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?5"êA^ ü? ~f o§u o?N?K F ―è ?5"êA^ ü?~f o§u ?L?? ê?X 2012c12 15F ?5"êA^ ü? ~f o§u o?N?K F ―è Eulero?N?K Eulero?N?K X?^o?N 8^c?L?§ N?? Figure: o?NOABC X?^l, m, n, p, q, rL?o?NOABC N?V ? ?5"êA^ ü? ~f o§u o?N?K F ―è )??{ O? :, A, B, Ck?I: A = (a1, b1, c1), B = (a2, b2, c2), C = (a3, b3, c3).

-V6? ? → OA, ? ? → OB, ? ? → OC|¤m?X?, ±§??c ?18 ?N N?. K 6V = V6 = a1 b1 c1 a2 b2 c2 a3 b3 c3 . ?5"êA^ ü? ~f o§u o?N?K F ―è )??{(Y) ¤±k 36V

2 = a1 b1 c1 a2 b2 c2 a3 b3 c3

2 = a1 b1 c1 a2 b2 c2 a3 b3 c3 a1 a2 a3 b1 b2 b3 c1 c2 c3 = ? → OA ・ ? → OA ? → OA ・ ? ? → OB ? → OA ・ ? ? → OC ? → OA ・ ? ? → OB ? ? → OB ・ ? ? → OB ? ? → OB ・ ? ? → OC ? → OA ・ ? ? → OC ? ? → OB ・ ? ? → OC ? ? → OC ・ ? ? → OC . ?5"êA^ ü? ~f o§u o?N?K F ―è )??{(Y) ・?k ? → OA ・ ? → OA = p2 , ? ? → OB ・ ? ? → OB = q2 , ? ? → OC ・ ? ? → OC = r2 . d{u?n, ?k ? → OA ・ ? ? → OB = p2 + q2 ? n2

2 , ? → OA ・ ? ? → OC = p2 + r2 ? m2

2 , ? ? → OB ・ ? ? → OC = q2 + r2 ? l2

2 . ?5"êA^ ü? ~f o§u o?N?K F ―è Eulero?NN?ú? ? Eulero?NN?ú? 36V

2 = p2 p2+q2?n2

2 p2+r2?m2

2 p2+q2?n2

2 q2 q2+r2?l2

2 p2+r2?m2

2 q2+r2?l2

2 r2 . y3\?±?L?? D9 7i? 8^c? ?§ N? . ?5"êA^ ü? ~f o§u o?N?K F ―è ―è? ? ?? \―(encryption, encode) ò ?(??E)?1?L§. )―(decryption, decode) ò\―L ??E? ?/?. ??(plaintext)?―?(ciphertext) ― (key) ?5"êA^ ü? ~f o§u o?N?K F ―è é?―èN? é?―èN? ??. Figure: é?―èN? ?5"êA^ ü? ~f o§u o?N?K F ―è ?é?―èN? ?é?―èN? y. \―L§?)―L§ Figure: ?é?―èN? ?5"êA^ ü? ~f o§u o?N?K F ―è F ―è n S . ò???¤n?n' ??. ???n ?_? A(― ). ^A?z? ^A?1?z? ?5"êA^ ü? ~f o§u o?N?K F ―è ??~f ~X, ?áéA A ? 1, B ? 2,Y ? 25, Z ? 0. ??: LIKE → 12, 9, 11, 5. n = 2, ? A =

1 1

1 0 , A?1 =

0 1

1 ?1 . d A

12 9 =

21 12 , A

11 5 =

16 11 , "?: 21, 12, 16,

11 → ULPK. ?5"êA^ ü? ~f o§u o?N?K F ―è ?K ? ?êi?i?m?±p?, 7L?^ g0?25? m ê. ~X

1 1

1 0

1 25 =

26 1 , 26éA i?? d?5"ê??, A?1 = A? det(A) , ??A??A ?'? . ¤±?U?y?ê. ~X,

1 1 ?1

1 ?1 =

1 2

1 ?1

1 1 . ?5"êA^ ü? ~f o§u o?N?K F ―è )??{ 38? Z26 = {0, 1,25} ?‰?5"ê. I???Z26? : \{ ?{ ?_±9_. ó? ?{, '{?{ ?? ên, ^26?, k n = 26q(n) + r(n), ??r(n)?{ê, ÷vr(n) ∈ Z26. ?5"êA^ ü? ~f o§u o?N?K F ―è )??{(Y) u?kN r : Z ?→ Z26;

n → r(n). éum, n ∈ Z26, ・??? m ? n = r(m + n) ∈ Z26, m n = r(mn) ∈ Z26. u?k m ?

0 =

0 ? m = m, m

1 =

1 m = m, ?m ∈ Z26. éum ∈ Z26, ?? ?m = r(26 ? m) ∈ Z26. u?k m ? (?m) = 0, ?m ∈ Z26. ¤±k~{ m n = m ? (?n) ∈ Z26, ?m, n ∈ Z26. ?5"êA^ ü? ~f o§u o?N?K F ―è )??{(Y) ?? m3Z26?_ ? ?3m ∈ Z26, ? m m = 1. ?_ d5 m ∈ Z26?_? m?26p?(=: m?26 êú?f? k±1). 7?5: d??, mm = 26q + 1, ?q ∈ Z. u?m?26 êú?f?mm ? 26q =

1 ?f, ?U ?±1. ?5"êA^ ü? ~f o§u o?N?K F ―è (Z26,5"ê m, n ∈ N, P Mm*n(Z26) = {A = (aij)m*n | aij ∈ Z26}. ?? \{: A = (aij), B = (bij) ∈ Mm*n(Z26), A ? B = (aij ? bij)m*n. ?{: A = (aij) ∈ Mm*p(Z26), B = (bij) ∈ Mp*n(Z26), A B = (cij)m*n, ?? cij = (ai1 b1j) ? (ai2 b2j)aip bpj). A ∈ Mn*n(Z26)?_?? ?3A ∈ Mn*n(Z26), ? A A = A A = En. ?5"êA^ ü? ~f o§u o?N?K F ―è (Z26,5"ê(Y) A ∈ Mn*n(Z26)

1 ?det(A) ∈ Z26. A ∈ Mn*n(Z26) ?'? A? ∈ Mn*n(Z26). éuA ∈ Mn*n(Z26), ?±y?: A?_ ? det(A)3Z26??_. … A?1 = det(A)?1 A? . ?K

1 ?XUì?ê? ?/, u?Z26? ?5"ê.

2 ò26?¤?? g,ê, ?/X?? ?5"êA^ ü? ~f o§u o?N?K F ―è F ―è UéA A ? 1, B ? 2,Y ? 25, Z ?