x3+x4+x3+x+1=(x2+1)(x°+x4+x2+x+1)+x4 冷x5+x4+x2+x+1=(x2+1)x4+x2+x+1 冷x4=(x2+x)(x2+x+1)+x 冷x2+x+1=(X+1)x+1 冷故1=(x2+x+1)-X+1)x (x2+x+1)-(x+1)(x4-(x2+x)(x2+x+1) 冷=(1+(X+1)(x2+x)(x2+x+1)+(X+1)x4 =(1+(x+1)(x2+x)(X6+x4+x2+x+1)-(x2+1)x4)+(x+1)x 冷=(x3+x+1)(x+x4+x2+x+1)+(x3+x+1)(x2+1)+(x+1)x4 冷=(x3+x+1)(x+x4+x2+x+1)+(x5+x2)(x3+x4+x3+x+1) (x2+1)(x6+x4+x2+x+1) (X3+x+1)+(x5+×2(x2+1)(x+x4+x2+x+1)+ (x+x2)(x3+x4+x3+x+1) =(x7+x5+x4+x3+x2+x+1)(x6+x4+x2+x+1)+ x5+x2)(x3+x4+x3+x+1) 所以x6+x4+x2+x+1关于模x8+x4x3+x+1的逆元是: 令X7+X5+X4+X3+X2+X+1❖ x 8+x4+x3+x+1=(x 2+1)(x6+x4+x2+x+1)+x 4 ❖ x 6+x4+x2+x+1=(x2+1)x4+x2+x+1 ❖ x 4=(x2+x)(x2+x+1)+x ❖ x 2+x+1=(x+1)x+1 ❖ 故1=(x2+x+1)-(x+1)x ❖ =(x2+x+1)-(x+1)(x4 -(x2+x)(x2+x+1)) ❖ =(1+(x+1)(x2+x))(x2+x+1)+(x+1)x4 ❖ =(1+(x+1)(x2+x))((x6+x4+x2+x+1)-(x2+1)x4 ) +(x+1)x4 ❖ =(x3+x+1)(x6+x4+x2+x+1)+((x3+x+1)(x2+1) +(x+1))x4 ❖ =(x3+x+1)(x6+x4+x2+x+1)+(x5+x2 )((x8+x4+x3+x+1)- (x2+1)(x6+x4+x2+x+1)) ❖ =((x3+x+1)+(x5+x2 )(x2+1))(x6+x4+x2+x+1)+ (x5+x2 )(x8+x4+x3+x+1) ❖ =(x7+x5+x4+x3+x2+x+1)(x6+x4+x2+x+1)+ (x5+x2 )(x8+x4+x3+x+1) ❖ 所以x 6+x4+x2+x+1关于模x 8+x4+x3+x+1的逆元是： ❖ x 7+x5+x4+x3+x2+x+1