P = 13 {\displaystyle P=13} 0 < M < P {\displaystyle 0<M<P} M = 9 {\displaystyle M=9} M C D ( R , P ) = 1 {\displaystyle MCD(R,P)=1} M C D ( R , P − 1 ) = 1 {\displaystyle MCD(R,P-1)=1} R = 5 {\displaystyle R=5} C ≡ M R ( mod P ) {\displaystyle C\equiv M^{R}{\pmod {P}}} C ≡ 95 ( mod 13 ) {\displaystyle C\equiv 95{\pmod {13}}} C = 3 {\displaystyle C=3} S ∗ R ≡ 1 ( mod P − 1 ) {\displaystyle S*R\equiv 1{\pmod {P-1}}} S ∗ 5 ≡ 1 ( mod 12 ) {\displaystyle S*5\equiv 1{\pmod {12}}} S = 17 {\displaystyle S=17} C S ≡ M ( mod P ) {\displaystyle C^{S}\equiv M{\pmod {P}}} 3 17 ≡ M ( mod 13 ) {\displaystyle 3^{17}\equiv M{\pmod {13}}} f ( N ) = N − 1 {\displaystyle f(N)=N-1} P = 23 {\displaystyle P=23} Q = 17 {\displaystyle Q=17} N = P ∗ Q = 391 {\displaystyle N=P*Q=391} f ( N ) {\displaystyle f(N)} f ( N ) = ( P − 1 ) ∗ ( Q − 1 ) = 22 ∗ 16 = 352 {\displaystyle f(N)=(P-1)*(Q-1)=22*16=352} M C D ( N , R ) = 1 {\displaystyle MCD(N,R)=1} M C D ( f ( N ) , R ) = 1 {\displaystyle MCD(f(N),R)=1} R = 25 {\displaystyle R=25} S ∗ R ≡ 1 ( mod f ( N ) ) {\displaystyle S*R\equiv 1{\pmod {f(N)}}} S ∗ 25 ≡ 1 ( mod 352 ) {\displaystyle S*25\equiv 1{\pmod {352}}} S = 169 {\displaystyle S=169} ( R , N ) → ( 25 , 391 ) {\displaystyle (R,N)\to (25,391)} ( S , N ) → ( 169 , 391 ) {\displaystyle (S,N)\to (169,391)} M = 37 {\displaystyle M=37} C ≡ M R ( mod N ) {\displaystyle C\equiv M^{R}{\pmod {N}}} C ≡ 37 25 ( mod 391 ) = 1603404513114153724313506335083015711557 ( mod 391 ) = 99 {\displaystyle C\equiv 37^{25}{\pmod {391}}=1603404513114153724313506335083015711557{\pmod {391}}=99} M ≡ C S ( mod N ) {\displaystyle M\equiv C^{S}{\pmod {N}}} M ≡ 99 169 ( mod 391 ) = 18295651830906112060192536872505514221599876994097090850246588464664834465981453818408289459755490780867781886155339776519170761066377072744603471399172568833972085967137143417811513821081158788834371307852764576449010417775472698276720528096596892892197928778696687265987371082463200584667598753085403762897279970485173492260857502056899 ( mod 391 ) = 37 = M {\displaystyle M\equiv 99^{169}{\pmod {391}}=18295651830906112060192536872505514221599876994097090850246588464664834465981453818408289459755490780867781886155339776519170761066377072744603471399172568833972085967137143417811513821081158788834371307852764576449010417775472698276720528096596892892197928778696687265987371082463200584667598753085403762897279970485173492260857502056899{\pmod {391}}=37=M}
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