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문제

계수가 0 또는 1로만 이루어지는 특수한 다항식들의 집합에 대해 생각해보자.

이 집합의 두 다항식을 더하는 방법은 다음과 같다.

  • 우선, 두 다항식을 더한다.
  • 결과 다항식의 각 계수를 2로 나눈 나머지를 각각의 계수에 대입한다.

이 때, (0 + 0) mod 2 = 0, (0 + 1) mod 2 = 1, (1 + 0) mod 2 = 1, and (1 + 1) mod 2 = 0 이므로

결국 결과 다항식의 각 차수의 계수는 더하고자 했던 두 다항식의 계수에 대한 XOR 연산의 결과가 된다.

(x 6 + x4 + x2 + x + 1) + (x7 + x + 1) = x7 + x6 + x4 + x2

이제 뺄셈을 정의하는데, 뺄셈의 경우에도 덧셈과 같이 (0 - 0) mod 2 = 0, (0 - 1) mod 2 = 1, (1 - 0) mod 2 = 1, and (1 - 1) mod 2 = 0 이므로
덧셈과 뺄셈은 이 다항식들의 집합에 대해 동일한 의미를 지닌다. (계수 간의 XOR 연산이 새 다항식의 계수가 됨)

예시는 다음과 같다.

(x 6 + x4 + x2 + x + 1) - (x7 + x + 1) = x7 + x6 + x4 + x2

다음으로 곱셈을 정의하는데, 곱셈 또한 일반적인 다항식 간의 곱셈 연산을 한 뒤 각각의 계수를 2로 나눈 나머지를 취하면 된다.

(x 6 + x4 + x2 + x + 1) (x7+ x + 1) = x13 + x11 + x9 + x8 + x6 + x5 + x4 + x3 + 1

두 다항식 f(x)와 g(x)의 곱을 h(x)로 나눈 나머지는 f(x)와 g(x)를 위와 같은 방법으로 곱한 뒤, h(x)로 나눈 나머지가 된다.

(x 6 + x4 + x2 + x + 1) (x7 + x + 1) modulo (x8 + x4 + x3 + x + 1) = x7 + x6 + 1

어떤 다항식의 최고차항의 차수를 degree, 줄여서 d라 하면,

f(x), g(x), h(x)의 계수가 0 또는 1이므로 각 다항식은 정수 d+1과 길이 d+1인 비트스트링의 쌍으로 유일하게 표기할 수 있다.

예를 들어 x 7 + x6 + 1은 다음과 같이 표기된다.

8 1 1 0 0 0 0 0 1

이 때, 8은 총 비트의 개수이며, 최고차항의 차수(7) + 1이다.

이제 위에서 설명한 성질을 만족하는 세 다항식 f(x), g(x), h(x)가 설명한 형식대로 주어지면 f(x)g(x) modulo h(x)를 계산하면 된다.

입력

입력은 여러 테스트 케이스로 이루어져 있다.

입력의 첫 줄에 테스트케이스의 수 T가 주어지며, 각 테스트 케이스마다 세 줄에 걸쳐 문제에서 설명한 형식대로 표기한 f(x), g(x), h(x)가 주어진다.

출력

각 테스트 케이스마다 한 줄에 f(x)g(x)를 h(x)로 나눈 나머지를 문제에서 설명한 형식대로 출력한다.

예제 입력 1

2
7 1 0 1 0 1 1 1
8 1 0 0 0 0 0 1 1
9 1 0 0 0 1 1 0 1 1
10 1 1 0 1 0 0 1 0 0 1
12 1 1 0 1 0 0 1 1 0 0 1 0
15 1 0 1 0 1 1 0 1 1 1 1 1 0 0 1

예제 출력 1

8 1 1 0 0 0 0 0 1
14 1 1 0 1 1 0 0 1 1 1 0 1 0 0
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(\uacc4\uc218 \uac04\uc758 XOR \uc5f0\uc0b0\uc774 \uc0c8 \ub2e4\ud56d\uc2dd\uc758 \uacc4\uc218\uac00 \ub428)<\/p>\r\n\r\n<p>\uc608\uc2dc\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<p>(x <sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) - (x<sup>7<\/sup> + x + 1) = x<sup>7<\/sup> + x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup><\/p>\r\n\r\n<p>\ub2e4\uc74c\uc73c\ub85c \uacf1\uc148\uc744 \uc815\uc758\ud558\ub294\ub370, \uacf1\uc148 \ub610\ud55c \uc77c\ubc18\uc801\uc778 \ub2e4\ud56d\uc2dd \uac04\uc758 \uacf1\uc148 \uc5f0\uc0b0\uc744 \ud55c \ub4a4 \uac01\uac01\uc758 \uacc4\uc218\ub97c 2\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c \ucde8\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n\r\n<p>(x <sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) (x<sup>7<\/sup>+ x + 1) = x<sup>13<\/sup> + x<sup>11<\/sup> + x<sup>9<\/sup> + x<sup>8<\/sup> + x<sup>6<\/sup> + x<sup>5<\/sup> + x<sup>4<\/sup> + x<sup>3<\/sup> + 1<\/p>\r\n\r\n<p>\ub450 \ub2e4\ud56d\uc2dd f(x)\uc640 g(x)\uc758 \uacf1\uc744 h(x)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub294 f(x)\uc640 g(x)\ub97c \uc704\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uacf1\ud55c \ub4a4, h(x)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\uac00 \ub41c\ub2e4.<\/p>\r\n\r\n<p>(x <sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) (x<sup>7<\/sup> + x + 1) modulo (x<sup>8<\/sup> + x<sup>4<\/sup> + x<sup>3<\/sup> + x + 1) = x<sup>7<\/sup> + x<sup>6<\/sup> + 1<\/p>\r\n\r\n<p>\uc5b4\ub5a4 \ub2e4\ud56d\uc2dd\uc758 \ucd5c\uace0\ucc28\ud56d\uc758 \ucc28\uc218\ub97c degree, \uc904\uc5ec\uc11c d\ub77c \ud558\uba74,<\/p>\r\n\r\n<p>f(x), g(x), h(x)\uc758 \uacc4\uc218\uac00 0 \ub610\ub294 1\uc774\ubbc0\ub85c \uac01 \ub2e4\ud56d\uc2dd\uc740&nbsp;\uc815\uc218 d+1\uacfc \uae38\uc774 d+1\uc778&nbsp;\ube44\ud2b8\uc2a4\ud2b8\ub9c1\uc758 \uc30d\uc73c\ub85c \uc720\uc77c\ud558\uac8c&nbsp;\ud45c\uae30\ud560 \uc218 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc608\ub97c \ub4e4\uc5b4&nbsp;x <sup>7<\/sup> + x<sup>6<\/sup> + 1\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\uae30\ub41c\ub2e4.<\/p>\r\n\r\n<p>8 1 1 0 0 0 0 0 1<\/p>\r\n\r\n<p>\uc774 \ub54c, 8\uc740 \ucd1d \ube44\ud2b8\uc758 \uac1c\uc218\uc774\uba70, \ucd5c\uace0\ucc28\ud56d\uc758 \ucc28\uc218(7) + 1\uc774\ub2e4.<\/p>\r\n\r\n<p>\uc774\uc81c \uc704\uc5d0\uc11c \uc124\uba85\ud55c \uc131\uc9c8\uc744 \ub9cc\uc871\ud558\ub294&nbsp;\uc138 \ub2e4\ud56d\uc2dd f(x), g(x), h(x)\uac00 \uc124\uba85\ud55c \ud615\uc2dd\ub300\ub85c&nbsp;\uc8fc\uc5b4\uc9c0\uba74 f(x)g(x) modulo h(x)\ub97c \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc785\ub825\uc758 \uccab \uc904\uc5d0 \ud14c\uc2a4\ud2b8\ucf00\uc774\uc2a4\uc758 \uc218 T\uac00 \uc8fc\uc5b4\uc9c0\uba70, \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \uc138 \uc904\uc5d0 \uac78\uccd0&nbsp;\ubb38\uc81c\uc5d0\uc11c \uc124\uba85\ud55c \ud615\uc2dd\ub300\ub85c \ud45c\uae30\ud55c f(x), g(x), h(x)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub9c8\ub2e4 \ud55c \uc904\uc5d0&nbsp;f(x)g(x)\ub97c h(x)\ub85c \ub098\ub208 \ub098\uba38\uc9c0\ub97c&nbsp;\ubb38\uc81c\uc5d0\uc11c \uc124\uba85\ud55c \ud615\uc2dd\ub300\ub85c \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"7345","problem_lang":"1","title":"Modular multiplication of polynomials","description":"<p>Consider polynomials whose coefficients are 0 and 1. Addition of two polynomials is achieved by &#39;adding&#39; the coefficients for the corresponding powers in the polynomials. The addition of coefficients is performed by addition modulo 2, i.e., (0 + 0) mod 2 = 0, (0 + 1) mod 2 = 1, (1 + 0) mod 2 = 1, and (1 + 1) mod 2 = 0. Hence, it is the same as the exclusive-or operation.<\/p>\r\n\r\n<p>(x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) + (x<sup>7<\/sup> + x + 1) = x<sup>7<\/sup> + x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup><\/p>\r\n\r\n<p>Subtraction of two polynomials is done similarly. Since subtraction of coefficients is performed by subtraction modulo 2 which is also the exclusive-or operation, subtraction of polynomials is identical to addition of polynomials.<\/p>\r\n\r\n<p>(x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) - (x<sup>7<\/sup> + x + 1) = x<sup>7<\/sup> + x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup><\/p>\r\n\r\n<p>Multiplication of two polynomials is done in the usual way (of course, addition of coefficients is performed by addition modulo 2).<\/p>\r\n\r\n<p>(x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) (x<sup>7<\/sup>+ x + 1) = x<sup>13<\/sup> + x<sup>11<\/sup> + x<sup>9<\/sup> + x<sup>8<\/sup> + x<sup>6<\/sup> + x<sup>5<\/sup> + x<sup>4<\/sup> + x<sup>3<\/sup> + 1<\/p>\r\n\r\n<p>Multiplication of two polynomials f(x) and g(x) modulo a polynomial h(x) is the remainder of f(x)g(x) divided by h(x).<\/p>\r\n\r\n<p>(x<sup>6<\/sup> + x<sup>4<\/sup> + x<sup>2<\/sup> + x + 1) (x<sup>7<\/sup> + x + 1) modulo (x<sup>8<\/sup> + x<sup>4<\/sup> + x<sup>3<\/sup> + x + 1) = x<sup>7<\/sup> + x<sup>6<\/sup> + 1<\/p>\r\n\r\n<p>The largest exponent of a polynomial is called its degree. For example, the degree of x<sup>7<\/sup> + x<sup>6<\/sup> + 1 is 7.<\/p>\r\n\r\n<p>Given three polynomials f(x), g(x), and h(x), you are to write a program that computes f(x)g(x) modulo h(x). We assume that the degrees of both f(x) and g(x) are less than the degree of h(x). The degree of a polynomial is less than 1000.<\/p>\r\n\r\n<p>Since coefficients of a polynomial are 0 or 1, a polynomial can be represented by d+1 and a bit string of length d+1, where d is the degree of the polynomial and the bit string represents the coefficients of the polynomial. For example, x<sup>7<\/sup> + x<sup>6<\/sup> + 1 can be represented by<\/p>\r\n\r\n<p>8 1 1 0 0 0 0 0 1.<\/p>\r\n","input":"<p>The input consists of T test cases. The number of test cases (T) is given in the first line of the input file. Each test case consists of three lines that contain three polynomials f(x), g(x), and h(x), one per line. Each polynomial is represented as described above.<\/p>\r\n","output":"<p>The output should contain the polynomial f(x)g(x) modulo h(x), one per line.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]