Federal Information Processing Standards Publications (FIPS PUBS) are issued by the National Institute of Standards and Technology (NIST) after approval by the Secretary of Commerce pursuant to Section 5131 of the Information Technology Management Reform Act of 1996 (Public Law 104-106) and the Computer Security Act of 1987 (Public Law 100-235).

### Name of Standard. Advanced Encryption Standard (AES) (FIPS PUB 197).

### Category of Standard. Computer Security Standard, Cryptography.

### Explanation. The Advanced Encryption Standard (AES) specifies a FIPS-approved cryptographic algorithm that can be used to protect electronic data. The AES algorithm is a symmetric block cipher that can encrypt (encipher) and decrypt (decipher) information. Encryption converts data to an unintelligible form called ciphertext; decrypting the ciphertext converts the data back into its original form, called plaintext.

The AES algorithm is capable of using cryptographic keys of 128, 192, and 256 bits to encrypt and decrypt data in blocks of 128 bits.

### Approving Authority. Secretary of Commerce.

### Maintenance Agency. Department of Commerce, National Institute of Standards and Technology, Information Technology Laboratory (ITL).

### Applicability. This standard may be used by Federal departments and agencies when an agency determines that sensitive (unclassified) information (as defined in P. L. 100-235) requires cryptographic protection.

Other FIPS-approved cryptographic algorithms may be used in addition to, or in lieu of, this standard. Federal agencies or departments that use cryptographic devices for protecting classified information can use those devices for protecting sensitive (unclassified) information in lieu of this standard.

In addition, this standard may be adopted and used by non-Federal Government organizations. Such use is encouraged when it provides the desired security for commercial and private organizations.

### Specifications. Federal Information Processing Standard (FIPS) 197, Advanced Encryption Standard (AES) (affixed).

**Implementations.**The algorithm specified in this standard may be implemented in software, firmware, hardware, or any combination thereof. The specific implementation may depend on several factors such as the application, the environment, the technology used, etc. The algorithm shall be used in conjunction with a FIPS approved or NIST recommended mode of operation. Object Identifiers (OIDs) and any associated parameters for AES used in these modes are available at the Computer Security Objects Register (CSOR), located at http://csrc.nist.gov/csor/ [2].Implementations of the algorithm that are tested by an accredited laboratory and validated will be considered as complying with this standard. Since cryptographic security depends on many factors besides the correct implementation of an encryption algorithm, Federal Government employees, and others, should also refer to NIST Special Publication 800-21,

*Guideline for Implementing Cryptography in the Federal Government*, for additional information and guidance (NIST SP 800-21 is available at http://csrc.nist.gov/publications/).### Implementation Schedule. This standard becomes effective on May 26, 2002.

### Patents. Implementations of the algorithm specified in this standard may be covered by

U.S. and foreign patents.

### Export Control. Certain cryptographic devices and technical data regarding them are subject to Federal export controls. Exports of cryptographic modules implementing this standard and technical data regarding them must comply with these Federal regulations and be licensed by the Bureau of Export Administration of the U.S. Department of Commerce. Applicable Federal government export controls are specified in Title 15, Code of Federal Regulations (CFR) Part 740.17; Title 15, CFR Part 742; and Title 15, CFR Part 774, Category 5, Part 2.

### Qualifications. NIST will continue to follow developments in the analysis of the AES algorithm. As with its other cryptographic algorithm standards, NIST will formally reevaluate this standard every five years.

Both this standard and possible threats reducing the security provided through the use of this standard will undergo review by NIST as appropriate, taking into account newly available analysis and technology. In addition, the awareness of any breakthrough in technology or any mathematical weakness of the algorithm will cause NIST to reevaluate this standard and provide necessary revisions.

### Waiver Procedure. Under certain exceptional circumstances, the heads of Federal agencies, or their delegates, may approve waivers to Federal Information Processing Standards (FIPS). The heads of such agencies may redelegate such authority only to a senior official designated pursuant to Section 3506(b) of Title 44, U.S. Code. Waivers shall be granted only when compliance with this standard would

adversely affect the accomplishment of the mission of an operator of Federal computer system or

cause a major adverse financial impact on the operator that is not offset by government- wide savings.

Agency heads may act upon a written waiver request containing the information detailed above. Agency heads may also act without a written waiver request when they determine that conditions for meeting the standard cannot be met. Agency heads may approve waivers only by a written decision that explains the basis on which the agency head made the required finding(s). A copy of each such decision, with procurement sensitive or classified portions clearly identified, shall be sent to: National Institute of Standards and Technology; ATTN: FIPS Waiver Decision, Information Technology Laboratory, 100 Bureau Drive, Stop 8900, Gaithersburg, MD 20899- 8900.

In addition, notice of each waiver granted and each delegation of authority to approve waivers shall be sent promptly to the Committee on Government Operations of the House of Representatives and the Committee on Government Affairs of the Senate and shall be published promptly in the Federal Register.

When the determination on a waiver applies to the procurement of equipment and/or services, a notice of the waiver determination must be published in the Commerce Business Daily as a part of the notice of solicitation for offers of an acquisition or, if the waiver determination is made after that notice is published, by amendment to such notice.

A copy of the waiver, any supporting documents, the document approving the waiver and any supporting and accompanying documents, with such deletions as the agency is authorized and decides to make under Section 552(b) of Title 5, U.S. Code, shall be part of the procurement documentation and retained by the agency.

**Where to obtain copies.**This publication is available electronically by accessing http://csrc.nist.gov/publications/. A list of other available computer security publications, including ordering information, can be obtained from NIST Publications List 91, which is available at the same web site. Alternatively, copies of NIST computer security publications are available from: National Technical Information Service (NTIS), 5285 Port Royal Road, Springfield, VA 22161.

INTRODUCTION 5

DEFINITIONS 5

GLOSSARY OF TERMS AND ACRONYMS 5

ALGORITHM PARAMETERS, SYMBOLS, AND FUNCTIONS 6

NOTATION AND CONVENTIONS 7

INPUTS AND OUTPUTS 7

BYTES 8

ARRAYS OF BYTES 8

THE STATE 9

THE STATE AS AN ARRAY OF COLUMNS 10

MATHEMATICAL PRELIMINARIES 10

ADDITION 10

MULTIPLICATION 10

Multiplication by x 11

POLYNOMIALS WITH COEFFICIENTS IN GF(28) 12

ALGORITHM SPECIFICATION 13

CIPHER 14

SubBytes()Transformation 15

ShiftRows() Transformation 17

MixColumns() Transformation 17

AddRoundKey() Transformation 18

KEY EXPANSION 19

INVERSE CIPHER 20

InvShiftRows() Transformation 21

InvSubBytes() Transformation 22

InvMixColumns() Transformation 23

Inverse of the AddRoundKey() Transformation 23

Equivalent Inverse Cipher 23

IMPLEMENTATION ISSUES 25

KEY LENGTH REQUIREMENTS 25

KEYING RESTRICTIONS 26

PARAMETERIZATION OF KEY LENGTH, BLOCK SIZE, AND ROUND NUMBER 26

IMPLEMENTATION SUGGESTIONS REGARDING VARIOUS PLATFORMS 26

APPENDIX A - KEY EXPANSION EXAMPLES 27

EXPANSION OF A 128-BIT CIPHER KEY 27

EXPANSION OF A 192-BIT CIPHER KEY 28

EXPANSION OF A 256-BIT CIPHER KEY 30

APPENDIX B – CIPHER EXAMPLE 33

APPENDIX C – EXAMPLE VECTORS 35

C.1 AES-128 (*N*K=4, *N*R=10) 35

C.2 AES-192 (*N*K=6, *N*R=12) 38

C.3 AES-256 (*N*K=8, *N*R=14) 42

APPENDIX D - REFERENCES 47

Figure 1. Hexadecimal representation of bit patterns. 8

Figure 2. Indices for Bytes and Bits. 9

Figure 3. State array input and output. 9

Figure 4. Key-Block-Round Combinations. 14

Figure 5. Pseudo Code for the Cipher. 15

Figure 6. SubBytes() applies the S-box to each byte of the State. 16

Figure 7. S-box: substitution values for the byte xy (in hexadecimal format). 16

Figure 8. ShiftRows() cyclically shifts the last three rows in the State 17

Figure 9. MixColumns() operates on the State column-by-column. 18

Figure 10. AddRoundKey() XORs each column of the State with a word from the key schedule. 19

Figure 11. Pseudo Code for Key Expansion. 20

Figure 12. Pseudo Code for the Inverse Cipher. 21

Figure 13. InvShiftRows()cyclically shifts the last three rows in the State. 22

Figure 14. Inverse S-box: substitution values for the byte xy (in hexadecimal format). 22

Figure 15. Pseudo Code for the Equivalent Inverse Cipher. 25

# Introduction

This standard specifies the

**Rijndael**algorithm ([3] and [4]), a symmetric block cipher that can process**data blocks**of**128 bits**, using cipher**keys**with lengths of**128**,**192**, and**256 bits**. Rijndael was designed to handle additional block sizes and key lengths, however they are not adopted in this standard.Throughout the remainder of this standard, the algorithm specified herein will be referred to as “the AES algorithm.” The algorithm may be used with the three different key lengths indicated above, and therefore these different “flavors” may be referred to as “AES-128”, “AES-192”, and “AES-256”.

This specification includes the following sections:

Definitions of terms, acronyms, and algorithm parameters, symbols, and functions;

Notation and conventions used in the algorithm specification, including the ordering and numbering of bits, bytes, and words;

Mathematical properties that are useful in understanding the algorithm;

Algorithm specification, covering the key expansion, encryption, and decryption routines;

Implementation issues, such as key length support, keying restrictions, and additional block/key/round sizes.

The standard concludes with several appendices that include step-by-step examples for Key Expansion and the Cipher, example vectors for the Cipher and Inverse Cipher, and a list of references.

# Definitions

## Glossary of Terms and Acronyms

The following definitions are used throughout this standard: AES Advanced Encryption Standard

Affine A transformation consisting of multiplication by a matrix followed by Transformation the addition of a vector.

Array An enumerated collection of identical entities (e.g., an array of bytes). Bit A binary digit having a value of 0 or 1.

Block Sequence of binary bits that comprise the input, output, State, and Round Key. The length of a sequence is the number of bits it contains. Blocks are also interpreted as arrays of bytes.

Byte A group of eight bits that is treated either as a single entity or as an array of 8 individual bits.

Cipher Series of transformations that converts plaintext to ciphertext using the Cipher Key.

Cipher Key Secret, cryptographic key that is used by the Key Expansion routine to generate a set of Round Keys; can be pictured as a rectangular array of bytes, having four rows and

columns.**Nk**Ciphertext Data output from the Cipher or input to the Inverse Cipher.

Inverse Cipher Series of transformations that converts ciphertext to plaintext using the Cipher Key.

Key Expansion Routine used to generate a series of Round Keys from the Cipher Key. Plaintext Data input to the Cipher or output from the Inverse Cipher.

Rijndael Cryptographic algorithm specified in this Advanced Encryption Standard (AES).

Round Key Round keys are values derived from the Cipher Key using the Key Expansion routine; they are applied to the State in the Cipher and Inverse Cipher.

State Intermediate Cipher result that can be pictured as a rectangular array of bytes, having four rows and

columns.**Nb**S-box Non-linear substitution table used in several byte substitution transformations and in the Key Expansion routine to perform a one- for-one substitution of a byte value.

Word A group of 32 bits that is treated either as a single entity or as an array of 4 bytes.

## Algorithm Parameters, Symbols, and Functions

The following algorithm parameters, symbols, and functions are used throughout this standard:

AddRoundKey() Transformation in the Cipher and Inverse Cipher in which a Round Key is added to the State using an XOR operation. The length of a Round Key equals the size of the State (i.e., for

*Nb*= 4, the Round Key length equals 128 bits/16 bytes).InvMixColumns()Transformation in

the

Inverse

Cipher

that

is

the

inverse

of

MixColumns().

InvShiftRows() Transformation in

the

Inverse

Cipher

that

is

the

inverse

of

ShiftRows().

InvSubBytes() Transformation in the Inverse Cipher that is the inverse of

SubBytes().

K Cipher Key.

MixColumns() Transformation in the Cipher that takes all of the columns of the State and mixes their data (independently of one another) to produce new columns.

Nb Number of columns (32-bit words) comprising the State. For this standard, Nb = 4. (Also see Sec. 6.3.)

Nk Number of 32-bit words comprising the Cipher Key. For this standard, Nk = 4, 6, or 8. (Also see Sec. 6.3.)

Nr Number of rounds, which is a function of Nk and Nb (which is fixed). For this standard, Nr = 10, 12, or 14. (Also see Sec. 6.3.)

### Rcon[] The round constant word array.

RotWord() Function used in the Key Expansion routine that takes a four-byte word and performs a cyclic permutation.

ShiftRows() Transformation in the Cipher that processes the State by cyclically shifting the last three rows of the State by different offsets.

SubBytes() Transformation in the Cipher that processes the State using a non- linear byte substitution table (S-box) that operates on each of the State bytes independently.

SubWord() Function used in the Key Expansion routine that takes a four-byte input word and applies an S-box to each of the four bytes to produce an output word.

XOR Exclusive-OR operation.

Exclusive-OR operation.

Multiplication of two polynomials (each with degree < 4) modulo

x4 + 1.

Finite field multiplication.

# Notation and Conventions

## Inputs and Outputs

The

**input**and**output**for the AES algorithm each consist of**sequences of 128 bits**(digits with values of 0 or 1). These sequences will sometimes be referred to as**blocks**and the number of bits they contain will be referred to as their length. The**Cipher Key**for the AES algorithm is a**sequence of 128, 192 or 256 bits**. Other input, output and Cipher Key lengths are not permitted by this standard.The bits within such sequences will be numbered starting at zero and ending at one less than the sequence length (block length or key length). The number

*i*attached to a bit is known as its index and will be in one of the ranges 0 *i*< 128, 0 *i*< 192 or 0 *i*< 256 depending on the block length and key length (specified above).## Bytes

The basic unit for processing in the AES algorithm is a

**byte,**a sequence of eight bits treated as a single entity. The input, output and Cipher Key bit sequences described in Sec. 3.1 are processed as arrays of bytes that are formed by dividing these sequences into groups of eight contiguous bits to form arrays of bytes (see Sec. 3.3). For an input, output or Cipher Key denoted by*a*, the bytes in the resulting array will be referenced using one of the two forms,*a*n or*a*[*n*], where*n*will be in one of the following ranges:Key length = 128 bits, 0

*n*< 16; Block length = 128 bits, 0 *n*< 16; Key length = 192 bits, 0 *n*< 24;Key length = 256 bits, 0

*n*< 32.All byte values in the AES algorithm will be presented as the concatenation of its individual bit values (0 or 1) between braces in the order {

*b*7,*b*6,*b*5,*b*4,*b*3,*b*2,*b*1,*b*0}. These bytes are interpreted as finite field elements using a polynomial representation:7

b7 x

6

b6 x

5

b5 x

4

b4 x

3

b3 x

2

b2 x

7

i

b1 x b0 bi x

. (3.1)

i0

For example, {01100011} identifies the specific finite field element

x6 x5 x 1.

It is also convenient to denote byte values using hexadecimal notation with each of two groups of four bits being denoted by a single character as in Fig. 1.

Bit Pattern

Character

0000

0

0001

1

0010

2

0011

3

Bit Pattern

Character

0100

4

0101

5

0110

6

0111

7

Bit Pattern

Character

1000

8

1001

9

1010

a

1011

b

Bit Pattern

Character

1100

c

1101

d

1110

e

1111

f

#### Figure 1. Hexadecimal representation of bit patterns.

Hence the element {01100011} can be represented as {63}, where the character denoting the four-bit group containing the higher numbered bits is again to the left.

Some finite field operations involve one additional bit (

*b*8) to the left of an 8-bit byte. Where this extra bit is present, it will appear as ‘{01}’ immediately preceding the 8-bit byte; for example, a 9-bit sequence will be presented as {01}{1b}.## Arrays of Bytes

Arrays of bytes will be represented in the following form:

a0 a1 a2 ...a15

The bytes and the bit ordering within bytes are derived from the 128-bit input sequence

input0 input1 input2 … input126 input127

as follows:

a0 = {input0, input1, …, input7}; a1 = {input8, input9, …, input15};

a15 = {input120, input121, …, input127}.

The pattern can be extended to longer sequences (i.e., for 192- and 256-bit keys), so that, in general,

Input bit sequence

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

…

Byte number

0

1

2

…

Bit numbers in byte

7

6

5

4

3

2

1

0

7

6

5

4

3

2

1

0

7

6

5

4

3

2

1

0

…

an = {input8n, input8n+1, …, input8n+7}. (3.2) Taking Sections 3.2 and 3.3 together, Fig. 2 shows how bits within each byte are numbered.

#### Figure 2. Indices for Bytes and Bits.

## The State

Internally, the AES algorithm’s operations are performed on a two-dimensional array of bytes called the

**State**. The State consists of four rows of bytes, each containingbytes, where**Nb**is the block length divided by 32. In the State array denoted by the symbol**Nb***s*, each individual byte has two indices, with its row number*r*in the range 0 *r*< 4 and its column number*c*in the range 0 *c*<. This allows an individual byte of the State to be referred to as either**Nb***s*r,c or*s*[*r*,*c*]. For this standard,=4, i.e., 0 **Nb***c*< 4 (also see Sec. 6.3).At the start of the Cipher and Inverse Cipher described in Sec. 5, the input – the array of bytes

*in*0,*in*1, …*in*15 – is copied into the State array as illustrated in Fig. 3. The Cipher or Inverse Cipher operations are then conducted on this State array, after which its final value is copied to the output – the array of bytes*out*0,*out*1, …*out*15.input bytes State array output bytes

in0

in4

in8

in12

in1

in5

in9

in13

in2

in6

in10

in14

in3

in7

in11

in15

s0,0

s0,1

s0,2

s0,3

s1,0

s1,1

s1,2

s1,3

s2,0

s2,1

s2,2

s2,3

s3,0

s3,1

s3,2

s3,3

out0

out4

out8

out12

out1

out5

out9

out13

out2

out6

out10

out14

out3

out7

out11

out15

多 多

#### Figure 3. State array input and output.

Hence, at the beginning of the Cipher or Inverse Cipher, the input array,

*in*, is copied to the State array according to the scheme:s[r, c] = in[r + 4c] for 0 r < 4 and 0 c < Nb, (3.3)

and at the end of the Cipher and Inverse Cipher, the State is copied to the output array

*out*as follows:out[r + 4c] = s[r, c] for 0 r < 4 and 0 c < Nb. (3.4)

## The State as an Array of Columns

The four bytes in each column of the State array form 32-bit

**words**, where the row number*r*provides an index for the four bytes within each word. The state can hence be interpreted as a one-dimensional array of 32 bit words (columns),*w*0...*w*3, where the column number*c*provides an index into this array. Hence, for the example in Fig. 3, the State can be considered as an array of four words, as follows:w0 = s0,0 s1,0 s2,0 s3,0

w2 = s0,2 s1,2 s2,2 s3,2

w1 = s0,1 s1,1 s2,1 s3,1

w3 = s0,3 s1,3 s2,3 s3,3 .

(3.5)

# Mathematical Preliminaries

All bytes in the AES algorithm are interpreted as finite field elements using the notation introduced in Sec. 3.2. Finite field elements can be added and multiplied, but these operations are different from those used for numbers. The following subsections introduce the basic mathematical concepts needed for Sec. 5.

## Addition

The addition of two elements in a finite field is achieved by “adding” the coefficients for the corresponding powers in the polynomials for the two elements. The addition is performed with

the XOR operation (denoted by ) - i.e., modulo 2 - so that 1 1 0 , 1 0 1, and Consequently, subtraction of polynomials is identical to addition of polynomials.

0 0 0 .

Alternatively, addition of finite field elements can be described as the modulo 2 addition of corresponding bits in the byte. For two bytes {a7a6a5a4a3a2a1a0} and {b7b6b5b4b3b2b1b0}, the sum is

{c7c6c5c4c3c2c1c0}, where each ci = ai bi (i.e., c7 = a7 b7, c6 = a6 b6, ...c0 = a0 b0).

For example, the following expressions are equivalent to one another:

(x6 x4 x2 x 1)

+ (x7 x 1) =

x7 x6 x4 x 2

(polynomial notation);

{01010111} {10000011} = {11010100} (binary notation);

{57} {83} = {d4} (hexadecimal notation).

## Multiplication

In the polynomial representation, multiplication in GF(28) (denoted by ) corresponds with the multiplication of polynomials modulo an

**irreducible polynomial**of degree 8. A polynomial is irreducible if its only divisors are one and itself.**For the AES algorithm, this****irreducible****polynomial is**m(x) x 8 x 4 x 3 x 1 , (4.1)

or {01}{1b} in hexadecimal notation.

For example, {57} {83} = {c1}, because

(x 6 x 4 x 2 x 1) (x 7 x 1) =

=

and

x13 x11 x 9 x8 x 7

x 7 x 5 x 3 x 2 x

x 6 x 4 x 2 x 1

x13 x11 x 9 x8 x 6 x 5 x 4 x 3 1

x13 x11 x 9 x8 x 6 x 5 x 4 x 3 1

=

modulo ( x8 x 4 x 3 x 1

**)**x 7 x 6 1 .

The modular reduction by

*m*(*x*) ensures that the result will be a binary polynomial of degree less than 8, and thus can be represented by a byte. Unlike addition, there is no simple operation at the byte level that corresponds to this multiplication.The multiplication defined above is associative, and the element {01} is the multiplicative identity. For any non-zero binary polynomial

*b*(*x*) of degree less than 8, the multiplicative inverse of*b*(*x*), denoted*b*-1(*x*), can be found as follows: the extended Euclidean algorithm [7] is used to compute polynomials*a*(*x*) and*c*(*x*) such that*b*(*x*)*a*(*x*) *m*(*x*)*c*(*x*) 1**.**(4.2)Hence,

*a*(*x*) *b*(*x*) mod*m*(*x*) 1, which means*b*1 (*x*) *a*(*x*) mod*m*(*x*)**.**(4.3)Moreover, for any

*a*(*x*),*b*(*x*) and*c*(*x*) in the field, it holds that*a*(*x*) (*b*(*x*) *c*(*x*)) *a*(*x*) *b*(*x*) *a*(*x*) *c*(*x*) .It follows that the set of 256 possible byte values, with XOR used as addition and the multiplication defined as above, has the structure of the finite field GF(28).

Multiplication by x

Multiplying the binary polynomial defined in equation (3.1) with the polynomial

*x*results in8

b7 x

7

b6 x

6

b5 x

5

b4 x

4

b3 x

3

b2 x

2

b1 x

b0 x . (4.4)

The result

x b(x) is obtained by reducing the above result modulo m(x), as defined in equation

(4.1). If

*b*7 = 0, the result is already in reduced form. If*b*7 = 1, the reduction is accomplished by subtracting (i.e., XORing) the polynomial*m*(*x*). It follows that multiplication by*x*(i.e.,{00000010} or {02}) can be implemented at the byte level as a left shift and a subsequent conditional bitwise XOR with {1b}. This operation on bytes is denoted by xtime(). Multiplication by higher powers of

*x*can be implemented by repeated application of xtime(). By adding intermediate results, multiplication by any constant can be implemented.For example, {57} {13} = {fe} because

thus,

{57} {02} = xtime({57}) = {ae}

{57} {04} = xtime({ae}) = {47}

{57} {08} = xtime({47}) = {8e}

{57} {10} = xtime({8e}) = {07},

{57} {13} = {57} ({01} {02} {10})

= {57} {ae} {07}

= {fe}.

## Polynomials with Coefficients in GF(28)

Four-term polynomials can be defined - with coefficients that are finite field elements - as:

3

a(x) a3 x

2

a2 x

a1 x a0

(4.5)

which will be denoted as a word in the form [

*a*0 ,*a*1 ,*a*2 ,*a*3 ]. Note that the polynomials in this section behave somewhat differently than the polynomials used in the definition of finite field elements, even though both types of polynomials use the same indeterminate,*x*. The coefficients in this section are themselves finite field elements, i.e., bytes, instead of bits; also, the multiplication of four-term polynomials uses a different reduction polynomial, defined below. The distinction should always be clear from the context.To illustrate the addition and multiplication operations, let

3

b(x) b3 x

2

b2 x

b1x b0

(4.6)

define a second four-term polynomial. Addition is performed by adding the finite field coefficients of like powers of

*x*. This addition corresponds to an XOR operation between the corresponding bytes in each of the words – in other words, the XOR of the complete word values.Thus, using the equations of (4.5) and (4.6),

3

a(x) b(x) (a3 b3 )x

2

(a2 b2 )x

(a1 b1 )x (a0 b0 )

(4.7)

Multiplication is achieved in two steps. In the first step, the polynomial product

*c*(*x*) =*a*(*x*) b(x) is algebraically expanded, and like powers are collected to give

where

6

c(x) c6 x

5

c5 x

4

c4 x

3

c3 x

2

c2 x

c1x c0

(4.8)

c0 a0 b0

c1 a1 b0 a0 b1

c2 a2 b0 a1 b1 a0 b2

c4 a3 b1 a2 b2 a1 b3 c5 a3 b2 a2 b3

c6 a3 b3

(4.9)

c3 a3 b0 a2 b1 a1 b2 a0 b3 .

The result,

*c*(*x*), does not represent a four-byte word. Therefore, the second step of the multiplication is to reduce*c*(*x*) modulo a polynomial of degree 4; the result can be reduced to a polynomial of degree less than 4.**For the AES algorithm, this is accomplished with the polynomial**4**x****+ 1**, so thatxi mod(x 4 1) xi mod 4 . (4.10)

The modular product of

*a*(*x*) and*b*(*x*), denoted by*a*(*x*) *b*(*x*), is given by the four-term polynomial*d*(*x*), defined as follows:with

3

d (x) d3 x

2

d2 x

d1 x

d0

(4.11)

d0 (a0 b0 ) (a3 b1 ) (a2 b2 ) (a1 b3 ) d1 (a1 b0 ) (a0 b1 ) (a3 b2 ) (a2 b3 ) d 2 (a2 b0 ) (a1 b1 ) (a0 b2 ) (a3 b3 ) d3 (a3 b0 ) (a2 b1 ) (a1 b2 ) (a0 b3 )

(4.12)

When

*a*(*x*) is a fixed polynomial, the operation defined in equation (4.11) can be written in matrix form as:d0

a0

a3 a2

a1 b0

d

a a

a a b

1 1 0

3 2 1

(4.13)

d 2

a2

a1 a0

a3 b2

d3

a3 a2

a1 a0 b3

Because

*x*4 1 is not an irreducible polynomial over GF(28), multiplication by a fixed four-termpolynomial is not necessarily invertible. However, the AES algorithm specifies a fixed four-term polynomial that

*does*have an inverse (see Sec. 5.1.3 and Sec. 5.3.3):*a*(*x*) = {03}*x*3 + {01}*x*2 + {01}*x*+ {02} (4.14)*a*-1(*x*) = {0b}*x*3 + {0d}*x*2 + {09}*x*+ {0e}. (4.15)Another polynomial used in the AES algorithm (see the RotWord() function in Sec. 5.2) has

*a*0=

*a*1 =*a*2 = {00} and*a*3 = {01}, which is the polynomial*x*3. Inspection of equation (4.13) above will show that its effect is to form the output word by rotating bytes in the input word. This means that [*b*0,*b*1,*b*2,*b*3] is transformed into [*b*1,*b*2,*b*3,*b*0].

# Algorithm Specification

For the AES algorithm

**, the length of the input block, the output block and the State is 128 bits.**This is represented by= 4, which reflects the number of 32-bit words (number of columns) in the State.**Nb**For the AES algorithm

**, the length of the Cipher Key,****K****, is 128, 192, or 256 bits.**The key length is represented by= 4, 6, or 8, which reflects the number of 32-bit words (number of columns) in the Cipher Key.**Nk**For the AES algorithm, the number of rounds to be performed during the execution of the algorithm is dependent on the key size. The number of rounds is represented by

, where**Nr**= 10 when**Nr**= 4,**Nk**= 12 when**Nr**= 6, and**Nk**= 14 when**Nr**= 8.**Nk**### The only Key-Block-Round combinations that conform to this standard are given in Fig. 4. For implementation issues relating to the key length, block size and number of rounds, see Sec. 6.3.

Key Length

(Nk words)

Block Size

(Nb words)

Number of Rounds

(Nr)

AES-128

4

4

10

AES-192

6

4

12

AES-256

8

4

14

#### Figure 4. Key-Block-Round Combinations.

For both its Cipher and Inverse Cipher, the AES algorithm uses a round function that is composed of four different byte-oriented transformations: 1) byte substitution using a substitution table (S-box), 2) shifting rows of the State array by different offsets, 3) mixing the data within each column of the State array, and 4) adding a Round Key to the State. These transformations (and their inverses) are described in Sec. 5.1.1-5.1.4 and 5.3.1-5.3.4.

The Cipher and Inverse Cipher are described in Sec. 5.1 and Sec. 5.3, respectively, while the Key Schedule is described in Sec. 5.2.

## Cipher

At the start of the Cipher, the input is copied to the State array using the conventions described in Sec. 3.4. After an initial Round Key addition, the State array is transformed by implementing a round function 10, 12, or 14 times (depending on the key length), with the final round differing

slightly from the first

**Nr**Sec. 3.4.

1 rounds. The final State is then copied to the output as described in

The round function is parameterized using a key schedule that consists of a one-dimensional array of four-byte words derived using the Key Expansion routine described in Sec. 5.2.

The Cipher is described in the pseudo code in Fig. 5. The individual transformations - SubBytes(), ShiftRows(), MixColumns(), and AddRoundKey() – process the State and are described in the following subsections. In Fig. 5, the array w[] contains the key schedule, which is described in Sec. 5.2.

As shown in Fig. 5, all

rounds are identical with the exception of the final round, which does not include the MixColumns() transformation.**Nr**Appendix B presents an example of the Cipher, showing values for the State array at the beginning of each round and after the application of each of the four transformations described in the following sections.

Cipher(byte in[4*Nb], byte out[4*Nb], word w[Nb*(Nr+1)])

begin

byte state[4,Nb]

state = in

AddRoundKey(state, w[0, Nb-1])

// See Sec. 5.1.4

for round = 1 step 1 to Nr–1

SubBytes(state) ShiftRows(state) MixColumns(state)

// See Sec. 5.1.1

// See Sec. 5.1.2

// See Sec. 5.1.3

AddRoundKey(state, w[round*Nb, (round+1)*Nb-1])

end for

SubBytes(state) ShiftRows(state)

AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb-1])

out = state end

#### Figure 5. Pseudo Code for the Cipher.1

SubBytes()Transformation

The SubBytes() transformation is a non-linear byte substitution that operates independently on each byte of the State using a substitution table (S-box). This S-box (Fig. 7), which is invertible, is constructed by composing two transformations:

Take the multiplicative inverse in the finite field GF(28), described in Sec. 4.2; the element {00} is mapped to itself.

Apply the following affine transformation (over GF(2) ):

b b b b b b c (5.1)

'

i i (i 4) mod 8 (i 5) mod 8 (i 6) mod 8 (i 7) mod 8 i

for

0 i 8 , where

*b*i is the*i*th bit of the byte, and*c*i is the*i*th bit of a byte*c*with thevalue {63} or {01100011}. Here and elsewhere, a prime on a variable (e.g., b ) indicates that the variable is to be updated with the value on the right.

In matrix form, the affine transformation element of the S-box can be expressed as:

1 The various transformations (e.g., SubBytes(), ShiftRows(), etc.) act upon the State array that is addressed by the ‘state’ pointer. AddRoundKey() uses an additional pointer to address the Round Key.

0

1

0

0

0

1

1

1

1 b0

1

'

1 1 1

0

0

0

1

1

1 b 1

1

1 1

2

1

0

0

0

1

1 b2 0

3 1 1

1

1

0

0

0

1 b 0

3

1 1

1

1

1

0

0

0 b4 0

5 0 1

1

1

1

1

0

0 b5 1

0 0

6

1

1

1

1

1

0 b 1

6

7 0 0

0

1

1

1

1

1 b7 0

b'

b

b'

'

b

. (5.2)

b'

4

'

b

b'

b'

Figure 6 illustrates the effect of the SubBytes() transformation on the State.

s0,0

s0,1

s0,2

s0,3

#### S-Box

s

'

0,0

s

'

0,1

s

'

0,2

s

'

0,3

s1,0

s1,1 s

sr ,c

1,2

s1,3

s

'

1,0

'

s

'

r ,c

'

'

s1,1

s

'

1,2

'

s

'

1,3

'

s2,0

s2,1

s2,2

s2,3

s2,0

s2,1

s2,2

s2,3

s3,0

s3,1

s3,2

s3,3

s

'

3,0

s

'

3,1

s

'

3,2

s

'

3,3

#### Figure 6. SubBytes() applies the S-box to each byte of the State.

The S-box used in the SubBytes() transformation is presented in hexadecimal form in Fig. 7.

For example, if

*s*1,1 {53}, then the substitution value would be determined by the intersectionof the row with index ‘5’ and the column with index ‘3’ in Fig. 7. This would result in a value of {ed}.

s1,1 having

y

0

1

2

3

4

5

6

7

8

9

a

b

c

d

e

f

x

0

63

7c

77

7b

f2

6b

6f

c5

30

01

67

2b

fe

d7

ab

76

1

ca

82

c9

7d

fa

59

47

f0

ad

d4

a2

af

9c

a4

72

c0

2

b7

fd

93

26

36

3f

f7

cc

34

a5

e5

f1

71

d8

31

15

3

04

c7

23

c3

18

96

05

9a

07

12

80

e2

eb

27

b2

75

4

09

83

2c

1a

1b

6e

5a

a0

52

3b

d6

b3

29

e3

2f

84

5

53

d1

00

ed

20

fc

b1

5b

6a

cb

be

39

4a

4c

58

cf

6

d0

ef

aa

fb

43

4d

33

85

45

f9

02

7f

50

3c

9f

a8

7

51

a3

40

8f

92

9d

38

f5

bc

b6

da

21

10

ff

f3

d2

8

cd

0c

13

ec

5f

97

44

17

c4

a7

7e

3d

64

5d

19

73

9

60

81

4f

dc

22

2a

90

88

46

ee

b8

14

de

5e

0b

db

a

e0

32

3a

0a

49

06

24

5c

c2

d3

ac

62

91

95

e4

79

b

e7

c8

37

6d

8d

d5

4e

a9

6c

56

f4

ea

65

7a

ae

08

c

ba

78

25

2e

1c

a6

b4

c6

e8

dd

74

1f

4b

bd

8b

8a

d

70

3e

b5

66

48

03

f6

0e

61

35

57

b9

86

c1

1d

9e

e

e1

f8

98

11

69

d9

8e

94

9b

1e

87

e9

ce

55

28

df

f

8c

a1

89

0d

bf

e6

42

68

41

99

2d

0f

b0

54

bb

16

#### Figure 7. S-box: substitution values for the byte xy (in hexadecimal format).

ShiftRows() Transformation

In the ShiftRows() transformation, the bytes in the last three rows of the State are cyclically shifted over different numbers of bytes (offsets). The first row,

*r*= 0, is not shifted.Specifically, the ShiftRows() transformation proceeds as follows:

s s

'

r ,c r ,(c shift (r , Nb)) mod Nb

for 0 <

*r*< 4 and 0 *c*<, (5.3)**Nb**where the shift value

*shift*(*r*,*Nb*) depends on the row number,*r*, as follows (recall that= 4):**Nb***shift*(1,4) 1 ;*shift*(2,4) 2 ;*shift*(3,4) 3 . (5.4)This has the effect of moving bytes to “lower” positions in the row (i.e., lower values of

*c*in a given row), while the “lowest” bytes wrap around into the “top” of the row (i.e., higher values of*c*in a given row).Figure 8 illustrates the ShiftRows() transformation.

ShiftRows()

sr ,0

sr ,1

sr ,2

sr ,3

s'

r ,0

s'

r ,1

s'

r ,2

s'

r ,3

S S ’

s0,0

s0,1

s0,2

s0,3

s1,0

s1,1

s1,2

s1,3

s2,0

s2,1

s2,2

s2,3

s3,0

s3,1

s3,2

s3,3

s0,0

s0,1

s0,2

s0,3

s1,1

s1,2

s1,3

s1,0

s2,2

s2,3

s2,0

s2,1

s3,3

s3,0

s3,1

s3,2

#### Figure 8. ShiftRows() cyclically shifts the last three rows in the State.

MixColumns() Transformation

The MixColumns() transformation operates on the State column-by-column, treating each column as a four-term polynomial as described in Sec. 4.3. The columns are considered as polynomials over GF(28) and multiplied modulo

*x*4 + 1 with a fixed polynomial*a*(*x*), given by*a*(*x*) = {03}*x*3 + {01}*x*2 + {01}*x*+ {02} . (5.5)As described in Sec. 4.3, this can be written as a matrix multiplication. Let

s(x) a(x) s(x) :

s

'

0,

*c*02 03

01 01 s0,c

'

s1,c 01 02 03

01 s1,c

for 0

*c*<. (5.6)**Nb**s

'

2,

*c*01

01 02

03 s2,c

'

s3,c

03

01 01

02 s3,c

As a result of this multiplication, the four bytes in a column are replaced by the following:

s0,c ({02}

*s*0,c ) ({03} s1,c )

s2,c

s3,c

s1,

*c**s*0,c ({02} *s*1,c ) ({03} s2,c )

s3,c

s2,c

s0,c

*s*1,c ({02} *s*2,c ) ({03} s3,c )

s3,

*c*({03}

s0,c )

s1,c

*s*2,c ({02} s3,c ).

Figure 9 illustrates the MixColumns() transformation.

MixColumns()

s0,0 s1,0 s2,0 s3,0

s0,1

s1,1

s2,1

s3,1

s0,

*c*s1,*c*s2,*c*s3,*c*s0,2

s1,2

s2,2

s3,2

s0,3 s1,3 s2,3 s3,3

s

'

0,0

s

'

1,0

s

'

2,0

s

'

3,0

'

s0,1

s

'

0,

*c*'

s1,1

s

'

1,

*c*'

s2,1

s

'

2,

*c*'

s3,1

s

'

3,

*c*s

'

0,2

s

'

1,2

s

'

2,2

s

'

3,2

s

'

0,3

s

'

1,3

s

'

2,3

s

'

3,3

#### Figure 9. MixColumns() operates on the State column-by-column.

AddRoundKey() Transformation

In the AddRoundKey() transformation, a Round Key is added to the State by a simple bitwise XOR operation. Each Round Key consists of

words from the key schedule (described in Sec. 5.2). Those**Nb**words are each added into the columns of the State, such that**Nb**[

*s*'0,c ,*s*'1,c ,*s*'2,c ,*s*'3,c ] [*s*0,c ,*s*1,c ,*s*2,c ,*s*3,c ] [*w*round Nbc ]for 0

*c*<, (5.7)**Nb**where [

*w*i] are the key schedule words described in Sec. 5.2, and*round*is a value in the range 0 *round*. In the Cipher, the initial Round Key addition occurs when**Nr***round*= 0, prior to the first application of the round function (see Fig. 5). The application of the AddRoundKey() transformation to therounds of the Cipher occurs when 1 **Nr***round*.**Nr**The action of this transformation is illustrated in Fig. 10, where

*l*=*round**. The byte address within words of the key schedule was described in Sec. 3.1.**Nb**s0,0

s1,0

s0,1 s0

s s

s0,c

,2

s1,c

1,1 1

s2,1 s2

,2

s0,3

s1,3

wl+c

wl +1 wl

l round * Nb

s0,0

s

'

1,0

'

s0,1 s

s' s

s0,c

s

'

1,

*c*0,2

'

1,1

'

s2,1 s

1,2

s0,3

s

'

1,3

s wl

2 wl 3

'

' s ' '

s2,0

s3,0

2,

*c*s3,1 s3

s3,c

,2 s2,3

,2 s3,3

s2,0 s3,0

'

s3,1 s

2,

*c*s3,c

2,2

3,2

s2,3 s3,3

#### Figure 10. AddRoundKey() XORs each column of the State with a word

#### from the key schedule.

## Key Expansion

The AES algorithm takes the Cipher Key,

, and performs a Key Expansion routine to generate a key schedule. The Key Expansion generates a total of**K**(**Nb**+ 1) words: the algorithm requires an initial set of**Nr**words, and each of the**Nb**rounds requires**Nr**words of key data. The resulting key schedule consists of a linear array of 4-byte words, denoted [**Nb***w*i ], with*i*in the range 0 *i*<(**Nb**+ 1).**Nr**The expansion of the input key into the key schedule proceeds according to the pseudo code in Fig. 11.

SubWord() is a function that takes a four-byte input word and applies the S-box (Sec. 5.1.1, Fig. 7) to each of the four bytes to produce an output word. The function RotWord() takes a word [

*a*0,*a*1,*a*2,*a*3] as input, performs a cyclic permutation, and returns the word [*a*1,*a*2,*a*3,*a*0]. The round constant word array, Rcon[i], contains the values given by [*x*i-1,{00},{00},{00}], with*x*i-1 being powers of*x*(*x*is denoted as {02}) in the field GF(28), as discussed in Sec. 4.2 (note that*i*starts at 1, not 0).From Fig. 11, it can be seen that the first Nk words of the expanded key are filled with the Cipher Key. Every following word, wi, is equal to the XOR of the previous word, wi-1, and the word Nk positions earlier, wi-Nk. For words in positions that are a multiple of Nk, a transformation is applied to wi-1 prior to the XOR, followed by an XOR with a round constant, Rcon[i]. This transformation consists of a cyclic shift of the bytes in a word (RotWord()), followed by the application of a table lookup to all four bytes of the word (SubWord()).

It is important to note that the Key Expansion routine for 256-bit Cipher Keys (

= 8) is slightly different than for 128- and 192-bit Cipher Keys. If**Nk**= 8 and i-4 is a multiple of**Nk**, then SubWord() is applied to wi-1 prior to the XOR.**Nk**KeyExpansion(byte key[4*Nk], word w[Nb*(Nr+1)], Nk) begin

word temp i = 0

while (i < Nk)

w[i] = word(key[4*i], key[4*i+1], key[4*i+2], key[4*i+3]) i = i+1

end while i = Nk

while (i < Nb * (Nr+1)] temp = w[i-1]

if (i mod Nk = 0)

temp = SubWord(RotWord(temp)) xor Rcon[i/Nk] else if (Nk > 6 and i mod Nk = 4)

temp = SubWord(temp) end if

w[i] = w[i-Nk] xor temp i = i + 1

end while end

Note that

*Nk*=4, 6, and 8 do not all have to be implemented; they are all included in the conditional statement above for conciseness. Specific implementation requirements for the Cipher Key are presented in Sec. 6.1.#### Figure 11. Pseudo Code for Key Expansion.2

Appendix A presents examples of the Key Expansion.

## Inverse Cipher

The Cipher transformations in Sec. 5.1 can be inverted and then implemented in reverse order to produce a straightforward Inverse Cipher for the AES algorithm. The individual transformations used in the Inverse Cipher - InvShiftRows(), InvSubBytes(),InvMixColumns(), and AddRoundKey() – process the State and are described in the following subsections.

The Inverse Cipher is described in the pseudo code in Fig. 12. In Fig. 12, the array w[] contains the key schedule, which was described previously in Sec. 5.2.

2 The functions SubWord() and RotWord() return a result that is a transformation of the function input, whereas the transformations in the Cipher and Inverse Cipher (e.g., ShiftRows(), SubBytes(), etc.) transform the State array that is addressed by the ‘state’ pointer.

InvCipher(byte in[4*Nb], byte out[4*Nb], word w[Nb*(Nr+1)])

begin

byte state[4,Nb] state = in

AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb-1]) // See Sec. 5.1.4

for round = Nr-1 step -1 downto 1

InvShiftRows(state) InvSubBytes(state)

// See Sec. 5.3.1

// See Sec. 5.3.2

AddRoundKey(state, w[round*Nb, (round+1)*Nb-1])

InvMixColumns(state) // See Sec. 5.3.3 end for

InvShiftRows(state) InvSubBytes(state) AddRoundKey(state, w[0, Nb-1])

out = state end

#### Figure 12. Pseudo Code for the Inverse Cipher.3

InvShiftRows() Transformation

InvShiftRows() is the inverse of the ShiftRows() transformation. The bytes in the last three rows of the State are cyclically shifted over different numbers of bytes (offsets). The first row,

*r*= 0, is not shifted. The bottom three rows are cyclically shifted by**Nb***shift*(*r*,*Nb*)bytes, where the shift value

*shift(r,Nb)*depends on the row number, and is given in equation (5.4) (see Sec. 5.1.2).Specifically, the InvShiftRows() transformation proceeds as follows:

s s

'

r ,(c shift ( r , Nb)) mod Nb r ,c

for 0 <

*r*< 4 and 0 *c*<(5.8)**Nb**Figure 13 illustrates the InvShiftRows() transformation.

3 The various transformations (e.g., InvSubBytes(), InvShiftRows(), etc.) act upon the State array that is addressed by the ‘state’ pointer. AddRoundKey() uses an additional pointer to address the Round Key.

InvShiftRows()

sr ,0

sr ,1

sr ,2

sr ,3

s'

r ,0

s'

r ,1

s'

r ,2

s'

r ,3

S S ’

s0,0

s0,1

s0,2

s0,3

s1,0

s1,1

s1,2

s1,3

s2,0

s2,1

s2,2

s2,3

s3,0

s3,1

s3,2

s3,3

s0,0

s0,1

s0,2

s0,3

s1,3

s1,0

s1,1

s1,2

s2,2

s2,3

s2,0

s2,1

s3,1

s3,2

s3,3

s3,0

#### Figure 13. InvShiftRows()cyclically shifts the last three rows in the State.

InvSubBytes() Transformation

InvSubBytes() is the inverse of the byte substitution transformation, in which the inverse S- box is applied to each byte of the State. This is obtained by applying the inverse of the affine transformation (5.1) followed by taking the multiplicative inverse in GF(28).

The inverse S-box used in the InvSubBytes() transformation is presented in Fig. 14:

y

0

1

2

3

4

5

6

7

8

9

a

b

c

d

e

f

x

0

52

09

6a

d5

30

36

a5

38

bf

40

a3

9e

81

f3

d7

fb

1

7c

e3

39

82

9b

2f

ff

87

34

8e

43

44

c4

de

e9

cb

2

54

7b

94

32

a6

c2

23

3d

ee

4c

95

0b

42

fa

c3

4e

3

08

2e

a1

66

28

d9

24

b2

76

5b

a2

49

6d

8b

d1

25

4

72

f8

f6

64

86

68

98

16

d4

a4

5c

cc

5d

65

b6

92

5

6c

70

48

50

fd

ed

b9

da

5e

15

46

57

a7

8d

9d

84

6

90

d8

ab

00

8c

bc

d3

0a

f7

e4

58

05

b8

b3

45

06

7

d0

2c

1e

8f

ca

3f

0f

02

c1

af

bd

03

01

13

8a

6b

8

3a

91

11

41

4f

67

dc

ea

97

f2

cf

ce

f0

b4

e6

73

9

96

ac

74

22

e7

ad

35

85

e2

f9

37

e8

1c

75

df

6e

a

47

f1

1a

71

1d

29

c5

89

6f

b7

62

0e

aa

18

be

1b

b

fc

56

3e

4b

c6

d2

79

20

9a

db

c0

fe

78

cd

5a

f4

c

1f

dd

a8

33

88

07

c7

31

b1

12

10

59

27

80

ec

5f

d

60

51

7f

a9

19

b5

4a

0d

2d

e5

7a

9f

93

c9

9c

ef

e

a0

e0

3b

4d

ae

2a

f5

b0

c8

eb

bb

3c

83

53

99

61

f

17

2b

04

7e

ba

77

d6

26

e1

69

14

63

55

21

0c

7d

#### Figure 14. Inverse S-box: substitution values for the byte xy (in hexadecimal format).

InvMixColumns() Transformation

InvMixColumns() is the inverse of the MixColumns() transformation. InvMixColumns() operates on the State column-by-column, treating each column as a four- term polynomial as described in Sec. 4.3. The columns are considered as polynomials over GF(28) and multiplied modulo

*x*4 + 1 with a fixed polynomial*a*-1(*x*), given by*a*-1(*x*) = {0b}*x*3 + {0d}*x*2 + {09}*x*+ {0e}. (5.9)As described in Sec. 4.3, this can be written as a matrix multiplication. Let

s(x) a 1 (x) s(x) :

s

'

0,

*c* 0e

0

*b*0*d*09 s0,c

'

s1,c 09 0e

0b 0d s1,c

for 0

*c*<. (5.10)**Nb**'

,

s2 c

0d

09 0

*e*0b s2,c

'

s3,c

0

*b*0*d*09 0e s3,c

As a result of this multiplication, the four bytes in a column are replaced by the following:

s0,c ({0e}

*s*0,c ) ({0b} *s*1,c ) ({0d} *s*2,c ) ({09} s3,c )

s1,

*c*({09}

*s*0,c ) ({0e} *s*1,c ) ({0b} *s*2,c ) ({0d} s3,c )

s2,c ({0d}

*s*0,c ) ({09} *s*1,c ) ({0e} *s*2,c ) ({0b} s3,c )

s3,

*c*({0b}

*s*0,c ) ({0d} *s*1,c ) ({09} *s*2,c ) ({0e} s3,c )

Inverse of the AddRoundKey() Transformation

AddRoundKey(), which was described in Sec. 5.1.4, is its own inverse, since it only involves an application of the XOR operation.

Equivalent Inverse Cipher

In the straightforward Inverse Cipher presented in Sec. 5.3 and Fig. 12, the sequence of the transformations differs from that of the Cipher, while the form of the key schedules for encryption and decryption remains the same. However, several properties of the AES algorithm allow for an Equivalent Inverse Cipher that has the same sequence of transformations as the Cipher (with the transformations replaced by their inverses). This is accomplished with a change in the key schedule.

The two properties that allow for this Equivalent Inverse Cipher are as follows:

The SubBytes() and ShiftRows() transformations commute; that is, a SubBytes() transformation immediately followed by a ShiftRows() transformation is equivalent to a ShiftRows() transformation immediately followed buy a SubBytes() transformation. The same is true for their inverses, InvSubBytes() and InvShiftRows.

The column mixing operations - MixColumns() and InvMixColumns() - are linear with respect to the column input, which means

InvMixColumns(state XOR Round Key) =

InvMixColumns(state) XOR InvMixColumns(Round Key).

These properties allow the order of InvSubBytes() and InvShiftRows() transformations to be reversed. The order of the AddRoundKey() and InvMixColumns() transformations can also be reversed, provided that the columns (words) of the decryption key schedule are modified using the InvMixColumns() transformation.

The equivalent inverse cipher is defined by reversing the order of the InvSubBytes() and InvShiftRows() transformations shown in Fig. 12, and by reversing the order of the AddRoundKey() and InvMixColumns() transformations used in the “round loop” after first modifying the decryption key schedule for

*round*= 1 to*Nr*-1 using the InvMixColumns() transformation. The first and last*Nb*words of the decryption key schedule shall*not*be modified in this manner.Given these changes, the resulting Equivalent Inverse Cipher offers a more efficient structure than the Inverse Cipher described in Sec. 5.3 and Fig. 12. Pseudo code for the Equivalent Inverse Cipher appears in Fig. 15. (The word array dw[] contains the modified decryption key schedule. The modification to the Key Expansion routine is also provided in Fig. 15.)

EqInvCipher(byte in[4*Nb], byte out[4*Nb], word dw[Nb*(Nr+1)])

begin

byte state[4,Nb] state = in

AddRoundKey(state, dw[Nr*Nb, (Nr+1)*Nb-1])

for round = Nr-1 step -1 downto 1 InvSubBytes(state) InvShiftRows(state) InvMixColumns(state)

AddRoundKey(state, dw[round*Nb, (round+1)*Nb-1]) end for

InvSubBytes(state) InvShiftRows(state) AddRoundKey(state, dw[0, Nb-1])

out = state end

For the Equivalent Inverse Cipher, the following pseudo

the end of the Key Expansion routine (Sec. 5.2):

for i = 0 step 1 to (Nr+1)*Nb-1 dw[i] = w[i]

end for

code

is added at

for round = 1 step 1 to Nr-1

InvMixColumns(dw[round*Nb, (round+1)*Nb-1])

type

end for

// note change of

Note that, since InvMixColumns operates on a two-dimensional array of bytes

while the Round Keys are held in an array of words, the call to InvMixColumns in this code sequence involves a change of type (i.e. the input to InvMixColumns() is normally the State array, which is considered to be a two-dimensional array of bytes, whereas the input here is a Round Key computed as a one-dimensional array of words).

#### Figure 15. Pseudo Code for the Equivalent Inverse Cipher.

# Implementation Issues

## Key Length Requirements

An implementation of the AES algorithm shall support

*at least one*of the three key lengths specified in Sec. 5: 128, 192, or 256 bits (i.e.,= 4, 6, or 8, respectively). Implementations**Nk**may optionally support two or three key lengths, which may promote the interoperability of algorithm implementations.

## Keying Restrictions

No weak or semi-weak keys have been identified for the AES algorithm, and there is no restriction on key selection.

## Parameterization of Key Length, Block Size, and Round Number

This standard explicitly defines the allowed values for the key length (

), block size (**Nk**), and number of rounds (**Nb**) – see Fig. 4. However, future reaffirmations of this standard could include changes or additions to the allowed values for those parameters. Therefore, implementers may choose to design their AES implementations with future flexibility in mind.**Nr**## Implementation Suggestions Regarding Various Platforms

Implementation variations are possible that may, in many cases, offer performance or other advantages. Given the same input key and data (plaintext or ciphertext), any implementation that produces the same output (ciphertext or plaintext) as the algorithm specified in this standard is an acceptable implementation of the AES.

Reference [3] and other papers located at Ref. [1] include suggestions on how to efficiently implement the AES algorithm on a variety of platforms.

This appendix shows the development of the key schedule for various key sizes. Note that multi- byte values are presented using the notation described in Sec. 3. The intermediate values produced during the development of the key schedule (see Sec. 5.2) are given in the following table (all values are in hexadecimal format, with the exception of the index column (i)).

## Expansion of a 128-bit Cipher Key

This section contains the key expansion of the following cipher key:

Cipher Key = 2b 7e 15 16 28 ae d2 a6 ab f7 15 88 09 cf 4f 3c

for

= 4, which results in**Nk**w0 = 2b7e1516 w1 = 28aed2a6 w2 = abf71588 w3 = 09cf4f3c

i

(dec)

temp

After

RotWord()

After

SubWord()

Rcon[i/Nk]

After XOR

with Rcon

w[i–Nk]

w[i]= temp XOR w[i-Nk]

4

09cf4f3c

cf4f3c09

8a84eb01

01000000

8b84eb01

2b7e1516

a0fafe17

5

a0fafe17

28aed2a6

88542cb1

6

88542cb1

abf71588

23a33939

7

23a33939

09cf4f3c

2a6c7605

8

2a6c7605

6c76052a

50386be5

02000000

52386be5

a0fafe17

f2c295f2

9

f2c295f2

88542cb1

7a96b943

10

7a96b943

23a33939

5935807a

11

5935807a

2a6c7605

7359f67f

12

7359f67f

59f67f73

cb42d28f

04000000

cf42d28f

f2c295f2

3d80477d

13

3d80477d

7a96b943

4716fe3e

14

4716fe3e

5935807a

1e237e44

15

1e237e44

7359f67f

6d7a883b

16

6d7a883b

7a883b6d

dac4e23c

08000000

d2c4e23c

3d80477d

ef44a541

17

ef44a541

4716fe3e

a8525b7f

18

a8525b7f

1e237e44

b671253b

19

b671253b

6d7a883b

db0bad00

20

db0bad00

0bad00db

2b9563b9

10000000

3b9563b9

ef44a541

d4d1c6f8

21

d4d1c6f8

a8525b7f

7c839d87

22

7c839d87

b671253b

caf2b8bc

23

caf2b8bc

db0bad00

11f915bc

24

11f915bc

f915bc11

99596582

20000000

b9596582

d4d1c6f8

6d88a37a

25

6d88a37a

7c839d87

110b3efd

26

110b3efd

caf2b8bc

dbf98641

27

dbf98641

11f915bc

ca0093fd

28

ca0093fd

0093fdca

63dc5474

40000000

23dc5474

6d88a37a

4e54f70e

29

4e54f70e

110b3efd

5f5fc9f3

30

5f5fc9f3

dbf98641

84a64fb2

31

84a64fb2

ca0093fd

4ea6dc4f

32

4ea6dc4f

a6dc4f4e

2486842f

80000000

a486842f

4e54f70e

ead27321

33

ead27321

5f5fc9f3

b58dbad2

34

b58dbad2

84a64fb2

312bf560

35

312bf560

4ea6dc4f

7f8d292f

36

7f8d292f

8d292f7f

5da515d2

1b000000

46a515d2

ead27321

ac7766f3

37

ac7766f3

b58dbad2

19fadc21

38

19fadc21

312bf560

28d12941

39

28d12941

7f8d292f

575c006e

40

575c006e

5c006e57

4a639f5b

36000000

7c639f5b

ac7766f3

d014f9a8

41

d014f9a8

19fadc21

c9ee2589

42

c9ee2589

28d12941

e13f0cc8

43

e13f0cc8

575c006e

b6630ca6

## Expansion of a 192-bit Cipher Key

This section contains the key expansion of the following cipher key:

Cipher Key = 8e 73 b0 f7 da 0e 64 52 c8 10 f3 2b

80 90 79 e5 62 f8 ea d2 52 2c 6b 7b

for

= 6, which results in**Nk**w0 = 8e73b0f7 w1 = da0e6452 w2 = c810f32b w3 = 809079e5

w4 = 62f8ead2 w5 = 522c6b7b

i

(dec)

temp

After

RotWord()

After

SubWord()

Rcon[i/Nk]

After XOR

with Rcon

w[i–Nk]

w[i]= temp XOR w[i-Nk]

6

522c6b7b

2c6b7b52

717f2100

01000000

707f2100

8e73b0f7

fe0c91f7

7

fe0c91f7

da0e6452

2402f5a5

8

2402f5a5

c810f32b

ec12068e

9

ec12068e

809079e5

6c827f6b

10

6c827f6b

62f8ead2

0e7a95b9

11

0e7a95b9

522c6b7b

5c56fec2

12

5c56fec2

56fec25c

b1bb254a

02000000

b3bb254a

fe0c91f7

4db7b4bd

13

4db7b4bd

2402f5a5

69b54118

14

69b54118

ec12068e

85a74796

15

85a74796

6c827f6b

e92538fd

16

e92538fd

0e7a95b9

e75fad44

17

e75fad44

5c56fec2

bb095386

18

bb095386

095386bb

01ed44ea

04000000

05ed44ea

4db7b4bd

485af057

19

485af057

69b54118

21efb14f

20

21efb14f

85a74796

a448f6d9

21

a448f6d9

e92538fd

4d6dce24

22

4d6dce24

e75fad44

aa326360

23

aa326360

bb095386

113b30e6

24

113b30e6

3b30e611

e2048e82

08000000

ea048e82

485af057

a25e7ed5

25

a25e7ed5

21efb14f

83b1cf9a

26

83b1cf9a

a448f6d9

27f93943

27

27f93943

4d6dce24

6a94f767

28

6a94f767

aa326360

c0a69407

29

c0a69407

113b30e6

d19da4e1

30

d19da4e1

9da4e1d1

5e49f83e

10000000

4e49f83e

a25e7ed5

ec1786eb

31

ec1786eb

83b1cf9a

6fa64971

32

6fa64971

27f93943

485f7032

33

485f7032

6a94f767

22cb8755

34

22cb8755

c0a69407

e26d1352

35

e26d1352

d19da4e1

33f0b7b3

36

33f0b7b3

f0b7b333

8ca96dc3

20000000

aca96dc3

ec1786eb

40beeb28

37

40beeb28

6fa64971

2f18a259

38

2f18a259

485f7032

6747d26b

39

6747d26b

22cb8755

458c553e

40

458c553e

e26d1352

a7e1466c

41

a7e1466c

33f0b7b3

9411f1df

42

9411f1df

11f1df94

82a19e22

40000000

c2a19e22

40beeb28

821f750a

43

821f750a

2f18a259

ad07d753

44

ad07d753

6747d26b

ca400538

45

ca400538

458c553e

8fcc5006

46

8fcc5006

a7e1466c

282d166a

47

282d166a

9411f1df

bc3ce7b5

48

bc3ce7b5

3ce7b5bc

eb94d565

80000000

6b94d565

821f750a

e98ba06f

49

e98ba06f

ad07d753

448c773c

50

448c773c

ca400538

8ecc7204

51

8ecc7204

8fcc5006

01002202

## Expansion of a 256-bit Cipher Key

This section contains the key expansion of the following cipher key:

Cipher Key = 60 3d eb 10 15 ca 71 be 2b 73 ae f0 85 7d 77 81

1f 35 2c 07 3b 61 08 d7 2d 98 10 a3 09 14 df f4

for * Nk *= 8, which results in

w0 = 603deb10 w1 = 15ca71be w2 = 2b73aef0 w3 = 857d7781

w4 = 1f352c07 w5 = 3b6108d7 w6 = 2d9810a3 w7 = 0914dff4

i (dec) | temp | After RotWord() | After SubWord() | Rcon[i/Nk] | After XOR with Rcon | w[i–Nk] | w[i]= temp XOR w[i-Nk] |

8 | 0914dff4 | 14dff409 | fa9ebf01 | 01000000 | fb9ebf01 | 603deb10 | 9ba35411 |

9 | 9ba35411 | 15ca71be | 8e6925af | ||||

10 | 8e6925af | 2b73aef0 | a51a8b5f | ||||

11 | a51a8b5f | 857d7781 | 2067fcde | ||||

12 | 2067fcde | b785b01d | 1f352c07 | a8b09c1a | |||

13 | a8b09c1a | 3b6108d7 | 93d194cd | ||||

14 | 93d194cd | 2d9810a3 | be49846e | ||||

15 | be49846e | 0914dff4 | b75d5b9a | ||||

16 | b75d5b9a | 5d5b9ab7 | 4c39b8a9 | 02000000 | 4e39b8a9 | 9ba35411 | d59aecb8 |

17 | d59aecb8 | 8e6925af | 5bf3c917 | ||||

18 | 5bf3c917 | a51a8b5f | fee94248 | ||||

19 | fee94248 | 2067fcde | de8ebe96 | ||||

20 | de8ebe96 | 1d19ae90 | a8b09c1a | b5a9328a | |||

21 | b5a9328a | 93d194cd | 2678a647 | ||||

22 | 2678a647 | be49846e | 98312229 |

23 | 98312229 | b75d5b9a | 2f6c79b3 | ||||

24 | 2f6c79b3 | 6c79b32f | 50b66d15 | 04000000 | 54b66d15 | d59aecb8 | 812c81ad |

25 | 812c81ad | 5bf3c917 | dadf48ba | ||||

26 | dadf48ba | fee94248 | 24360af2 | ||||

27 | 24360af2 | de8ebe96 | fab8b464 | ||||

28 | fab8b464 | 2d6c8d43 | b5a9328a | 98c5bfc9 | |||

29 | 98c5bfc9 | 2678a647 | bebd198e | ||||

30 | bebd198e | 98312229 | 268c3ba7 | ||||

31 | 268c3ba7 | 2f6c79b3 | 09e04214 | ||||

32 | 09e04214 | e0421409 | e12cfa01 | 08000000 | e92cfa01 | 812c81ad | 68007bac |

33 | 68007bac | dadf48ba | b2df3316 | ||||

34 | b2df3316 | 24360af2 | 96e939e4 | ||||

35 | 96e939e4 | fab8b464 | 6c518d80 | ||||

36 | 6c518d80 | 50d15dcd | 98c5bfc9 | c814e204 | |||

37 | c814e204 | bebd198e | 76a9fb8a | ||||

38 | 76a9fb8a | 268c3ba7 | 5025c02d | ||||

39 | 5025c02d | 09e04214 | 59c58239 | ||||

40 | 59c58239 | c5823959 | a61312cb | 10000000 | b61312cb | 68007bac | de136967 |

41 | de136967 | b2df3316 | 6ccc5a71 | ||||

42 | 6ccc5a71 | 96e939e4 | fa256395 | ||||

43 | fa256395 | 6c518d80 | 9674ee15 | ||||

44 | 9674ee15 | 90922859 | c814e204 | 5886ca5d | |||

45 | 5886ca5d | 76a9fb8a | 2e2f31d7 | ||||

46 | 2e2f31d7 | 5025c02d | 7e0af1fa | ||||

47 | 7e0af1fa | 59c58239 | 27cf73c3 | ||||

48 | 27cf73c3 | cf73c327 | 8a8f2ecc | 20000000 | aa8f2ecc | de136967 | 749c47ab |

49 | 749c47ab | 6ccc5a71 | 18501dda | ||||

50 | 18501dda | fa256395 | e2757e4f | ||||

51 | e2757e4f | 9674ee15 | 7401905a | ||||

52 | 7401905a | 927c60be | 5886ca5d | cafaaae3 | |||

53 | cafaaae3 | 2e2f31d7 | e4d59b34 | ||||

54 | e4d59b34 | 7e0af1fa | 9adf6ace | ||||

55 | 9adf6ace | 27cf73c3 | bd10190d | ||||

56 | bd10190d | 10190dbd | cad4d77a | 40000000 | 8ad4d77a | 749c47ab | fe4890d1 |

57 | fe4890d1 | 18501dda | e6188d0b |

58 | e6188d0b | e2757e4f | 046df344 | ||||

59 | 046df344 | 7401905a | 706c631e |

The following diagram shows the values in the State array as the Cipher progresses for a block length and a Cipher Key length of 16 bytes each (i.e., *Nb *= 4 and *Nk *= 4).

Input = 32 43 f6 a8 88 5a 30 8d 31 31 98 a2 e0 37 07 34

Cipher Key = 2b 7e 15 16 28 ae d2 a6 ab f7 15 88 09 cf 4f 3c

The Round Key values are taken from the Key Expansion example in Appendix A.

Round | Start of | After | After | After | Round Key |

Number | Round | SubBytes | ShiftRows | MixColumns | Value |

32 | 88 | 31 | e0 |

43 | 5a | 31 | 37 |

f6 | 30 | 98 | 07 |

a8 | 8d | a2 | 34 |

2b | 28 | ab | 09 |

7e | ae | f7 | cf |

15 | d2 | 15 | 4f |

16 | a6 | 88 | 3c |

input =

19 | a0 | 9a | e9 |

3d | f4 | c6 | f8 |

e3 | e2 | 8d | 48 |

be | 2b | 2a | 08 |

d4 | e0 | b8 | 1e |

27 | bf | b4 | 41 |

11 | 98 | 5d | 52 |

ae | f1 | e5 | 30 |

d4 | e0 | b8 | 1e |

bf | b4 | 41 | 27 |

5d | 52 | 11 | 98 |

30 | ae | f1 | e5 |

04 | e0 | 48 | 28 |

66 | cb | f8 | 06 |

81 | 19 | d3 | 26 |

e5 | 9a | 7a | 4c |

a0 | 88 | 23 | 2a |

fa | 54 | a3 | 6c |

fe | 2c | 39 | 76 |

17 | b1 | 39 | 05 |

1 =

a4 | 68 | 6b | 02 |

9c | 9f | 5b | 6a |

7f | 35 | ea | 50 |

f2 | 2b | 43 | 49 |

49 | 45 | 7f | 77 |

de | db | 39 | 02 |

d2 | 96 | 87 | 53 |

89 | f1 | 1a | 3b |

49 | 45 | 7f | 77 |

db | 39 | 02 | de |

87 | 53 | d2 | 96 |

3b | 89 | f1 | 1a |

58 | 1b | db | 1b |

4d | 4b | e7 | 6b |

ca | 5a | ca | b0 |

f1 | ac | a8 | e5 |

f2 | 7a | 59 | 73 |

c2 | 96 | 35 | 59 |

95 | b9 | 80 | f6 |

f2 | 43 | 7a | 7f |

2 =

aa | 61 | 82 | 68 |

8f | dd | d2 | 32 |

5f | e3 | 4a | 46 |

03 | ef | d2 | 9a |

ac | ef | 13 | 45 |

73 | c1 | b5 | 23 |

cf | 11 | d6 | 5a |

7b | df | b5 | b8 |

ac | ef | 13 | 45 |

c1 | b5 | 23 | 73 |

d6 | 5a | cf | 11 |

b8 | 7b | df | b5 |

75 | 20 | 53 | bb |

ec | 0b | c0 | 25 |

09 | 63 | cf | d0 |

93 | 33 | 7c | dc |

3d | 47 | 1e | 6d |

80 | 16 | 23 | 7a |

47 | fe | 7e | 88 |

7d | 3e | 44 | 3b |

3 =

48 | 67 | 4d | d6 |

6c | 1d | e3 | 5f |

4e | 9d | b1 | 58 |

ee | 0d | 38 | e7 |

52 | 85 | e3 | f6 |

50 | a4 | 11 | cf |

2f | 5e | c8 | 6a |

28 | d7 | 07 | 94 |

52 | 85 | e3 | f6 |

a4 | 11 | cf | 50 |

c8 | 6a | 2f | 5e |

94 | 28 | d7 | 07 |

0f | 60 | 6f | 5e |

d6 | 31 | c0 | b3 |

da | 38 | 10 | 13 |

a9 | bf | 6b | 01 |

ef | a8 | b6 | db |

44 | 52 | 71 | 0b |

a5 | 5b | 25 | ad |

41 | 7f | 3b | 00 |

4 =

e0 | c8 | d9 | 85 |

92 | 63 | b1 | b8 |

7f | 63 | 35 | be |

e8 | c0 | 50 | 01 |

e1 | e8 | 35 | 97 |

4f | fb | c8 | 6c |

d2 | fb | 96 | ae |

9b | ba | 53 | 7c |

e1 | e8 | 35 | 97 |

fb | c8 | 6c | 4f |

96 | ae | d2 | fb |

7c | 9b | ba | 53 |

25 | bd | b6 | 4c |

d1 | 11 | 3a | 4c |

a9 | d1 | 33 | c0 |

ad | 68 | 8e | b0 |

d4 | 7c | ca | 11 |

d1 | 83 | f2 | f9 |

c6 | 9d | b8 | 15 |

f8 | 87 | bc | bc |

5 =

f1 | c1 | 7c | 5d |

00 | 92 | c8 | b5 |

6f | 4c | 8b | d5 |

55 | ef | 32 | 0c |

a1 | 78 | 10 | 4c |

63 | 4f | e8 | d5 |

a8 | 29 | 3d | 03 |

fc | df | 23 | fe |

a1 | 78 | 10 | 4c |

4f | e8 | d5 | 63 |

3d | 03 | a8 | 29 |

fe | fc | df | 23 |

4b | 2c | 33 | 37 |

86 | 4a | 9d | d2 |

8d | 89 | f4 | 18 |

6d | 80 | e8 | d8 |

6d | 11 | db | ca |

88 | 0b | f9 | 00 |

a3 | 3e | 86 | 93 |

7a | fd | 41 | fd |

6 =

26 | 3d | e8 | fd |

0e | 41 | 64 | d2 |

2e | b7 | 72 | 8b |

17 | 7d | a9 | 25 |

f7 | 27 | 9b | 54 |

ab | 83 | 43 | b5 |

31 | a9 | 40 | 3d |

f0 | ff | d3 | 3f |

f7 | 27 | 9b | 54 |

83 | 43 | b5 | ab |

40 | 3d | 31 | a9 |

3f | f0 | ff | d3 |

14 | 46 | 27 | 34 |

15 | 16 | 46 | 2a |

b5 | 15 | 56 | d8 |

bf | ec | d7 | 43 |

4e | 5f | 84 | 4e |

54 | 5f | a6 | a6 |

f7 | c9 | 4f | dc |

0e | f3 | b2 | 4f |

7 =

5a | 19 | a3 | 7a |

41 | 49 | e0 | 8c |

42 | dc | 19 | 04 |

b1 | 1f | 65 | 0c |

be | d4 | 0a | da |

83 | 3b | e1 | 64 |

2c | 86 | d4 | f2 |

c8 | c0 | 4d | fe |