리젠다엘 믹스콜럼스

Rijndael 암호에 의해 수행되는 MixColumns 작전은 ShiftRows 스텝과 함께 Rijndael의 주요 확산원이다.각 열은 4항 다항 $b{\displaystyle (x}$)${\displaystyle }$= b ${\displaystyle {3\mathrm{}}_{}}$ ${\displaystyle {x\mathrm{}}^{}}$ 3${\displaystyle {}^{}}$+ ${\displaystyle {b\mathrm{}}_{}}$ ${\displaystyle {2\mathrm{}}^{}}$+ b ${\displaystyle {}_{}}$+ ${\displaystyle {}_{}}$ ${\displaystyle {0\mathrm{}}_{}}$ ${\$}로 처리된다.$}x^{2}+b_{1$$}x+b_$${\displaystyle \{}$$0}$ 필드 내 요소인$$ ${\displaystyle \}{x\mathrm{}}^{}}$+${\displaystyle {\mathrm{}}^{}}${0${\displaystyle {}^{}}$ ${\displaystyle \operatorname {GF}(2^{8})$.다항식의 계수는 기본 하위 필드 ${\displaystyle \mathrm{GF}}$ ${\displaystyle }$( ${\displaystyle 2}$) ${\displaystyle \operatorname {GF}(2)}$ 내의 요소들이다$.$

Each column is multiplied with a fixed polynomial ${\displaystyle a(x)=3x^{3}+x^{2}+x+2}$ modulo ${\displaystyle x^{4}+1}$; the inverse of this polynomial is ${\displaystyle a^{-1}(x)=11x^{3}+13x^{2}+9x+14}$.

믹스콜럼스

연산은 계수가 ${\displaystyle \mathrm{GF}}$⁡${\displaystyle }$( 2 ${\displaystyle {8\mathrm{}}^{}}$ )${\$인 4항 다항식 두 개의 모듈식 곱셈으로 이루어진다$.$이 작업에 사용되는 계수는 $x{\displaystyle {}^{}}$ ${\displaystyle {4\mathrm{}}^{}}$+ ${\displaystyle 1}$ ${\displaystyle$ x$^{4}+1}$ 입니다$.$

첫 번째 4항 다항식 계수는 4바이트를 포함하는 상태 열 3 2 $][\$}&$b_{0}\end{bmatrix}}}}}$에 의해 정의된다각 바이트는 다음과 같이 4개 항의 계수다.

${\displaystyle b(x)=b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}.}$

두 번째 4항 다항식은 ${\displaystyle \mathrm{상수}}$ 다항식 $a{\displaystyle (}$x${\displaystyle }$) = ${\displaystyle {}^{}}$ x ${\displaystyle {3\mathrm{}}^{}}$+ ${\displaystyle x}$ ${\displaystyle {}^{}}$+ ${\displaystyle 2}$+ 2 ${\displaystyle a(x)=3x^{3}+x^{2}+x+2}$입니다$.$계수 또한 ${\displaystyle \mathrm{GF}}$ ${\displaystyle }$( 2 ${\displaystyle {8\mathrm{}}^{}}$) ${\$의 요소들이다$.$그것의 역은 $a{\displaystyle {}^{}}$- ${\displaystyle 1^{}}$(${\displaystyle x}$) = ${\displaystyle {}^{}}$ ${\displaystyle {x\mathrm{}}^{}}$3 + ${\displaystyle {}^{}}$ x${\displaystyle {}^{}2}$ + ${\displaystyle 9}$x + ${\displaystyle 14}$ $(\displaystyle$ a$^{-1}(x)=11x^{3}+13x^{2}+9x+14}$이다$.$

우리는 몇 가지 표기법을 정의해야 한다.

${\displaystyle \otimes }$ ${\displaystyle \otimes }$은(는) 곱셈모듈로 ${\displaystyle {x\mathrm{}}^{}}$ 4${\displaystyle {}^{}}$+ ${\displaystyle 1}$을 의미한다$$. $\{$ {\$displaystyle$ x$^{4}+1}.$

${\displaystyle \oplus }$ ${\displaystyle \oplus }$은(는) ${\displaystyle \mathrm{GF}}$ ${\displaystyle }$( 2 ${\displaystyle {8\mathrm{}}^{}}$) ${\$를$$통한 추가를 의미한다.

${\displaystyle \bullet }$ ${\displaystyle \bullet }$은(는) 곱셈(계수에 대해$$ GF$$ (${\displaystyle }$2 ${\displaystyle {8\mathrm{}}^{}}$ $displaystyle \operatorname {GF} \left(2^{8}\오른쪽)]$을 의미한다.

계수가 ${\displaystyle \mathrm{GF}}$ ${\displaystyle }$( ${\displaystyle {\mathrm{}}^{}}$ ${\displaystyle {8\mathrm{}}^{}}$) ${\$인 두 다항식을 추가하면 다음과$$ 같은 규칙이 있다.

${\displaystyle {\displaysty}&\왼쪽(a_{3}x^{3}+a_{2)}x^{2}+a_{1}x+a_{0}\오른쪽)+\왼쪽(b_{3}x^{3}+b_{2)}x^{2}+b_{1}x+b_{0}\오른쪽)\\={}&\left(a_{3}\oplus b_{3}\right)x^{3}+\left(a_{2}\oplus b_{2}\right)x^{2}+\left(a_{1}\oplus b_{1}\right)x+\left(a_{0}\oplus b_{0}\right)\end{aligned}}}$

데모

다항식 $a{\displaystyle (x}$ ${\displaystyle )}$= 3 ${\displaystyle \mathrm{x}}$ ${\displaystyle {3\mathrm{}}^{}}$+ ${\displaystyle {x\mathrm{}}^{}}$ ${\displaystyle {}^{}}$+ ${\displaystyle x2}$ + 2 ${\displaystyle a(x)=3x^{3}+x^{2}+x+2}$는 a${\displaystyle }$( x${\displaystyle }$)${\displaystyle }$= ${\displaystyle {}_{}}$ ${\displaystyle {x\mathrm{}}^{}}$ ${\displaystyle {}_{}{}^{}{}^{}}$+ ${\displaystyle {2\mathrm{}}^{}}$ + ${\displaystyle {}_{}{x\mathrm{}}_{}}$ ${\displaystyle {}_{}{}^{}}$+ ${\displaystyle {0\mathrm{}}_{}}$ + 0으로 표현된다$$. $a(x)=a_{3}x^{3}+a_{2a$}$}x^{2}+a_{1$$}}x+a_{0}}$.

다항식 곱하기

${\displaystyle {\displaysty}a(x)\b(x)=c(x)&=\좌(a_{3}x^{3}+a_{2)}x^{2}+a_{1}x+a_{0}\right)\nows \left(b_{3}x^{3}+b_{2}}x^{2}+b_{1}x+b_{0}\오른쪽)\\&=c_{6}x^{6}+c_{5}x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\end{aigned}}}$

여기서:

${\displaystyle{\begin{정렬}c_{0}&, =a_{0}\bullet b_{0}\\c_{1}&, =a_{1}\bullet b_{0}\oplus a_{0}\bullet b_{1}\\c_{2}&, =a_{2}\bullet b_{0}\oplus a_{1}\bullet b_{1}\oplus a_{0}\bullet b_{2}\\c_{3}&, =a_{3}\bullet b_{0}\oplus a_{2}\bullet b_{1}\oplus a_{1}\bullet b_{2}\oplus a_{0}\bullet b_{3}\\c_{4}&, =a_{3}\bullet b_{1}\oplus a_{2}\bul.{1b_{2}\oplus a_ 봅시다}\cHB_{3}\\c_{5}>a_{3}&#{3}\oplus a_{2}\oplus a_{3}\{3}\c_{6}&=a_{3}\nd{3}\c_{3}\c_{3}\ned}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

모듈식감소

결과 $c{\displaystyle }$( ${\displaystyle x}$) ${\displaystyle c(x)}$은 7항 다항식인데, 4바이트 단어로 줄여야 하는데, 곱셈모듈로 $x{\displaystyle {}^{}}$ ${\displaystyle {4\mathrm{}}^{}}$+ ${\displaystyle 1}$ ${\displaysty$ x$^{4}+1}$을 하면 된다$.$

몇 가지 기본적인 다항식 모듈러 연산을 수행하면 다음과 같은 것을 알 수 있다.

${\displaystyle {\begin}x^{6}{\bmod {\좌측(x^{4}+1\우측)}&=-x^{2}=x^{2}}{\text{{{{}}}{}}}}}}}}}}}}}}}}}{{{{우측)\\x^{5}{\bmod {\좌측(x^{4}+1\우측)}}}&=-x=x{\텍스트{{{}{GF} \좌측(2^{8}\우측)\\x^{4}{\bmod {\좌측(x^{4}+1\우측)}}&=-1=1=1\텍스트{{\text{over }\{GF} \좌측(2^{8}\우측)\end{arged}}}}$

일반적으로 $x{\displaystyle {}^{}}$ ${\displaystyle i^{}}$ ${\displaystyle mod({x\mathrm{}}^{}}$ 4${\displaystyle {}^{}}$+ ${\displaystyle 1}$)${\displaystyle }$= x ${\displaystyle {}^{}}$ ${\displaystyle {mod\mathrm{}}^{}}$ 4${\displaystyle {}^{}}$. ${\displaystyle x^{i}{\bmod}{\\$bmod}{\\$ft{4}+1\오른쪽$)}}=x$^{i{\bmod{$4}}}라고 말할 수 있다$.$$}$

그렇게

${\displaystyle {\displaysty}a(x)\otimes b(x)&=c(x){\bmod {\좌측(x^{4}+1\우측)}}\\&=\왼쪽(c_{6}x^{6}+c_{5)}x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\오른쪽){\bmod {\좌측(x^{4}+1\오른쪽)}}\\&=c_{6}x^{6{\bmod{4}}+c_{5}}x^{5{\bmod{4}}+c_{4}}x^{4{\bmod{4}}+c_{3}x^{3{\bmod{4}}+c_{2}x^{2}{\bmod{4}}+c_{1}x^{1}{\bmod{4}}+c_{0}x^{0{\bmod {4}\&=c_{6}x^{2}+c_{5}x+c_{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x+c_{0}\\&=c_{3}x^{3}+\좌측(c_{2}\oplus c_{6}\우측)x^{2}+\좌측(c_{1}\oplus c_{5}\우측)x+{0}\oplus c_{4}\\&=d_{3}x^{3}+d_{2}x^{2}+d_{1}x+d_{0}\end{aigned}}}$

어디에

${\displaystyle d_{0}=c_{0}\oplus c_{4}}$

${\displaystyle d_{1}=c_{1}\oplus c_{5}}$

${\displaystyle d_{2}=c_{2}\oplus c_{6}}$

${\displaystyle d_{3}=c_{3}}}$

행렬 표현

계수 $d{\displaystyle {}_{}}$ ${\displaystyle {3\mathrm{}}_{}}$ ${\$ ${\displaystyle {d\mathrm{}}_{}}$ 2 ${\$ $d{\displaystyle {}_{}}$ ${\displaystyle {1\mathrm{}}_{}}$ ${\$ d$$ ${\displaystyle {0\mathrm{}}_{}}$ ${\$ 등도 다음과 같이 표현할 수 있다$$.

${\displaystyle d_{0}=a_{0}\nft b_{0}\oplus a_{3}\oplus b_{1}\oplus a_{2}\oplus a_{1}\nft b_{3}}}\oplus a_{1}\no3}}$

${\displaystyle d_{1}=a_{1}\cHB_{0}\oplus a_{0}\oplus b_{1}\oplus a_{3}\oplus a_{2}\oplus a_{2}\cHB_{3}}}\oplus a_{3}}}\oplus b_{3}$

${\displaystyle d_{2}=a_{2}\cHB_{0}\oplus a_{1}\oplus b_{1}\oplus a_{0}\oplus a_{3}\oplus b_{3}\{3}}}}}\oplus b_{3}}}}}}}}}}}}\oplus b_{{{3}}}}}}}}}}}}}}}$

${\displaystyle d_{3}=a_{3}\noplus b_{0}\oplus a_{2}\noplus b_{1}\noplus a_{0}\noplus b_{3}}\oplus a_{0}\no}\no3}$

${\displaystyle \mathrm{그리고}}$ 암호에 사용된 (${\displaystyle }x{\displaystyle }$ )${\displaystyle a(x)}$의 계수를 [ 1 2 ${\$ 상수로$$ 바꾸면 다음과 같은 결과를 얻는다.

${\displaystyle d_{0}=2\b_{0}\oplus 3\oplus b_{1}\oplus 1\oplus b_{3}$

${\displaystyle d_{1}=1\b_{0}\oplus 2\oplus b_{1}\oplus 3\oplus b_{2}\oplus 1\oplus b_{3}}}$

${\displaystyle d_{2}=1\noplus b_{0}\oplus 1\oplus b_{1}\oplus 2\oplus b_{2}\oplus 3\oplus b_{3}}}$

${\displaystyle d_{3}=3\noplus b_{0}\oplus 1\oplus b_{1}\oplus 1\oplus b_{2}\oplus 2\oplus b_{3}}}$

이는 작전 자체가 힐 암호와 유사하다는 것을 보여준다.Rijndael의 Galois 필드에서 4개의 숫자의 좌표 벡터에 다음과 같은 순환 MDS 행렬을 곱하여 수행할 수 있다.

${\displaystyle {\begin{bmatrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{bmatrix}}={\begin{bmatrix}2&3&1&1\\1&2&3&1\\1&1&2&3\\3&1&1&2\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}}$

구현 예

이것은 실제 구현에서 2 곱하기 2를 1교대 및 조건부 배타적 또는 3 곱하기 2를 배타적 또는 배타적 또는 배타적 곱하기 2로 대체함으로써 다소 단순화할 수 있다.그러한 구현의 C 예는 다음과 같다.

공허하게 하다 gmix_column(서명이 없는 마를 뜨다 *r) {     서명이 없는 마를 뜨다 a[4];     서명이 없는 마를 뜨다 b[4];     서명이 없는 마를 뜨다 c;     서명이 없는 마를 뜨다 h;     /* 어레이 'a'는 입력 어레이 'r'의 복사본일 뿐임 * 'b' 배열이 'a' 배열에 2를 곱한 각 요소임 * Rijndael's Galois 분야에서 * a[n] ^ b[n]는 Rijndael's Galois 필드에서 n에 3을 곱한 원소 */      을 위해 (c = 0; c < 4; c++) {         a[c] = r[c];         /* h는 r[c]의 높은 비트가 설정된 경우 0xff이고, 그렇지 않은 경우 0 */         h = (r[c] >> 7) & 1; /* 산술적 우측 이동, 즉 0 또는 1로 이동 */         b[c] = r[c] << 1; /* b[c]는 8비트 char이므로 다음 줄에서 0x11b가 아니라 0x1b로 xor하기 때문에 높은 비트를 암시적으로 제거함 */         b[c] ^= h * 0x1B; /* Rijndael's Galois 필드 */     }     r[0] = b[0] ^ a[3] ^ a[2] ^ b[1] ^ a[1]; /* 2 * a0 + a3 + a2 + 3 * a1 */     r[1] = b[1] ^ a[0] ^ a[3] ^ b[2] ^ a[2]; /* 2 * a1 + a0 + a3 + 3 * a2 */     r[2] = b[2] ^ a[1] ^ a[0] ^ b[3] ^ a[3]; /* 2 * a2 + a1 + a0 + 3 * a3 */     r[3] = b[3] ^ a[2] ^ a[1] ^ b[0] ^ a[0]; /* 2 * a3 + a2 + a1 + 3 * a0 */ } 

C# 예제

사유의 바이트 지물(바이트 a, 바이트 b) { // 갈루아 필드(26) 2바이트 곱하기     바이트 p = 0;      을 위해 (인트로 반격하다 = 0; 반격하다 < 8; 반격하다++) {         만일 ((b & 1) != 0) {             p ^= a;         }          바가지 긁다 hi_bit_set = (a & 0x80) != 0;         a <<= 1;         만일 (hi_bit_set) {             a ^= 0x1B; /* x^8 + x^4 + x^3 + x + 1 */         }         b >>= 1;     }      돌아오다 p; }  사유의 공허하게 하다 믹스콜럼스() { // 's'는 주 상태 행렬이고, 'ss'는 's'와 같은 차원의 온도 행렬이다.     배열.분명한(ss, 0, ss.길이);      을 위해 (인트로 c = 0; c < 4; c++) {         ss[0, c] = (바이트)(지물(0x02, s[0, c]) ^ 지물(0x03, s[1, c]) ^ s[2, c] ^ s[3, c]);         ss[1, c] = (바이트)(s[0, c] ^ 지물(0x02, s[1, c]) ^ 지물(0x03, s[2, c]) ^ s[3,c]);         ss[2, c] = (바이트)(s[0, c] ^ s[1, c] ^ 지물(0x02, s[2, c]) ^ 지물(0x03, s[3, c]));         ss[3, c] = (바이트)(지물(0x03, s[0,c]) ^ s[1, c] ^ s[2, c] ^ 지물(0x02, s[3, c]));     }      ss.카피토(s, 0); } 

MixColumn()에 대한 벡터 검정

InverseMixColumns

MixColumns 연산에는 다음과 같은 역행(숫자는 소수)이 있다.

${\displaystyle {\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}={\begin{bmatrix}14&11&13&9\\9&14&11&13\\13&9&14&11\\11&13&9&14\end{bmatrix}}{\begin{bmatrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{bmatrix}}}$

또는:

${\displaystyle{\begin{정렬}b_{0}&,=14\bullet d_{0}\oplus 11\bullet d_{1}\oplus 13\bullet d_{2}\oplus 9\bullet d_{3}\\b_{1}&, =9\bullet d_{0}\oplus 14\bullet d_{1}\oplus 11\bullet d_{2}\oplus 13\bullet d_{3}\\b_{2}&, =13\bullet d_{0}\oplus 9\bullet d_{1}\oplus 14\bullet d_{2}\oplus 11\bullet d_{3}\\b_{3}&,=11\bullet d_{0}\oplus 13\bull.(9d_{1}\oplus\cHB d_{2}\oplus 14\d_{3}\ended}}}}$

Galois 곱셈 조회 테이블

일반적으로, Rijndael 구현은 Galois 곱셈을 구현하기 보다는 사전 계산된 룩업 테이블을 사용하여 2, 3, 9, 11, 13, 14 바이트 곱셈을 수행한다.

예를 들어, C#에서는 이러한 테이블을 바이트[256] 배열로 저장할 수 있다.계산하기 위해

p * 3

결과는 다음과 같은 방법으로 얻는다.

결과 = table_3[(int)p]

이러한 룩업 테이블에서 가장 일반적인 예는 다음과 같다.

곱하기 2:

0x00,0x02,0x04,0x06,0x08,0x0a,0x0c,0x0e,0x10,0x12,0x14,0x16,0x18,0x1a,0x1c,0x1e, 0x20,0x22,0x24,0x26,0x28,0x2a,0x2c,0x2e,0x30,0x32,0x34,0x36,0x38,0x3a,0x3c,0x3e, 0x40,0x42,0x44,0x46,0x48,0x4a,0x4c,0x4e,0x50,0x52,0x54,0x56,0x58,0x5a,0x5c,0x5e, 0x60,0x62,0x64,0x66,0x68,0x6a,0x6c,0x6e,0x70,0x72,0x74,0x76,0x78,0x7a,0x7c,0x7e,  0x80,0x82,0x84,0x86,0x88,0x8a,0x8c,0x8e,0x90,0x92,0x94,0x96,0x98,0x9a,0x9c,0x9e, 0xa0,0xa2,0xa4,0xa6,0xa8,0xaa,0xac,0xae,0xb0,0xb2,0xb4,0xb6,0xb8,0xba,0xbc,0xbe, 0xc0,0xc2,0xc4,0xc6,0xc8,0xca,0xcc,0xce,0xd0,0xd2,0xd4,0xd6,0xd8,0xda,0xdc,0xde, 0xe0,0xe2,0xe4,0xe6,0xe8,0xea,0xec,0xee,0xf0,0xf2,0xf4,0xf6,0xf8,0xfa,0xfc,0xfe, 0x1b,0x19,0x1f,0x1d,0x13,0x11,0x17,0x15,0x0b,0x09,0x0f,0x0d,0x03,0x01,0x07,0x05, 0x3b,0x39,0x3f,0x3d,0x33,0x31,0x37,0x35,0x2b,0x29,0x2f,0x2d,0x23,0x21,0x27,0x25, 0x5b,0x59,0x5f,0x5d,0x53,0x51,0x57,0x55,0x4b,0x49,0x4f,0x4d,0x43,0x41,0x47,0x45, 0x7b,0x79,0x7f,0x7d,0x73,0x71,0x77,0x75,0x6b,0x69,0x6f,0x6d,0x63,0x61,0x67,0x65, 0x9b,0x99,0x9f,0x9d,0x93,0x91,0x97,0x95,0x8b,0x89,0x8f,0x8d,0x83,0x81,0x87,0x85, 0xbb,0xb9,0xbf,0xbd,0xb3,0xb1,0xb7,0xb5,0xab,0xa9,0xaf,0xad,0xa3,0xa1,0xa7,0xa5, 0xdb,0xd9,0xdf,0xdd,0xd3,0xd1,0xd7,0xd5,0xcb,0xc9,0xcf,0xcd,0xc3,0xc1,0xc7,0xc5, 0xfb,0xf9,0xff,0xfd,0xf3,0xf1,0xf7,0xf5,0xeb,0xe9,0xef,0xed,0xe3,0xe1,0xe7,0xe5

곱하기 3:

0x00,0x03,0x06,0x05,0x0c,0x0f,0x0a,0x09,0x18,0x1b,0x1e,0x1d,0x14,0x17,0x12,0x11, 0x30,0x33,0x36,0x35,0x3c,0x3f,0x3a,0x39,0x28,0x2b,0x2e,0x2d,0x24,0x27,0x22,0x21, 0x60,0x63,0x66,0x65,0x6c,0x6f,0x6a,0x69,0x78,0x7b,0x7e,0x7d,0x74,0x77,0x72,0x71, 0x50,0x53,0x56,0x55,0x5c,0x5f,0x5a,0x59,0x48,0x4b,0x4e,0x4d,0x44,0x47,0x42,0x41, 0xc0,0xc3,0xc6,0xc5,0xcc,0xcf,0xca,0xc9,0xd8,0xdb,0xde,0xdd,0xd4,0xd7,0xd2,0xd1, 0xf0,0xf3,0xf6,0xf5,0xfc,0xff,0xfa,0xf9,0xe8,0xeb,0xee,0xed,0xe4,0xe7,0xe2,0xe1, 0xa0,0xa3,0xa6,0xa5,0xac,0xaf,0xaa,0xa9,0xb8,0xbb,0xbe,0xbd,0xb4,0xb7,0xb2,0xb1, 0x90,0x93,0x96,0x95,0x9c,0x9f,0x9a,0x99,0x88,0x8b,0x8e,0x8d,0x84,0x87,0x82,0x81,  0x9b,0x98,0x9d,0x9e,0x97,0x94,0x91,0x92,0x83,0x80,0x85,0x86,0x8f,0x8c,0x89,0x8a, 0xab,0xa8,0xad,0xae,0xa7,0xa4,0xa1,0xa2,0xb3,0xb0,0xb5,0xb6,0xbf,0xbc,0xb9,0xba, 0xfb,0xf8,0xfd,0xfe,0xf7,0xf4,0xf1,0xf2,0xe3,0xe0,0xe5,0xe6,0xef,0xec,0xe9,0xea,  0xcb,0xc8,0xcd,0xce,0xc7,0xc4,0xc1,0xc2,0xd3,0xd0,0xd5,0xd6,0xdf,0xdc,0xd9,0xda,  0x5b,0x58,0x5d,0x5e,0x57,0x54,0x51,0x52,0x43,0x40,0x45,0x46,0x4f,0x4c,0x49,0x4a, 0x6b,0x68,0x6d,0x6e,0x67,0x64,0x61,0x62,0x73,0x70,0x75,0x76,0x7f,0x7c,0x79,0x7a,  0x3b,0x38,0x3d,0x3e,0x37,0x34,0x31,0x32,0x23,0x20,0x25,0x26,0x2f,0x2c,0x29,0x2a, 0x0b,0x08,0x0d,0x0e,0x07,0x04,0x01,0x02,0x13,0x10,0x15,0x16,0x1f,0x1c,0x19,0x1a

9로 곱하기:

0x00,0x09,0x12,0x1b,0x24,0x2d,0x36,0x3f,0x48,0x41,0x5a,0x53,0x6c,0x65,0x7e,0x77, 0x90,0x99,0x82,0x8b,0xb4,0xbd,0xa6,0xaf,0xd8,0xd1,0xca,0xc3,0xfc,0xf5,0xee,0xe7, 0x3b,0x32,0x29,0x20,0x1f,0x16,0x0d,0x04,0x73,0x7a,0x61,0x68,0x57,0x5e,0x45,0x4c, 0xab,0xa2,0xb9,0xb0,0x8f,0x86,0x9d,0x94,0xe3,0xea,0xf1,0xf8,0xc7,0xce,0xd5,0xdc, 0x76,0x7f,0x64,0x6d,0x52,0x5b,0x40,0x49,0x3e,0x37,0x2c,0x25,0x1a,0x13,0x08,0x01, 0xe6,0xef,0xf4,0xfd,0xc2,0xcb,0xd0,0xd9,0xae,0xa7,0xbc,0xb5,0x8a,0x83,0x98,0x91, 0x4d,0x44,0x5f,0x56,0x69,0x60,0x7b,0x72,0x05,0x0c,0x17,0x1e,0x21,0x28,0x33,0x3a, 0xdd,0xd4,0xcf,0xc6,0xf9,0xf0,0xeb,0xe2,0x95,0x9c,0x87,0x8e,0xb1,0xb8,0xa3,0xaa,  0xec,0xe5,0xfe,0xf7,0xc8,0xc1,0xda,0xd3,0xa4,0xad,0xb6,0xbf,0x80,0x89,0x92,0x9b,  0x7c,0x75,0x6e,0x67,0x58,0x51,0x4a,0x43,0x34,0x3d,0x26,0x2f,0x10,0x19,0x02,0x0b, 0xd7,0xde,0xc5,0xcc,0xf3,0xfa,0xe1,0xe8,0x9f,0x96,0x8d,0x84,0xbb,0xb2,0xa9,0xa0, 0x47,0x4e,0x55,0x5c,0x63,0x6a,0x71,0x78,0x0f,0x06,0x1d,0x14,0x2b,0x22,0x39,0x30, 0x9a,0x93,0x88,0x81,0xbe,0xb7,0xac,0xa5,0xd2,0xdb,0xc0,0xc9,0xf6,0xff,0xe4,0xed, 0x0a,0x03,0x18,0x11,0x2e,0x27,0x3c,0x35,0x42,0x4b,0x50,0x59,0x66,0x6f,0x74,0x7d,  0xa1,0xa8,0xb3,0xba,0x85,0x8c,0x97,0x9e,0xe9,0xe0,0xfb,0xf2,0xcd,0xc4,0xdf,0xd6, 0x31,0x38,0x23,0x2a,0x15,0x1c,0x07,0x0e,0x79,0x70,0x6b,0x62,0x5d,0x54,0x4f,0x46

곱하기 11(0xB):

0x00,0x0b,0x16,0x1d,0x2c,0x27,0x3a,0x31,0x58,0x53,0x4e,0x45,0x74,0x7f,0x62,0x69, 0xb0,0xbb,0xa6,0xad,0x9c,0x97,0x8a,0x81,0xe8,0xe3,0xfe,0xf5,0xc4,0xcf,0xd2,0xd9, 0x7b,0x70,0x6d,0x66,0x57,0x5c,0x41,0x4a,0x23,0x28,0x35,0x3e,0x0f,0x04,0x19,0x12, 0xcb,0xc0,0xdd,0xd6,0xe7,0xec,0xf1,0xfa,0x93,0x98,0x85,0x8e,0xbf,0xb4,0xa9,0xa2, 0xf6,0xfd,0xe0,0xeb,0xda,0xd1,0xcc,0xc7,0xae,0xa5,0xb8,0xb3,0x82,0x89,0x94,0x9f, 0x46,0x4d,0x50,0x5b,0x6a,0x61,0x7c,0x77,0x1e,0x15,0x08,0x03,0x32,0x39,0x24,0x2f, 0x8d,0x86,0x9b,0x90,0xa1,0xaa,0xb7,0xbc,0xd5,0xde,0xc3,0xc8,0xf9,0xf2,0xef,0xe4, 0x3d,0x36,0x2b,0x20,0x11,0x1a,0x07,0x0c,0x65,0x6e,0x73,0x78,0x49,0x42,0x5f,0x54, 0xf7,0xfc,0xe1,0xea,0xdb,0xd0,0xcd,0xc6,0xaf,0xa4,0xb9,0xb2,0x83,0x88,0x95,0x9e, 0x47,0x4c,0x51,0x5a,0x6b,0x60,0x7d,0x76,0x1f,0x14,0x09,0x02,0x33,0x38,0x25,0x2e, 0x8c,0x87,0x9a,0x91,0xa0,0xab,0xb6,0xbd,0xd4,0xdf,0xc2,0xc9,0xf8,0xf3,0xee,0xe5, 0x3c,0x37,0x2a,0x21,0x10,0x1b,0x06,0x0d,0x64,0x6f,0x72,0x79,0x48,0x43,0x5e,0x55, 0x01,0x0a,0x17,0x1c,0x2d,0x26,0x3b,0x30,0x59,0x52,0x4f,0x44,0x75,0x7e,0x63,0x68, 0xb1,0xba,0xa7,0xac,0x9d,0x96,0x8b,0x80,0xe9,0xe2,0xff,0xf4,0xc5,0xce,0xd3,0xd8, 0x7a,0x71,0x6c,0x67,0x56,0x5d,0x40,0x4b,0x22,0x29,0x34,0x3f,0x0e,0x05,0x18,0x13, 0xca,0xc1,0xdc,0xd7,0xe6,0xed,0xf0,0xfb,0x92,0x99,0x84,0x8f,0xbe,0xb5,0xa8,0xa3

13을 곱하기(0xD):

0x00,0x0d,0x1a,0x17,0x34,0x39,0x2e,0x23,0x68,0x65,0x72,0x7f,0x5c,0x51,0x46,0x4b, 0xd0,0xdd,0xca,0xc7,0xe4,0xe9,0xfe,0xf3,0xb8,0xb5,0xa2,0xaf,0x8c,0x81,0x96,0x9b, 0xbb,0xb6,0xa1,0xac,0x8f,0x82,0x95,0x98,0xd3,0xde,0xc9,0xc4,0xe7,0xea,0xfd,0xf0, 0x6b,0x66,0x71,0x7c,0x5f,0x52,0x45,0x48,0x03,0x0e,0x19,0x14,0x37,0x3a,0x2d,0x20, 0x6d,0x60,0x77,0x7a,0x59,0x54,0x43,0x4e,0x05,0x08,0x1f,0x12,0x31,0x3c,0x2b,0x26, 0xbd,0xb0,0xa7,0xaa,0x89,0x84,0x93,0x9e,0xd5,0xd8,0xcf,0xc2,0xe1,0xec,0xfb,0xf6, 0xd6,0xdb,0xcc,0xc1,0xe2,0xef,0xf8,0xf5,0xbe,0xb3,0xa4,0xa9,0x8a,0x87,0x90,0x9d, 0x06,0x0b,0x1c,0x11,0x32,0x3f,0x28,0x25,0x6e,0x63,0x74,0x79,0x5a,0x57,0x40,0x4d, 0xda,0xd7,0xc0,0xcd,0xee,0xe3,0xf4,0xf9,0xb2,0xbf,0xa8,0xa5,0x86,0x8b,0x9c,0x91, 0x0a,0x07,0x10,0x1d,0x3e,0x33,0x24,0x29,0x62,0x6f,0x78,0x75,0x56,0x5b,0x4c,0x41, 0x61,0x6c,0x7b,0x76,0x55,0x58,0x4f,0x42,0x09,0x04,0x13,0x1e,0x3d,0x30,0x27,0x2a, 0xb1,0xbc,0xab,0xa6,0x85,0x88,0x9f,0x92,0xd9,0xd4,0xc3,0xce,0xed,0xe0,0xf7,0xfa, 0xb7,0xba,0xad,0xa0,0x83,0x8e,0x99,0x94,0xdf,0xd2,0xc5,0xc8,0xeb,0xe6,0xf1,0xfc, 0x67,0x6a,0x7d,0x70,0x53,0x5e,0x49,0x44,0x0f,0x02,0x15,0x18,0x3b,0x36,0x21,0x2c, 0x0c,0x01,0x16,0x1b,0x38,0x35,0x22,0x2f,0x64,0x69,0x7e,0x73,0x50,0x5d,0x4a,0x47, 0xdc,0xd1,0xc6,0xcb,0xe8,0xe5,0xf2,0xff,0xb4,0xb9,0xae,0xa3,0x80,0x8d,0x9a,0x97

14를 곱하기(0xE):

0x00,0x0e,0x1c,0x12,0x38,0x36,0x24,0x2a,0x70,0x7e,0x6c,0x62,0x48,0x46,0x54,0x5a, 0xe0,0xee,0xfc,0xf2,0xd8,0xd6,0xc4,0xca,0x90,0x9e,0x8c,0x82,0xa8,0xa6,0xb4,0xba, 0xdb,0xd5,0xc7,0xc9,0xe3,0xed,0xff,0xf1,0xab,0xa5,0xb7,0xb9,0x93,0x9d,0x8f,0x81, 0x3b,0x35,0x27,0x29,0x03,0x0d,0x1f,0x11,0x4b,0x45,0x57,0x59,0x73,0x7d,0x6f,0x61, 0xad,0xa3,0xb1,0xbf,0x95,0x9b,0x89,0x87,0xdd,0xd3,0xc1,0xcf,0xe5,0xeb,0xf9,0xf7, 0x4d,0x43,0x51,0x5f,0x75,0x7b,0x69,0x67,0x3d,0x33,0x21,0x2f,0x05,0x0b,0x19,0x17, 0x76,0x78,0x6a,0x64,0x4e,0x40,0x52,0x5c,0x06,0x08,0x1a,0x14,0x3e,0x30,0x22,0x2c, 0x96,0x98,0x8a,0x84,0xae,0xa0,0xb2,0xbc,0xe6,0xe8,0xfa,0xf4,0xde,0xd0,0xc2,0xcc, 0x41,0x4f,0x5d,0x53,0x79,0x77,0x65,0x6b,0x31,0x3f,0x2d,0x23,0x09,0x07,0x15,0x1b, 0xa1,0xaf,0xbd,0xb3,0x99,0x97,0x85,0x8b,0xd1,0xdf,0xcd,0xc3,0xe9,0xe7,0xf5,0xfb, 0x9a,0x94,0x86,0x88,0xa2,0xac,0xbe,0xb0,0xea,0xe4,0xf6,0xf8,0xd2,0xdc,0xce,0xc0, 0x7a,0x74,0x66,0x68,0x42,0x4c,0x5e,0x50,0x0a,0x04,0x16,0x18,0x32,0x3c,0x2e,0x20, 0xec,0xe2,0xf0,0xfe,0xd4,0xda,0xc8,0xc6,0x9c,0x92,0x80,0x8e,0xa4,0xaa,0xb8,0xb6, 0x0c,0x02,0x10,0x1e,0x34,0x3a,0x28,0x26,0x7c,0x72,0x60,0x6e,0x44,0x4a,0x58,0x56, 0x37,0x39,0x2b,0x25,0x0f,0x01,0x13,0x1d,0x47,0x49,0x5b,0x55,0x7f,0x71,0x63,0x6d, 0xd7,0xd9,0xcb,0xc5,0xef,0xe1,0xf3,0xfd,0xa7,0xa9,0xbb,0xb5,0x9f,0x91,0x83,0x8d

참조

• FIPS PUB 197: 공식 AES 표준(PDF 파일)

참고 항목

• 고급 암호화 표준