɛɹɡɚɬɟɥɶɧɨ ɜɵɛɢɪɚɬɶ ɩɪɨɫɬɵɟ ɱɢɫɥɚ, ɱɬɨ ɞɚɟɬ ɟɳɟ ɛɨɥɟɟ
ɲɢɪɨɤɢɟ ɜɨɡɦɨɠɧɨɫɬɢ ɲɢɮɪɨɜɚɧɢɹ. ɀɟɥɚɟɦ ɭɫɩɟɯɚ!
ȼ ɤɚɱɟɫɬɜɟ ɞɨɦɚɲɧɟɝɨ ɡɚɞɚɧɢɹ ɩɪɟɞɥɚɝɚɟɬɫɹ ɜ ɩɪɢɜɟɞɟɧɧɨɦ ɩɪɢɦɟɪɟ ɡɚɤɨɞɢɪɨɜɚɬɶ ɭɤɚɡɚɧɧɭɸ ɮɪɚɡɭ ɢ, ɩɨɥɭɱɢɜ ɯ, ɧɚɣɬɢ ɲɢɮɪɨɜɚɧɧɨɟ ɫɨɨɛɳɟɧɢɟ
f(x). ɇɚɩɢɫɚɬɶ ɚɥɝɨɪɢɬɦ ɲɢɮɪɨɜɚɧɢɹ ɢ ɞɟɲɢɮɪɨɜɚɧɢɹ ɫ p
ɢ q ɢɡ ɷɬɨɝɨ ɩɪɢɦɟ-
ɪɚ (RSA-129).
Ʉɨɧɟɱɧɨ, ɩɪɢ ɧɚɩɢɫɚɧɢɢ ɚɥɝɨɪɢɬɦɚ ɧɚ ɹɡɵɤɚɯ, ɧɟ ɩɪɢɫɩɨɫɨɛɥɟɧɧɵɯ ɞɥɹ
ɪɚɛɨɬɵ ɫ ɰɟɥɵɦɢ ɱɢɫɥɚɦɢ, ɞɥɹ ɚɪɢɮɦɟɬɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ ɩɨɬɪɟɛɭɸɬɫɹ ɫɜɨɢ
ɚɥɝɨɪɢɬɦɵ. ɇɚɩɪɢɦɟɪ,
Ⱥɥɝɨɪɢɬɦ 15 (ɫɥɨɠɟɧɢɟ ɞɥɢɧɧɵɯ ɱɢɫɟɥ).
a='2985771110987643098534'; b='23847777774447777272727272';
u=a.split('').reverse(); v=b.split('').reverse();
for(i=v.length; i<u.length; i++)v[i]=0; for(i=u.length; i<v.length; i++)u[i]=0;
for(i=0; i<u.length; i++){v[i]=+v[i]; u[i]=+u[i]}r=0; t='';
for(j=0; j<v.length; j++){x=u[j]+v[j]+r; t=(x%10)+t; r=floor(x/10)}
if(r)t=r+t; a+' + '+b+' = '+t+' = '+(-(-a-b))
Ɂɞɟɫɶ ɱɢɫɥɚ a ɢ b ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɤɚɤ ɦɚɫɫɢɜɵ ɫɜɨɢɯ ɰɢɮɪ ɢ ɪɟɡɭɥɶɬɚɬ
ɫɪɚɜɧɢɜɚɟɬɫɹ ɫ ɪɟɡɭɥɶɬɚɬɨɦ, ɩɨɥɭɱɟɧɧɵɦ
ɨɛɵɱɧɵɦ ɫɥɨɠɟɧɢɟɦ
.
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 15.
2985771110987643098534 + 23847777774447777272727272 =
23850763545558764915825806 = 2.3850763545558765e+25
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ ɚɥɝɨɪɢɬɦ ɭɦɧɨɠɟɧɢɹ ɞɥɢɧɧɨɝɨ ɱɢɫɥɚ ɚ ɧɚ
ɰɢɮɪɭ b:
Ⱥɥɝɨɪɢɬɦ 16 (ɭɦɧɨɠɟɧɢɟ ɞɥɢɧɧɨɝɨ ɱɢɫɥɚ ɧɚ ɰɢɮɪɭ).
a='2985771110987643098534'; b=7;
u=a.split('').reverse(); r=0; t='';
for(j=0; j<u.length; j++){x=u[j]*b +r; t=(x%10)+t; r=floor(x/10)}
if(r)t=r+t; a+'*'+b+' = '+t+' = '+(–(–a*b))
Ⱥ ɡɚɬɟɦ, ɨɛɴɟɞɢɧɹɹ ɷɬɢ ɚɥɝɨɪɢɬɦɵ, ɫɨɫɬɚɜɢɬɶ ɚɥɝɨɪɢɬɦ ɭɦɧɨɠɟɧɢɹ ɞɜɭɯ
ɞɥɢɧɧɵɯ ɱɢɫɟɥ.
Ɉɫɬɚɧɨɜɢɦɫɹ ɩɨɞɪɨɛɧɟɟ ɧɚ ɨɩɟɪɚɰɢɢ ɜɨɡɜɟɞɟɧɢɹ ɜ ɧɚɬɭɪɚɥɶɧɭɸ ɫɬɟɩɟɧɶ. Ⱦɥɹ ɩɨɹɫɧɟɧɢɹ ɪɚɫɫɦɨɬɪɢɦ ɜɵɱɢɫɥɟɧɢɹ
15
. Ɇɵ ɧɟ ɛɭɞɟɦ ɭɦɧɨɠɚɬɶ
ɯ
15 ɪɚɡ ɯ ɧɚ ɯ, ɚ ɛɭɞɟɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜɵɩɨɥɧɹɬɶ ɫɥɟɞɭɸɳɢɟ ɞɟɣɫɬɜɢɹ:
34
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɯ2, ɯ3 = ɯ·ɯ2, ɯ6 = ɯ3·ɯ3, ɯ12 = ɯ6·ɯ6, ɯ15 = ɯ12·ɯ3. Ɍɨ ɟɫɬɶ ɜɫɟɝɨ 5 ɞɟɣɫɬɜɢɣ. ȼ ɫɜɹɡɢ
ɫ ɷɬɢɦ ɩɟɪɟɣɞɟɦ ɤ ɫɥɟɞɭɸɳɟɦɭ ɩɚɪɚɝɪɚɮɭ.
§ 4. Ⱥɞɞɢɬɢɜɧɵɟ ɰɟɩɨɱɤɢ
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ 1, 2, ɚ
, ɚ3, …, ɚn = m ɧɚɡɵɜɚɟɬɫɹ ɚɞɞɢɬɢɜɧɨɣ ɰɟ-
2
ɩɨɱɤɨɣ ɞɥɹ ɱɢɫɥɚ m, ɟɫɥɢ ɤɚɠɞɵɣ ɟɟ ɫɥɟɞɭɸɳɢɣ ɱɥɟɧ ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɞɜɭɯ
ɤɚɤɢɯ-ɧɢɛɭɞɶ ɩɪɟɞɵɞɭɳɢɯ: ɚ
= 1, ɚ1 = 2, ɚk = ɚi + ɚj, 1 i, j < k n. ɉɪɢ ɷɬɨɦ
0
ɱɢɫɥɨ n ɧɚɡɵɜɚɸɬ ɞɥɢɧɨɣ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɤɢ, ɧɨ, ɱɚɫɬɨ, ɩɨɞ ɞɥɢɧɨɣ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɤɢ ɩɨɧɢɦɚɸɬ ɧɚɢɦɟɧɶɲɟɟ ɢɡ ɬɚɤɢɯ ɱɢɫɟɥ n ɢ ɨɛɨɡɧɚɱɚɸɬ ɱɟɪɟɡ
l(m). ɇɚɩɪɢɦɟɪ, l(15) = 5, ɬɚɤ ɤɚɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɰɟɩɨɱɤɚ ɦɨɠɟɬ ɢɦɟɬɶ ɜɢɞ:
1, 2, 3, 6, 12, 15 ɢɥɢ 1, 2, 3, 5, 10, 15. Ⱥ ɰɟɩɨɱɤɚ 1, 2, 4, 8, 12, 14, 15 – ɧɟ ɦɢɧɢ-
ɦɚɥɶɧɚɹ, ɬɚɤ ɤɚɤ ɟɟ
ɞɥɢɧɚ ɪɚɜɧɚ 6. Ⱦɪɭɝɢɦ ɩɪɢɦɟɪɨɦ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɤɢ ɹɜɥɹɟɬɫɹ ɢɡɜɟɫɬɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ Ɏɢɛɨɧɚɱɱɢ, ɝɞɟ ɤɚɠɞɵɣ ɫɥɟɞɭɸɳɢɣ
ɱɥɟɧ ɟɫɬɶ ɫɭɦɦɚ ɞɜɭɯ ɩɪɟɞɵɞɭɳɢɯ: 1, 2, 3, 5, 8, 13, 21, … (ɜ ɧɚɲɢɯ ɨɛɨɡɧɚɱɟ-
= 1, ɚ1 = 2, ɚk = ɚ
ɧɢɹɯ ɚ
0
k–1
+ ɚ
, k = 2, 3, …). Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɷɬɢ ɩɨɫɥɟɞɨ-
k–2
ɜɚɬɟɥɶɧɨɫɬɢ ɧɟ ɜɫɟɝɞɚ ɦɢɧɢɦɚɥɶɧɵ: l(8) = 3 (1, 2, 4, 8), ɚ ɞɥɢɧɚ ɰɟɩɨɱɤɢ
1, 2, 3, 5, 8 ɪɚɜɧɚ 4.
Ɂɚɞɚɱɚ ɨ ɧɚɯɨɠɞɟɧɢɢ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɱɢɫɥɚ m ɦɢɧɢɦɚɥɶɧɨɣ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɤɢ ɢ ɞɚɠɟ ɟɟ ɞɥɢɧɵ l(m) ɹɜɥɹɟɬɫɹ ɨɞɧɨɣ ɢɡ ɧɟɪɟɲɟɧɧɵɯ ɡɚɞɚɱ
ɬɟɨɪɢɢ ɱɢɫɟɥ. ɉɨɷɬɨɦɭ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɪɚɡɥɢɱɧɵɟ ɰɟɩɨɱɤɢ, ɧɚɩɪɢɦɟɪ, ɯɨɪɨɲɨ
ɢɡɜɟɫɬɧɵ ɫɥɟɞɭɸɳɢɟ.
ɐɟɩɨɱɤɚ Ȼɪɚɭɷɪɚ – ɚɞɞɢɬɢɜɧɚɹ
ɰɟɩɨɱɤɚ, ɜ ɤɨɬɨɪɨɣ ɤɚɠɞɵɣ ɫɥɟɞɭɸɳɢɣ
ɱɥɟɧ ɨɛɪɚɡɭɟɬɫɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɩɪɟɞɵɞɭɳɟɝɨ:
= ɚ
+ ɚk, k < i, i = 1, 2, …, n.
ɚ
i
i-1
ɐɟɩɨɱɤɚ ɏɚɧɫɟɧɚ – ɚɞɞɢɬɢɜɧɚɹ ɰɟɩɨɱɤɚ, ɞɥɹ ɤɨɬɨɪɨɣ ɫɭɳɟɫɬɜɭɟɬ ɩɨɞɦɧɨɠɟɫɬɜɨ ɇ = {1, 2, b
…, bm}, ɫɨɫɬɨɹɳɟɟ ɢɡ ɷɥɟɦɟɧɬɨɜ, ɬɚɤɢɯ, ɱɬɨ ɤɚɠɞɵɣ
1
ɱɥɟɧ ɰɟɩɨɱɤɢ ɨɛɪɚɡɭɟɬɫɹ ɢɡ ɧɚɢɛɨɥɶɲɟɝɨ ɷɥɟɦɟɧɬɚ ɢɡ ɇ, ɤɨɬɨɪɵɣ ɦɟɧɶɲɟ
ɷɬɨɝɨ ɱɥɟɧɚ.
= bp + ɚk, bp = max{bH, b < ɚi}, i = 1, 2, …, n.
ɚ
i
Ɇɢɧɢɦɚɥɶɧɵɟ ɚɞɞɢɬɢɜɧɵɟ ɰɟɩɨɱɤɢ Ȼɪɚɭɷɪɚ ɢ ɏɚɧɫɟɧɚ ɞɥɹ ɱɢɫɥɚ m ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ l
(m) ɢ lH(m) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
B
Ⱦɥɹ ɨɰɟɧɤɢ ɞɥɢɧɵ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɟɤ ɩɨɥɶɡɭɸɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ: Ȝ(m) = [log
m] – ɭɦɟɧɶɲɟɧɧɚɹ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɚ ɞɜɨɢɱɧɨɣ ɡɚɩɢ-
2
ɫɢ ɱɢɫɥɚ m ɢ ȝ(m) – ɫɭɦɦɚ ɰɢɮɪ ɢɥɢ ɱɢɫɥɨ ɟɞɢɧɢɰ ɜ ɞɜɨɢɱɧɨɣ ɡɚɩɢɫɢ ɱɢɫɥɚ m. ɇɚɩɪɢɦɟɪ, Ȝ(19) = Ȝ(10011
) = 4, ɚ ȝ(19) = 3.
2
Ʉ ɱɢɫɥɭ ɧɟɪɟɲɟɧɧɵɯ ɡɚɞɚɱ ɨɬɧɨɫɹɬɫɹ, ɧɚɩɪɢɦɟɪ, ɫɥɟɞɭɸɳɢɟ:
ȼɟɪɧɨ ɥɢ, ɱɬɨ
l(m) = l
(m), m,
H
35
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
d
ɬɨ ɟɫɬɶ ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ m ɟɝɨ ɚɞɞɢɬɢɜɧɚɹ ɰɟɩɨɱɤɚ ɦɢɧɢɦɚɥɶɧɨɣ ɞɥɢɧɵ ɦɨɠɟɬ
ɛɵɬɶ ɰɟɩɨɱɤɨɣ ɏɚɧɫɟɧɚ?
1. ȼɟɪɧɨ ɥɢ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ m ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ:
l(m) Ȝ(m)+ Ȝ(ȝ(m)).
2. Ƚɢɩɨɬɟɡɚ ɒɨɥɶɰɚ ɨ ɬɨɦ, ɱɬɨ
m
–1) m + l(m) –1.
l(2
Ɂɚɞɚɱɢ 2 ɢ 3 ɞɨɤɚɡɚɧɵ ɞɥɹ ɰɟɩɨɱɟɤ Ȼɪɚɭɷɪɚ ɢ ɏɚɧɫɟɧɚ ɢɦɢ ɠɟ. ɏɨɪɨɲɨ
ɬɚɤɠɟ ɢɡɜɟɫɬɧɵ ɫɥɟɞɭɸɳɢɟ ɬɟɨɪɟɦɵ.
Ɍɟɨɪɟɦɚ 3 (ɛɢɧɚɪɧɵɣ ɦɟɬɨɞ). ɋɩɪɚɜɟɞɥɢɜɨ ɧɟɪɚɜɟɧɫɬɜɨ
l(m) Ȝ(m) + ȝ(m) – 1.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɟɪɟɣɞɟɦ ɤ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɢɫɱɢɫɥɟɧɢɹ, ɬɨɝɞɚ ɧɚ-
ɱɚɥɨ ɰɟɩɨɱɤɢ ɛɭɞɟɬ 1, 10. Ⱦɚɥɟɟ ɦɵ ɛɭɞɟɦ ɞɨɩɢɫɵɜɚɬɶ ɧɭɥɢ, ɬɨ ɟɫɬɶ
ɭɞɜɚɢɜɚɬɶ
ɩɪɟɞɵɞɭɳɟɟ ɱɢɫɥɨ, ɢ ɡɚɦɟɧɹɬɶ ɩɨɫɥɟɞɧɢɣ ɧɨɥɶ ɟɞɢɧɢɰɟɣ, ɬɨ ɟɫɬɶ ɩɪɢɛɚɜɥɹɬɶ
ɟɞɢɧɢɰɭ. ɇɚɩɪɢɦɟɪ, ɞɥɹ m = 19 = 10011
ɢɦɟɟɦ: 1, 10, 100, 1000, 1001, 10010,
2
10011 ɢɥɢ ɜ ɞɟɫɹɬɢɱɧɨɣ ɡɚɩɢɫɢ: 1, 2, 4, 8, 9, 18, 19. Ɂɧɚɱɢɬ, l(19) = 6 (Ȝ(19) = 4,
ȝ(19) = 3). Ⱥɧɚɥɨɝɢɱɧɨ ɩɨɫɬɭɩɚɟɦ ɞɥɹ ɥɸɛɨɝɨ m.
Ɍɟɨɪɟɦɚ 5 (Ⱥ. Ȼɪɚɭɷɪ). ɉɪɢ k < log
l(m) (1+1/k)Ȝ(m) + 2
log2m ɫɩɪɚɜɟɞɥɢɜɨ ɧɟɪɚɜɟɧɫɬɜɨ
2
k–1
– k +2.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɷɬɨɣ ɬɟɨɪɟɦɵ ɩɪɨɜɨɞɢɬɶ ɧɟ ɛɭɞɟɦ. Ɉɧɨ ɨɫɧɨɜɚɧɨ ɧɚ
ɪɚɡɛɢɟɧɢɢ ɱɢɫɥɚ m ɧɚ ɛɥɨɤɢ.
ɉɪɢɜɟɞɟɦ ɧɟɤɨɬɨɪɵɟ ɫɜɨɣɫɬɜɚ ɚɞɞɢɬɢɜɧɵɯ ɰɟɩɨɱɟɤ ɜ ɜɢɞɟ ɭɩɪɚɠɧɟɧɢɣ
ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ.
ɚ) Ⱦɨɤɚɠɢɬɟ ɧɟɪɚɜɟɧɫɬɜɨ log
m l(m) 2log2m.
2
ɛ) Ⱦɨɤɚɠɢɬɟ ɧɟɪɚɜɟɧɫɬɜɨ l(mn) l(m) + l(n) (ɦɟɬɨɞ ɦɧɨɠɢɬɟɥɟɣ).
ɜ) ɉɨɤɚɠɢɬɟ, ɩɪɢɜɟɞɹ ɩɪɢɦɟɪɵ, ɱɬɨ ɢɧɨɝɞɚ ɦɟɬɨɞ ɦɧɨɠɢɬɟɥɟɣ ɥɭɱɲɟ, ɱɟɦ
ɛɢɧɚɪɧɵɣ ɦɟɬɨɞ, ɚ ɢɧɨɝɞɚ ɧɚɨɛɨɪɨɬ. ɂ ɬɚɤɢɯ ɩɪɢɦɟɪɨɜ ɛɟɫɤɨɧɟɱɧɨ ɦɧɨɝɨ.
ɝ) ɉɨɤɚɠɢɬɟ, ɤɚɤ ɫ ɩɨɦɨɳɶɸ ɛɢɧɚɪɧɨɝɨ ɦɟɬɨɞɚ ɦɨɠɧɨ ɧɚ ɩɪɨɫɬɟɣɲɟɦ
ɤɚɥɶɤɭɥɹɬɨɪɟ ɫ ɨɩɟɪɚɰɢɟɣ
ɛɥɢɠɟɧɧɨ ɜɵɱɢɫɥɢɬɶ ɮɭɧɤɰɢɸ ɯ
ɞ) ɇɚɣɞɢɬɟ
10
10
10
, ɧɨ ɛɟɡ ɨɩɟɪɚɰɢɢ ɜɨɡɜɟɞɟɧɢɹ ɜ ɫɬɟɩɟɧɶ, ɩɪɢ-
ɭ
, ɝɞɟ ɭ – ɞɜɨɢɱɧɨ-ɪɚɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɨ.
.
2013mo
ɟ) Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ȝ(m) ɦɨɠɧɨ ɪɟɤɭɪɪɟɧɬɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
ȝ(1) = 1, ȝ(2m) = ȝ(m), ȝ(2m+1) = ȝ(m),
36
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɚ ɮɭɧɤɰɢɸ Ȝ(m) ɦɨɠɧɨ ɪɟɤɭɪɪɟɧɬɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
Ȝ(1) = 0, Ȝ(2m) = Ȝ(2m + 1) = Ȝ(m) + 1.
ɠ) Ⱦɨɤɚɠɢɬɟ ɧɟɪɚɜɟɧɫɬɜɨ
ȝ(m+n) ȝ(m+n) + ȝ(m+n) – ȝ(mn),
ɝɞɟ ȝ(mn) – ɱɢɫɥɨ ɟɞɢɧɢɰ, ɫɬɨɹɳɢɯ ɧɚ ɨɞɢɧɚɤɨɜɵɯ ɦɟɫɬɚɯ ɜ ɞɜɨɢɱɧɨɣ ɡɚɩɢɫɢ ɱɢɫɟɥ m ɢ n. (ɇɟɪɚɜɟɧɫɬɜɨ ɩɨɥɭɱɟɧɨ
ɭɱɟɧɢɰɟɣ 10-ɝɨ ɤɥɚɫɫɚ Ʉɨɫɬɸɪɢɧɨɣ
Ʉɚɬɟɣ ɜ 2010–2011 ɝɨɞɭ, ɢ ɛɵɥɚ ɞɨɤɚɡɚɧɚ ɡɚɞɚɱɚ 3 (ɝɢɩɨɬɟɡɚ ɒɨɥɶɰɚ) ɞɥɹ ɰɟɩɨɱɟɤ ɏɚɧɫɟɧɚ ɞɪɭɝɢɦ, ɨɬɥɢɱɧɵɦ ɨɬ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɏɚɧɫɟɧɚ, ɫɩɨɫɨɛɨɦ).
ɡ) ɇɚɣɬɢ ɚɞɞɢɬɢɜɧɭɸ ɰɟɩɨɱɤɭ ɦɢɧɢɦɚɥɶɧɨɣ ɞɥɢɧɵ ɞɥɹ ɱɢɫɥɚ ɟ = 9007
ɢɡ ɩɪɢɦɟɪɚ RSA-129.
Ⱦɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɫɜɨɣɫɬɜ ɚɞɞɢɬɢɜɧɵɯ ɰɟɩɨɱɟɤ ɩɪɢɜɟɞɟɦ ɫɥɟɞɭɸɳɢɣ
ɚɥɝɨɪɢɬɦ:
Ⱥɥɝɨɪɢɬɦ 17 (ɚɞɞɢɬɢɜɧɵɟ ɰɟɩɨɱɤɢ).
g='19,1,2, '
//19=10011: 19=10011,la=4,mu=3
u=g.split(','); n=u.length-1; if(u[n]<=0)n--; for(i=0; i<=n; i++)u[i]=eval(u[i]);
x=+u[n]; for(i=1; i<n; i++){for(j=i; j<n; j++)if(x-u[i]-u[j]==0)break; if(j<n)break; }
if(i==n)n--; g=''; for(i=0; i<=n; i++)g+=u[i]+',';
t= dv(u[n]); y=(t.length-1); z=t.split('1').length-1;
if(u[0]==u[n])alert('ɉɨɛɟɞɚ
ɡɚ '+(n-1)+' ɲɚɝɨɜ
! ɉɪɨɬɢɜ '+(y+z-1))
t='//'+u[0] +'='+ dv(u[0])+': '+u[n]+'='+t+', la='+y+', mu='+z; ;
u=new Array(); u=forma.value.split('\n'); u[0]="g='"+g+"'"; u[1]=t
g=u[0]; for(i=1; i<u.length; i++)g+='\n'+u[i]; forma.value=g;
function dv(z){var t=''; while(z){if(z%2){t=1+t; z--}else t=0+t; z/=2; }return t}
Ɂɞɟɫɶ ɜ ɩɟɪɟɦɟɧɧɭɸ g (ɜ ɜɟɪɯɧɟɦ ɨɤɧɟ, ɝɞɟ ɧɚɩɢɫɚɧ ɤɨɞ ɩɪɨɝɪɚɦɦɵ) ɞɨɩɢɫɵɜɚɟɬɫɹ ɨɱɟɪɟɞɧɨɣ ɱɥɟɧ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɤɢ (ɩɟɪɟɞ ɤɚɜɵɱɤɨɣ!))
ɢ ɧɚɠɢɦɚɟɬɫɹ ɤɧɨɩɤɚ "calculator". Ⱥɥɝɨɪɢɬɦ ɩɪɨɜɟɪɹɟɬ, ɹɜɥɹɟɬɫɹ ɥɢ ɩɨɥɭɱɟɧɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɚɞɞɢɬɢɜɧɨɣ ɰɟɩɨɱɤɨɣ. ɉɟɪɜɵɦ ɱɢɫɥɨɦ ɷɬɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ (ɩɟɪɜɚɹ ɡɚɩɢɫɶ ɜ g) ɹɜɥɹɟɬɫɹ ɱɢɫɥɨ m, ɞɥɹ ɤɨɬɨɪɨɝɨ ɫɬɪɨɢɬɫɹ
ɞɢɬɢɜɧɚɹ ɰɟɩɨɱɤɚ. ȿɫɥɢ
m ɞɨɫɬɢɝɧɭɬɨ, ɬɨ ɜɵɞɚɟɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɫɨɨɛɳɟ-
ɚɞ-
ɧɢɟ. Ⱦɚɥɟɟ ɬɟɤɫɬ ɤɨɞɚ ɦɟɧɹɟɬɫɹ ɚɥɝɨɪɢɬɦɨɦ ɢ ɜ ɨɤɧɟ ɩɨɹɜɥɹɟɬɫɹ ɧɨɜɵɣ ɤɨɞ,
ɝɨɬɨɜɵɣ ɤ ɢɫɩɨɥɧɟɧɢɸ. (Ɍɟɤɫɬ ɤɨɞɚ – ɜ ɨɛɴɟɤɬɟ forma = document.forms[0].cls,
ɝɞɟ cls – ɢɦɹ ɨɤɧɚ).
ɉɪɟɞɫɬɚɜɥɟɧɧɵɣ ɚɥɝɨɪɢɬɦ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɚɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ
ɩɨɡɧɚɜɚɬɟɥɶɧɭɸ ɢɝɪɭ ɢɥɢ ɩɨɥɨɠɢɬɶ ɜ ɨɫɧɨɜɭ ɩɨɫɥɟɞɧɟɣ.
ɗɤɫɩɟɪɢɦɟɧɬɢɪɭɹ ɫ ɷɬɨɣ ɩɪɨɝɪɚɦɦɨɣ, ɩɨɥɭɱɢɥɢ ɚɞɞɢɬɢɜɧɭɸ ɰɟɩɨɱɤɭ
ɞɥɹ ɱɢɫɥɚ ɟ
= 9007 = 100011001011112 ɢɡ ɩɪɢɦɟɪɚ RSA-129:
37
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
1, 2, 3, 5, 7, 14, 28, 35, 70, 140, 280, 560, 1120, 1125, 2250, 4500, 9000, 9007.
Ɉɞɧɚɤɨ ɧɟ ɞɨɤɚɡɚɧɨ, ɱɬɨ l(9007) = 17, ɬɨ ɟɫɬɶ ɱɬɨ ɷɬɚ ɰɟɩɨɱɤɚ ɦɢɧɢɦɚɥɶɧɚɹ.
ȼ ɡɚɤɥɸɱɟɧɢɟ ɩɪɟɞɥɨɠɢɦ ɫɥɟɞɭɸɳɢɣ ɩɨɞɯɨɞ. Ɋɚɫɫɦɨɬɪɢɦ ɥɢɧɟɣɧɨɟ
ɩɪɨɫɬɪɚɧɫɬɜɨ
L
ɧɚɞ ɩɨɥɟɦ Z2 = {0,1}, ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɨɝɨ ɹɜɥɹɸɬɫɹ ɛɟɫɤɨɧɟɱɧɵɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɢɡ ɧɭɥɟɣ ɢ ɟɞɢɧɢɰ, ɩɪɢɱɟɦ ɟɞɢɧɢɰ ɤɨɧɟɱɧɨɟ
ɱɢɫɥɨ. Ɉɩɟɪɚɰɢɸ ɫɥɨɠɟɧɢɹ ɨɩɪɟɞɟɥɢɦ ɫ ɩɨɦɨɳɶɸ ɫɥɨɠɟɧɢɹ ɜ ɞɜɨɢɱɧɨɣ
ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ, ɚ ɢɦɟɧɧɨ, ɩɭɫɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɚ ɢ b
L
, ɬɚɤ ɤɚɤ
ɨɧɢ ɫɨɫɬɨɹɬ ɢɡ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɟɞɢɧɢɰ, ɬɨ ɤɚɠɞɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
ɦɨɠɧɨ ɩɨɫɬɚɜɢɬɶ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨ ɤɨɧɟɱɧɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ – ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ, ɡɚɩɢɫɚɧɧɭɸ ɧɚɨɛɨɪɨɬ, ɧɚɱɢɧɚɹ ɫ ɩɨɫɥɟɞɧɟɣ ɟɞɢɧɢɰɵ.
ɉɨɥɭɱɢɦ ɡɚɩɢɫɢ ɞɜɭɯ ɧɟɤɨɬɨɪɵɯ ɱɢɫɟɥ ɜ ɞɜɨɢɱɧɨɣ ɫɢɫɬɟɦɟ ɫɱɢɫɥɟɧɢɹ.
ɋɥɨɠɢɦ ɢɯ ɩɨ ɨɛɵɱɧɨɦɭ ɩɪɚɜɢɥɭ, ɪɟɡɭɥɶɬɚɬ ɡɚɩɢɲɟɦ ɜ ɨɛɪɚɬɧɨɦ ɩɨɪɹɞɤɟ
ɢ ɞɨɛɚɜɢɦ ɛɟɫɤɨɧɟɱɧɨ ɧɭɥɟɣ. ɇɚɩɪɢɦɟɪ, ɞɥɹ
ɢ b = (0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, …) ɧɚɣɞɟɦ: 1100101
= 111001101
ɢ ɚ + b = (1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, …).
2
ɚ = (1, 0, 1, 0, 0, 1, 1, 0, 0, …)
+ 1011010002 =
2
Ɇɨɠɧɨ ɜɜɟɫɬɢ ɧɨɪɦɭ ɢ ɩɪɨɢɡɜɟɞɟɧɢɟ, ɤɚɤ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɟɤɬɨ-
L
ɪɨɜ ɚ ɢ b
– ɚ·b ɢ ɧɨɪɦɭ: |ɚ| = ɚ·a. ȼ ɧɚɲɟɦ ɩɪɢɦɟɪɟ, ɚ·b = 2, |ɚ| = 4, |b| = 4,
|ɚb| = 6. ɇɟ ɫɥɨɠɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɩɪɨɢɡɜɟɞɟɧɢɟ ɚ·b – ɧɟ ɹɜɥɹɟɬɫɹ ɫɤɚɥɹɪɧɵɦ
ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɜɜɢɞɭ ɧɟɜɵɩɨɥɧɟɧɢɹ ɨɞɧɨɣ ɢɡ ɚɤɫɢɨɦ ɫɤɚɥɹɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ:
ɚ·(c + b) = ɚ·c+ ɚ·b. ɉɪɨɢɡɜɟɞɟɧɢɟ ɚ·b – ɷɬɨ ɱɢɫɥɨ ɟɞɢɧɢɰ ɚ ɢ b
L
, ɫɬɨɹɳɢɟ ɧɚ
ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɦɟɫɬɚɯ ɢ ɩɨɷɬɨɦɭ ɜ ɫɢɥɭ ɫɜɨɣɫɬɜɚ ɠ) ɚɞɞɢɬɢɜɧɵɯ ɰɟɩɨɱɟɤ ɥɟɝɤɨ
ɩɪɨɜɟɪɢɬɶ ɜɵɩɨɥɧɟɧɢɹ ɧɟɪɚɜɟɧɫɬɜɚ ɬɪɟɭɝɨɥɶɧɢɤɚ: |ɚ + b| |ɚ | + |b|. Ɍɨ ɟɫɬɶ
L
ɹɜɥɹɟɬɫɹ ɥɢɧɟɣɧɵɦ ɧɨɪɦɢɪɨɜɚɧɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɢ ɟɝɨ ɷɥɟɦɟɧɬɵ ɩɪɟɞɫɬɚɜɥɹɸɬ
ɰɟɥɵɟ ɱɢɫɥɚ ɜ ɞɜɨɢɱɧɨɣ ɡɚɩɢɫɢ. ɍɛɟɞɢɬɟɫɶ ɜ ɜɵɩɨɥɧɟɧɢɢ ɜɫɟɯ ɚɤɫɢɨɦ ɥɢɧɟɣɧɨɝɨ ɧɨɪɦɢɪɨɜɚɧɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɚɞ ɤɨɧɟɱɧɵɦ ɩɨɥɟɦ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ
ɢ ɩɨɩɵɬɚɣɬɟɫɶ ɩɪɢɦɟɧɢɬɶ ɷɬɨ ɤ ɚɞɞɢɬɢɜɧɵɦ ɰɟɩɨɱɤɚɦ. ɀɟɥɚɟɦ ɭɫɩɟɯɚ!
§ 5. Ɂɚɞɚɱɚ Ɏɪɨɛɟɧɢɭɫɚ
ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɭɥɭɱɲɟɧɧɨɦɭ ɚɥɝɨɪɢɬɦɭ ȿɜɤɥɢɞɚ (ɚɥɝɨɪɢɬɦ 12) ɪɚɫ-
ɫɤɚɠɟɦ ɨ ɫɥɟɞɭɸɳɟɣ ɩɪɨɛɥɟɦɟ – ɡɚɞɚɱɟ
«ɨ ɪɚɡɦɟɧɟ ɦɨɧɟɬ». ɗɬɚ ɡɚɞɚɱɚ ɢɡɜɟɫɬɧɚ ɬɚɤ ɠɟ, ɤɚɤ ɩɪɨɛɥɟɦɚ Ɏɪɨɛɟɧɢɭɫɚ (ɧɟɦɟɰɤɨɝɨ ɦɚɬɟɦɚɬɢɤɚ Ɏɟɪɞɢɧɚɧɞɚ Ɏɪɨɛɟɧɢɭɫɚ (1849–1917), ɜ ɤɨɬɨɪɨɣ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɱɢɫɥɨ, ɹɜɥɹɸɳɟɟɫɹ ɤɪɭɩɧɟɣɲɟɣ ɞɟɧɟɠɧɨɣ ɫɭɦɦɨɣ, ɧɟ ɧɚɛɢɪɚɟɦɨɣ ɦɨɧɟɬɚɦɢ ɭɤɚɡɚɧɧɵɯ
ɧɨɦɢɧɚɥɨɜ. ɇɚɩɪɢɦɟɪ, ɤɪɭɩɧɟɣɲɚɹ ɫɭɦɦɚ, ɤɨɬɨɪɚɹ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɚ, ɢɫɩɨɥɶɡɭɹ ɬɨɥɶɤɨ ɦɨɧɟɬɵ ɜ 3 ɢ 5 ɟɞɢɧɢɰ, ɫɨɫɬɚɜɥɹɟɬ 7 ɟɞɢɧɢɰ. ȼ
ɞɚɧɧɨɦ
ɫɥɭɱɚɟ ɱɢɫɥɨ 7 ɦɵ ɧɟ ɫɦɨɠɟɦ ɩɨɥɭɱɢɬɶ, ɚ ɥɸɛɨɟ ɛɨɥɶɲɟɟ 7 ɫɦɨɠɟɦ, ɬɨ ɟɫɬɶ
F2(3, 5) = 7. Ɍɨ ɟɫɬɶ ɪɟɲɟɧɢɟɦ ɷɬɨɣ ɡɚɞɚɱɢ ɹɜɥɹɟɬɫɹ ɱɢɫɥɨ 7 ɢɥɢ ɱɢɫɥɨ
Ɏɪɨɛɟɧɢɭɫɚ ɧɚɛɨɪɚ (3, 5).
ɋɭɳɟɫɬɜɭɟɬ ɹɜɧɚɹ ɮɨɪɦɭɥɚ ɞɥɹ ɱɢɫɥɚ Ɏɪɨɛɟɧɢɭɫɚ, ɤɨɝɞɚ ɟɫɬɶ ɬɨɥɶɤɨ
ɞɜɟ ɦɨɧɟɬɵ. ȿɫɥɢ ɤɨɥɢɱɟɫɬɜɨ ɦɨɧɟɬ ɬɪɢ ɢɥɢ ɛɨɥɶɲɟ, ɮɨɪɦɭɥɚ ɧɟɢɡɜɟɫɬɧɚ, ɧɨ
ɞɥɹ ɥɸɛɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɱɢɫɥɚ ɦɨɧɟɬ ɫɭɳɟɫɬɜɭɟɬ
ɚɥɝɨɪɢɬɦ ɜɵɱɢɫɥɟɧɢɹ
ɱɢɫɥɚ Ɏɪɨɛɟɧɢɭɫɚ ɡɚ ɩɨɥɢɧɨɦɢɚɥɶɧɨɟ ɜɪɟɦɹ. Ɉɞɧɚɤɨ ɧɢ ɨɞɢɧ ɢɡ ɷɬɢɯ ɚɥɝɨ-
38
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɪɢɬɦɨɜ ɧɟ ɦɨɠɟɬ ɪɟɲɢɬɶ ɨɛɳɭɸ ɡɚɞɚɱɭ. Ɍɨ ɟɫɬɶ, ɤɨɝɞɚ ɱɢɫɥɨ ɦɨɧɟɬ ɦɨɠɟɬ
ɛɵɬɶ ɫɤɨɥɶ ɭɝɨɞɧɨ ɛɨɥɶɲɨɟ, ɡɚɞɚɱɚ ɫɬɚɧɨɜɢɬɫɹ NP-ɬɪɭɞɧɨɣ.
ɉɪɢɜɟɞɟɦ ɛɨɥɟɟ ɱɟɬɤɭɸ ɮɨɪɦɭɥɢɪɨɜɤɭ ɡɚɞɚɱɢ. ɉɭɫɬɶ n
ɛɨɪ ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɯ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ (ɇɈȾ(n
ɧɚɣɬɢ ɬɚɤɨɟ ɱɢɫɥɨ F = F(n
, n2, …, nk), ɱɬɨɛɵ ɞɥɹ ɥɸɛɨɝɨ f >F ɞɢɨɮɚɧɬɨɜɨ
1
, n2, …, nk) = 1). Ɍɪɟɛɭɟɬɫɹ
1
, n2, …, nk – ɧɚ-
1
ɭɪɚɜɧɟɧɢɟ
n
+ n2ɯ2 + …+ nkɯk = f
1ɯ1
ɢɦɟɥɨ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ ɯ
, ɯ2, …, ɯk (ɯ
1
0, j = 1, … , k), ɚ ɩɪɢ f = F –
j
ɪɟɲɟɧɢɣ ɛɵ ɧɟ ɢɦɟɥɨ. Ɍɨ ɟɫɬɶ F – ɦɚɤɫɢɦɚɥɶɧɨɟ ɱɢɫɥɨ, ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɟɣ ɱɢɫɟɥ n
ɜɚɸɬ ɱɢɫɥɨɦ Ɏɪɨɛɟɧɢɭɫɚ ɧɚɛɨɪɚ (n
, n2, …, nk ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ. ɗɬɨ ɱɢɫɥɨ ɧɚɡɵ-
1
, n2, … , nk).
1
ɂɡɜɟɫɬɧɚ ɬɟɨɪɟɦɚ.
Ɍɟɨɪɟɦɚ 6 (ɬɟɨɪɟɦɚ Ɏɪɨɛɟɧɢɭɫɚ). ɉɭɫɬɶ ɚ ɢ b – ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɟ ɱɢɫ-
ɥɚ, ɬɨɝɞɚ F(ɚ, b) = ɚb – ɚ – b.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɱɢɫɥɨ F(ɚ,b) ɧɟɥɶɡɹ ɩɪɟɞɫɬɚɜɢɬɶ
ɜ ɜɢɞɟ ɫɭɦɦɵ ɚɯ + bɭ, ɜ ɫɚɦɨɦ ɞɟɥɟ, ɚb – ɚ – b = ɚɯ + bɭ, ɬɨ ɚ(b – ɯ – 1) =
b(y + 1). Ɍɚɤ
ɤɚɤ ɚ ɢ b – ɜɡɚɢɦɧɨ ɩɪɨɫɬɵ, ɬɨ ɱɢɫɥɨ (b – ɯ – 1) ɞɨɥɠɧɨ ɞɟɥɢɬɶɫɹ
ɧɚ b, ɚ ɨɧɨ ɦɟɧɶɲɟ b ɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨ. ɉɪɨɬɢɜɨɪɟɱɢɟ.
Ɍɟɩɟɪɶ ɞɨɤɚɠɟɦ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɫ > 0 ɭɪɚɜɧɟɧɢɟ:
ɚɯ + bɭ = ɚb – ɚ – b + ɫ, (10)
ɢɦɟɟɬ ɪɟɲɟɧɢɟ. Ɍɚɤ ɤɚɤ ɇɈȾ(a, b) = 1, ɬɨ ɧɚɣɞɭɬɫɹ aɯ
bkxx
1
akyy
1
k
... ,1 ,0,
r
ɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ:
ɫɱɢɬɚɬɶ, ɱɬɨ ɯ
ɭɪɚɜɧɟɧɢɹ aɯ
ɬɚɤɢɟ ɯ
2
ɢ ɭ1 ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ. Ɉɬɫɸɞɚ ɩɨɥɭɱɢɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ
1
– bɭ1 =ɫ:
1
1
®
¯
1
ɢ ɭ2, ɱɬɨ ɭ2 < ɚ, ɚ ɯ2 > 0. ɉɨɞɫɬɚɜɥɹɹ ɢɯ ɜ (10), ɢɦɟɟɦ
®
¯
bkcxx
akcyy
– bɭ1 = 1 ɢ ɨɛɳɟɟ ɪɟ-
1
... , 1 ,0,
r
k
, ɩɨɷɬɨɦɭ ɦɨɠɧɨ
. ɂɡ ɷɬɢɯ ɪɟɲɟɧɢɣ ɜɵɛɟɪɟɦ
ɚ(ɯ + 1 – ɯ
) + b(ɭ – ɚ + ɭ2 + 1) = 0.
2
Ɉɬɤɭɞɚ ɧɚɯɨɞɢɦ ɯ = ɯ
– 1 ɢ ɭ = ɚ – ɭ2 – 1. ɉɨ ɩɨɫɬɪɨɟɧɢɸ ɨɧɢ ɰɟɥɵɟ
2
ɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɟ. Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɋɚɫɫɦɨɬɪɢɦ ɱɢɫɥɚ ɫɩɟɰɢɚɥɶɧɨɝɨ ɜɢɞɚ. ɉɭɫɬɶ ɪ
ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɟ ɱɢɫɥɚ. ȼ ɡɚɞɚɱɟ Ɏɪɨɛɟɧɢɭɫɚ ɜ ɤɚɱɟɫɬɜɟ ɱɢɫɟɥ n
, ɪ2, …, ɪk – ɩɨɩɚɪɧɨ
1
, n2, …, nk
1
ɜɨɡɶɦɟɦ ɬɚɤɢɟ:
kipn
..., ,1 ,0,
ji
ij
z
(11)
39
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɇɈȾ (n
, n2, …, nk) = 1.
1
ȼ 2011 ɝɨɞɭ ɭɱɟɧɢɰɚ 11-ɝɨ ɤɥɚɫɫɚ Ɍɨɤɦɚɤɨɜɚ Ⱥɥɢɧɚ ɞɨɤɚɡɚɥɚ ɫɥɟɞɭɸɳɭɸ ɬɟɨɪɟɦɭ, ɨɛɨɛɳɢɜ ɬɟɦ ɫɚɦɵɦ ɪɟɡɭɥɶɬɚɬ Ɏɪɨɛɟɧɢɭɫɚ.
Ɍɟɨɪɟɦɚ 7. ɉɭɫɬɶ ɱɢɫɥɚ ɪ
, n2, …, nk – ɨɩɪɟɞɟɥɹɸɬɫɹ ɮɨɪɦɭɥɚɦɢ (11). Ɍɨɝɞɚ
ɢ n
1
, ɪ2, …, ɪk – ɩɨɩɚɪɧɨ ɜɡɚɢɦɧɨ ɩɪɨɫɬɵ
1
k
k
21
)1() ..., , ,(
j
npknnnF
¦
jk
j
j
11
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɵɜɚɟɬɫɹ ɦɟɬɨɞɨɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɢɧɞɭɤɰɢɢ, ɝɞɟ
ɜ ɤɚɱɟɫɬɜɟ ɛɚɡɵ (k = 2) ɢɫɩɨɥɶɡɨɜɚɥɚɫɶ ɬɟɨɪɟɦɚ Ɏɪɨɛɟɧɢɭɫɚ.
ɑɬɨɛɵ ɨɰɟɧɢɬɶ ɫɥɨɠɧɨɫɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɜɨɩɪɨɫɨɜ, ɩɨɩɪɨɛɭɣɬɟ ɪɟɲɢɬɶ ɡɚɞɚɱɭ.
Ɂɚɞɚɱɚ. Ɇɨɧɟɬɧɵɣ ɞɜɨɪ ɱɟɤɚɧɢɬ ɦɨɧɟɬɵ ɬɪɟɯ ɞɨɫɬɨɢɧɫɬɜ: 6, 10
ɢ 15. ɇɚɣɬɢ ɫɭɦɦɵ, ɤɨɬɨɪɵɟ ɧɟ ɦɨɝɭɬ ɛɵɬɶ ɧɚɛɪɚɧɵ ɦɨɧɟɬɚɦɢ
ɢ ɦɚɤɫɢɦɚɥɶɧɭɸ ɢɡ ɧɢɯ?
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜ ɷɬɨɣ ɡɚɞɚɱɟ ɱɢɫɥɚ 6, 10 ɢ 15 ɢɦɟɸɬ
ɬɟɥɢ. ɉɨɥɨɠɢɦ ɪ
= ɇɈȾ(6, 10) = 2, ɪ2 = ɇɈȾ(6, 15) = 3, ɪ3 = ɇɈȾ(10, 15) = 5.
1
ɩɪɨɫɬɵɟ ɨɛɳɢɟ ɞɟɥɢ-
Ɍɨ ɟɫɬɶ ɱɢɫɥɚ 6, 10 ɢ 15 ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɸ ɬɟɨɪɟɦɵ 6, ɩɨɷɬɨɦɭ
F(6, 10, 15) = (3 – 1)·2·3·5 – 6 – 10 – 15 = 29. Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɜɫɟ s > 29 – ɧɚɛɢ-
ɪɚɟɦɵ.
ȼɨɡɜɪɚɳɚɹɫɶ ɟɳɟ ɪɚɡ ɤ ɚɥɝɨɪɢɬɦɚɦ ȿɜɤɥɢɞɚ (ɚɥɝɨɪɢɬɦɵ 11 ɢ 12), ɡɚɞɚɞɢɦɫɹ ɜɨɩɪɨɫɨɦ ɨ ɫɥɨɠɧɨɫɬɢ ɢ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɷɬɢɯ ɚɥɝɨɪɢɬɦɨɜ. ɉɨɦɨɝɭɬ ɧɚɦ
ɪɚɡɨɛɪɚɬɶɫɹ ɜ ɷɬɨɦ ɜɨɩɪɨɫɟ ɰɟɩɧɵɟ ɞɪɨɛɢ.
§ 6. ɐɟɩɧɵɟ ɞɪɨɛɢ
ɐɟɩɧɨɣ (ɢɥɢ ɧɟɩɪɟɪɵɜɧɨɣ) ɞɪɨɛɶɸ ɧɚɡɵɜɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟ ɜɢɞɚ
:
,...],...,,,[
n
aaaaa
0210
a
1
a
1
1
2
a
1
1
3
1
1
a
n
.
40
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