Protocol++® (Protocolpp®)  v5.6.2
jconfidentsa Class Reference

#include "include/jconfidentsa.h"

Detailed Description

See also
jmodes for encryption modes of operation

Data Encryption Standard (DES)

Description

For brevity, the following description omits the exact transformations and permutations which specify the algorithm; for reference, the details can be found in DES supplementary material

DES is the archetypal block cipher—an algorithm that takes a fixed-length string of plaintext bits and transforms it through a series of complicated operations into another ciphertext bitstring of the same length. In the case of DES, the block size is 64 bits. DES also uses a key to customize the transformation, so that decryption can supposedly only be performed by those who know the particular key used to encrypt. The key ostensibly consists of 64 bits; however, only 56 of these are actually used by the algorithm. Eight bits are used solely for checking parity, and are thereafter discarded. Hence the effective key length is 56 bits

The key is nominally stored or transmitted as 8 bytes, each with odd parity. According to ANSI X3.92-1981 (Now, known as ANSI INCITS 92-1981), section 3.5:

One bit in each 8-bit byte of the KEY may be utilized for error detection in key generation, distribution, and storage. Bits 8, 16,..., 64 are for use in ensuring that each byte is of odd parity

Like other block ciphers, DES by itself is not a secure means of encryption, but must instead be used in a mode of operation. FIPS-81 specifies several modes for use with DES. Further comments on the usage of DES are contained in FIPS-74. Decryption uses the same structure as encryption, but with the keys used in reverse order. (This has the advantage that the same hardware or software can be used in both directions

Overall structure

The algorithm's overall structure is shown in the figure below: there are 16 identical stages of processing, termed rounds. There is also an initial and final permutation, termed IP and FP, which are inverses (IP "undoes" the action of FP, and vice versa). IP and FP have no cryptographic significance, but were included in order to facilitate loading blocks in and out of mid-1970s 8-bit based hardware

Before the main rounds, the block is divided into two 32-bit halves and processed alternately; this criss-crossing is known as the Feistel scheme. The Feistel structure ensures that decryption and encryption are very similar processes—the only difference is that the subkeys are applied in the reverse order when decrypting. The rest of the algorithm is identical. This greatly simplifies implementation, particularly in hardware, as there is no need for separate encryption and decryption algorithms

The $ \oplus $ symbol denotes the exclusive-OR (XOR) operation. The F-function scrambles half a block together with some of the key. The output from the F-function is then combined with the other half of the block, and the halves are swapped before the next round. After the final round, the halves are swapped; this is a feature of the Feistel structure which makes encryption and decryption similar processes

The Feistel (F) function

The F-function, depicted in the figure below, operates on half a block (32 bits) at a time and consists of four stages:

  1. Expansion: the 32-bit half-block is expanded to 48 bits using the expansion permutation, denoted E in the diagram, by duplicating half of the bits. The output consists of eight 6-bit (8 * 6 = 48 bits) pieces, each containing a copy of 4 corresponding input bits, plus a copy of the immediately adjacent bit from each of the input pieces to either side
  2. Key mixing: the result is combined with a subkey using an XOR operation. Sixteen 48-bit subkeys—one for each round—are derived from the main key using the key schedule (described below)
  3. Substitution: after mixing in the subkey, the block is divided into eight 6-bit pieces before processing by the S-boxes, or substitution boxes. Each of the eight S-boxes replaces its six input bits with four output bits according to a non-linear transformation, provided in the form of a lookup table. The S-boxes provide the core of the security of DES—without them, the cipher would be linear, and trivially breakable
  4. Permutation: finally, the 32 outputs from the S-boxes are rearranged according to a fixed permutation, the P-box. This is designed so that, after permutation, the bits from the output of each S-box in this round are spread across four different S-boxes in the next round

The alternation of substitution from the S-boxes, and permutation of bits from the P-box and E-expansion provides so-called "confusion and diffusion" respectively, a concept identified by Claude Shannon in the 1940s as a necessary condition for a secure yet practical cipher

Key schedule

The figure below illustrates the key schedule for encryption—the algorithm which generates the subkeys. Initially, 56 bits of the key are selected from the initial 64 by Permuted Choice 1 (PC-1)—the remaining eight bits are either discarded or used as parity check bits. The 56 bits are then divided into two 28-bit halves; each half is thereafter treated separately. In successive rounds, both halves are rotated left by one or two bits (specified for each round), and then 48 subkey bits are selected by Permuted Choice 2 (PC-2)—24 bits from the left half, and 24 from the right. The rotations (denoted by "<<<" in the diagram) mean that a different set of bits is used in each subkey; each bit is used in approximately 14 out of the 16 subkeys

The key schedule for decryption is similar—the subkeys are in reverse order compared to encryption. Apart from that change, the process is the same as for encryption. The same 28 bits are passed to all rotation boxes

Security and cryptanalysis

Although more information has been published on the cryptanalysis of DES than any other block cipher, the most practical attack to date is still a brute-force approach. Various minor cryptanalytic properties are known, and three theoretical attacks are possible which, while having a theoretical complexity less than a brute-force attack, require an unrealistic number of known or chosen plaintexts to carry out, and are not a concern in practice

Brute-force attack

For any cipher, the most basic method of attack is brute force—trying every possible key in turn. The length of the key determines the number of possible keys, and hence the feasibility of this approach. For DES, questions were raised about the adequacy of its key size early on, even before it was adopted as a standard, and it was the small key size, rather than theoretical cryptanalysis, which dictated a need for a replacement algorithm. As a result of discussions involving external consultants including the NSA, the key size was reduced from 128 bits to 56 bits to fit on a single chip

The EFF's US$250,000 DES cracking machine contained 1,856 custom chips and could brute-force a DES key in a matter of days—the photo shows a DES Cracker circuit board fitted with several Deep Crack chips

In academia, various proposals for a DES-cracking machine were advanced. In 1977, Diffie and Hellman proposed a machine costing an estimated US$20 million which could find a DES key in a single day.[1][31] By 1993, Wiener had proposed a key-search machine costing US$1 million which would find a key within 7 hours. However, none of these early proposals were ever implemented—or, at least, no implementations were publicly acknowledged. The vulnerability of DES was practically demonstrated in the late 1990s. In 1997, RSA Security sponsored a series of contests, offering a $10,000 prize to the first team that broke a message encrypted with DES for the contest. That contest was won by the DESCHALL Project, led by Rocke Verser, Matt Curtin, and Justin Dolske, using idle cycles of thousands of computers across the Internet. The feasibility of cracking DES quickly was demonstrated in 1998 when a custom DES-cracker was built by the Electronic Frontier Foundation (EFF), a cyberspace civil rights group, at the cost of approximately US$250,000 (see EFF DES cracker). Their motivation was to show that DES was breakable in practice as well as in theory: "There are many people who will not believe a truth until they can see it with their own eyes. Showing them a physical machine that can crack DES in a few days is the only way to convince some people that they really cannot trust their security to DES." The machine brute-forced a key in a little more than 2 days' worth of searching

The next confirmed DES cracker was the COPACOBANA machine built in 2006 by teams of the Universities of Bochum and Kiel, both in Germany. Unlike the EFF machine, COPACOBANA consists of commercially available, reconfigurable integrated circuits. 120 of these field-programmable gate arrays (FPGAs) of type XILINX Spartan-3 1000 run in parallel. They are grouped in 20 DIMM modules, each containing 6 FPGAs. The use of reconfigurable hardware makes the machine applicable to other code breaking tasks as well. One of the more interesting aspects of COPACOBANA is its cost factor. One machine can be built for approximately $10,000. The cost decrease by roughly a factor of 25 over the EFF machine is an example of the continuous improvement of digital hardware—see Moore's law. Adjusting for inflation over 8 years yields an even higher improvement of about 30x. Since 2007, SciEngines GmbH, a spin-off company of the two project partners of COPACOBANA has enhanced and developed successors of COPACOBANA. In 2008 their COPACOBANA RIVYERA reduced the time to break DES to less than one day, using 128 Spartan-3 5000's. SciEngines RIVYERA held the record in brute-force breaking DES, having utilized 128 Spartan-3 5000 FPGAs. Their 256 Spartan-6 LX150 model has further lowered this time

In 2012, David Hulton and Moxie Marlinspike announced a system with 48 Xilinx Virtex-6 LX240T FPGAs, each FPGA containing 40 fully pipelined DES cores running at 400 MHz, for a total capacity of 768 gigakeys/sec. The system can exhaustively search the entire 56-bit DES key space in about 26 hours and this service is offered for a fee online

The Advanced Encryption Standard (AES)

Also known by its original name Rijndael is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001. AES is a subset of the Rijndael block cipher developed by two Belgian cryptographers, Vincent Rijmen and Joan Daemen, who submitted a proposal to NIST during the AES selection process. Rijndael is a family of ciphers with different key and block sizes

For AES, NIST selected three members of the Rijndael family, each with a block size of 128 bits, but three different key lengths: 128, 192 and 256 bits

AES has been adopted by the U.S. government and is now used worldwide. It supersedes the Data Encryption Standard (DES), which was published in 1977. The algorithm described by AES is a symmetric-key algorithm, meaning the same key is used for both encrypting and decrypting the data

In the United States, AES was announced by the NIST as U.S. FIPS PUB 197 (FIPS 197) on November 26,

  1. This announcement followed a five-year standardization process in which fifteen competing designs were presented and evaluated, before the Rijndael cipher was selected as the most suitable (see Advanced Encryption Standard process for more details)

AES became effective as a federal government standard on May 26, 2002, after approval by the Secretary of Commerce. AES is included in the ISO/IEC 18033-3 standard. AES is available in many different encryption packages, and is the first (and only) publicly accessible cipher approved by the National Security Agency (NSA) for top secret information when used in an NSA approved cryptographic module (see Security of AES, below)

Description of the ciphers

AES is based on a design principle known as a substitution–permutation network, and is efficient in both software and hardware. Unlike its predecessor DES, AES does not use a Feistel network. AES is a variant of Rijndael which has a fixed block size of 128 bits, and a key size of 128, 192, or 256 bits. By contrast, Rijndael per se is specified with block and key sizes that may be any multiple of 32 bits, with a minimum of 128 and a maximum of 256 bits

AES operates on a 4 × 4 column-major order array of bytes, termed the state. Most AES calculations are done in a particular finite field

For instance, if there are 16 bytes, $ b_0, b_1,...,b_15,$ these bytes are represented as this two-dimensional array:

The key size used for an AES cipher specifies the number of transformation rounds that convert the input, called the plaintext, into the final output, called the ciphertext. The number of rounds are as follows:

  • 10 rounds for 128-bit keys
  • 12 rounds for 192-bit keys
  • 14 rounds for 256-bit keys

Each round consists of several processing steps, including one that depends on the encryption key itself. A set of reverse rounds are applied to transform ciphertext back into the original plaintext using the same encryption key

High-level description of the algorithm

  1. KeyExpansion—round keys are derived from the cipher key using Rijndael's key schedule. AES requires a separate 128-bit round key block for each round plus one more
  2. Initial round key addition:
    1. AddRoundKey—each byte of the state is combined with a block of the round key using bitwise xor
  3. 9, 11 or 13 rounds:
    1. SubBytes—a non-linear substitution step where each byte is replaced with another according to a lookup table
    2. ShiftRows—a transposition step where the last three rows of the state are shifted cyclically a certain number of steps
    3. MixColumns—a linear mixing operation which operates on the columns of the state, combining the four bytes in each column.
    4. AddRoundKey
  4. Final round (making 10, 12 or 14 rounds in total):
    1. SubBytes
    2. ShiftRows
    3. AddRoundKey

The SubBytes step

In the SubBytes step, each byte $ a_i,j $ in the state array is replaced with a SubByte $ S(a_i,j) $ using an 8-bit substitution box. This operation provides the non-linearity in the cipher. The S-box used is derived from the multiplicative inverse over GF(28), known to have good non-linearity properties. To avoid attacks based on simple algebraic properties, the S-box is constructed by combining the inverse function with an invertible affine transformation. The S-box is also chosen to avoid any fixed points (and so is a derangement), i.e., $ S(a_i,j) ≠ a_i,j $, and also any opposite fixed points, i.e., $ S(a_i,j) \oplus a_i,j ≠ 0xFF_16 $. While performing the decryption, the InvSubBytes step (the inverse of SubBytes) is used, which requires first taking the inverse of the affine transformation and then finding the multiplicative inverse

The ShiftRows step

In the ShiftRows step, bytes in each row of the state are shifted cyclically to the left. The number of places each byte is shifted differs for each row. The ShiftRows step operates on the rows of the state; it cyclically shifts the bytes in each row by a certain offset. For AES, the first row is left unchanged. Each byte of the second row is shifted one to the left. Similarly, the third and fourth rows are shifted by offsets of two and three respectively. In this way, each column of the output state of the ShiftRows step is composed of bytes from each column of the input state. The importance of this step is to avoid the columns being encrypted independently, in which case AES degenerates into four independent block ciphers

The MixColumns step

In the MixColumns step, each column of the state is multiplied with a fixed polynomial $ c(x)$

In the MixColumns step, the four bytes of each column of the state are combined using an invertible linear transformation. The MixColumns function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes. Together with ShiftRows, MixColumns provides diffusion in the cipher

During this operation, each column is transformed using a fixed matrix (matrix left-multiplied by column gives new value of column in the state):

Matrix multiplication is composed of multiplication and addition of the entries. Entries are 8-bit bytes treated as coefficients of polynomial of order $ {x}^{7} $. Addition is simply XOR. Multiplication is modulo irreducible polynomial $ {x}^{8}+{x}^{4}+{x}^{3}+x+1 $. If processed bit by bit, then, after shifting, a conditional XOR with $ 1B_16 $ should be performed if the shifted value is larger than $ FF_16 $ (overflow must be corrected by subtraction of generating polynomial). These are special cases of the usual multiplication in $ GF({2}^{8}) $

In more general sense, each column is treated as a polynomial over $ GF({2}^{8}) $ and is then multiplied modulo $ 01_16 \cdot {z}^{4} + 01_16 $ with a fixed polynomial $ c(z) = 03_16 \cdot {z}^{3} + 01_16 \cdot {z}^{2} + 01_16 \cdot z + 02_16 $ The coefficients are displayed in their hexadecimal equivalent of the binary representation of bit polynomials from $ GF(2)[x] $. The MixColumns step can also be viewed as a multiplication by the shown particular MDS matrix in the finite field $ GF({2}^{8}) $. This process is described further in the article Rijndael MixColumns

The AddRoundKey step

In the AddRoundKey step, the subkey is combined with the state. For each round, a subkey is derived from the main key using Rijndael's key schedule; each subkey is the same size as the state. The subkey is added by combining each byte of the state with the corresponding byte of the subkey using bitwise XOR

Optimization of the cipher

On systems with 32-bit or larger words, it is possible to speed up execution of this cipher by combining the SubBytes and ShiftRows steps with the MixColumns step by transforming them into a sequence of table lookups. This requires four 256-entry 32-bit tables (together occupying 4096 bytes). A round can then be performed with 16 table lookup operations and 12 32-bit exclusive-or operations, followed by four 32-bit exclusive-or operations in the AddRoundKey step. Alternatively, the table lookup operation can be performed with a single 256-entry 32-bit table (occupying 1024 bytes) followed by circular rotation operations

Using a byte-oriented approach, it is possible to combine the SubBytes, ShiftRows, and MixColumns steps into a single round operation

Security

Until May 2009, the only successful published attacks against the full AES were side-channel attacks on some specific implementations. The National Security Agency (NSA) reviewed all the AES finalists, including Rijndael, and stated that all of them were secure enough for U.S. Government non-classified data. In June 2003, the U.S. Government announced that AES could be used to protect classified information:

The design and strength of all key lengths of the AES algorithm (i.e., 128, 192 and 256) are sufficient to protect classified information up to the SECRET level. TOP SECRET information will require use of either the 192 or 256 key lengths. The implementation of AES in products intended to protect national security systems and/or information must be reviewed and certified by NSA prior to their acquisition and use

AES has 10 rounds for 128-bit keys, 12 rounds for 192-bit keys, and 14 rounds for 256-bit keys

By 2006, the best known attacks were on 7 rounds for 128-bit keys, 8 rounds for 192-bit keys, and 9 rounds for 256-bit keys

Performance

High speed and low RAM requirements were criteria of the AES selection process. As the chosen algorithm, AES performed well on a wide variety of hardware, from 8-bit smart cards to high-performance computers

On a Pentium Pro, AES encryption requires 18 clock cycles per byte, equivalent to a throughput of about 11 Mbit/s for a 200 MHz processor. On a 1.7 GHz Pentium M throughput is about 60 Mbit/s

On Intel Core i3/i5/i7 and AMD Ryzen CPUs supporting AES-NI instruction set extensions, throughput can be multiple GB/s (even over 10 GB/s)

Serpent Block Cipher

Serpent is a symmetric key block cipher that was a finalist in the Advanced Encryption Standard (AES) contest, where it was ranked second to Rijndael. Serpent was designed by Ross Anderson, Eli Biham, and Lars Knudsen

Like other AES submissions, Serpent has a block size of 128 bits and supports a key size of 128, 192 or 256 bits. The cipher is a 32-round substitution–permutation network operating on a block of four 32-bit words. Each round applies one of eight 4-bit to 4-bit S-boxes 32 times in parallel. Serpent was designed so that all operations can be executed in parallel, using 32 bit slices. This maximizes parallelism, but also allows use of the extensive cryptanalysis work performed on DES

Serpent took a conservative approach to security, opting for a large security margin: the designers deemed 16 rounds to be sufficient against known types of attack, but specified 32 rounds as insurance against future discoveries in cryptanalysis. The official NIST report on AES competition classified Serpent as having a high security margin along with MARS and Twofish, in contrast to the adequate security margin of RC6 and Rijndael (currently AES). In final voting, Serpent had the fewest negative votes among the finalists, but scored second place overall because Rijndael had substantially more positive votes, the deciding factor being that Rijndael allowed for a far more efficient software implementation

The Serpent cipher algorithm is in the public domain and has not been patented. The reference code is public domain software and the optimized code is under GPL. There are no restrictions or encumbrances whatsoever regarding its use. As a result, anyone is free to incorporate Serpent in their software (or hardware implementations) without paying license fees

Rijndael vs. Serpent

Rijndael is a substitution-linear transformation network with ten, twelve, or fourteen rounds, depending on the key size, and with block sizes of 128 bits, 192 bits, or 256 bits, independently specified. Serpent is a substitution–permutation network which has thirty-two rounds, plus an initial and a final permutation to simplify an optimized implementation. The round function in Rijndael consists of three parts: a nonlinear layer, a linear mixing layer, and a key-mixing XOR layer. The round function in Serpent consists of key-mixing XOR, thirty-two parallel applications of the same 4×4 S-box, and a linear transformation, except in the last round, wherein another key-mixing XOR replaces the linear transformation. The nonlinear layer in Rijndael uses an 8×8 S-box whereas Serpent uses eight different 4×4 S-boxes. The 32 rounds means that Serpent has a higher security margin than Rijndael; however, Rijndael with 10 rounds is faster and easier to implement for small blocks. Hence, Rijndael was selected as the winner in the AES competition

Serpent-0 vs. Serpent-1

The original Serpent, Serpent-0, was presented at the 5th workshop on Fast Software Encryption, but a somewhat tweaked version, Serpent-1, was submitted to the AES competition. The AES submission paper discuss the changes, which include key scheduling differences

Security

The XSL attack, if effective, would weaken Serpent (though not as much as it would weaken Rijndael, which became AES). However, many cryptanalysts believe that once implementation considerations are taken into account the XSL attack would be more expensive than a brute force attack

In 2000, a paper by Kohno et al. presents a meet-in-the-middle attack against 6 of 32 rounds of Serpent and an amplified boomerang attack against 9 of 32 rounds in Serpent

A 2001 attack by Eli Biham, Orr Dunkelman and Nathan Keller presents a linear cryptanalysis attack that breaks 10 of 32 rounds of Serpent-128 with 2118 known plaintexts and 289 time, and 11 rounds of Serpent-192/256 with 2118 known plaintexts and 2187 time

A 2009 paper has noticed that the nonlinear order of Serpent S-boxes were not 3 as was claimed by the designers

A 2011 attack by Hongjun Wu, Huaxiong Wang and Phuong Ha Nguyen, also using linear cryptanalysis, breaks 11 rounds of Serpent-128 with 2116 known plaintexts, 2107.5 time and 2104 memory

The same paper also describes two attacks which break 12 rounds of Serpent-256. The first requires 2118 known plaintexts, 2228.8 time and 2228 memory. The other attack requires 2116 known plaintexts and 2121 memory but also requires 2237.5 time

Serpent Linear Function [1]

ARIA Block Cipher (Korea)

ARIA is a general-purpose block cipher algorithm developed by Korean cryptographers in 2003. It is an iterated block cipher with 128-, 192-, and 256-bit keys and encrypts 128-bit blocks in 12, 14, and 16 rounds, depending on the key size. It is secure and suitable for most software and hardware implementations on 32-bit and 8-bit processors. It was established as a Korean standard block cipher algorithm in 2004 [ARIAKS] and has been widely used in Korea, especially for government-to-public services. It was included in PKCS #11 in 2007 [ARIAPKCS]. The algorithm specification and object identifiers are described in [RFC5794].

ARIA-CTR

Section 4.1.1 of [RFC3711] defines AES-128 counter mode encryption, which it refers to as "AES_CM". Section 2 of [RFC6188] defines "AES_192_CM" and "AES_256_CM" in SRTP. ARIA counter modes are defined in the same manner except that each invocation of AES is replaced by that of ARIA [RFC5794], and are denoted by ARIA_128_CTR, ARIA_192_CTR and ARIA_256_CTR respectively, according to the key lengths. The plaintext inputs to the block cipher are formed as in AES-CTR(AES_CM, AES_192_CM, AES_256_CM) and the block cipher outputs are processed as in AES-CTR. Note that, ARIA-CTR MUST be used only in conjunction with an authentication transform.

ARIA-GCM

GCM (Galois Counter Mode) [GCM][RFC5116] is an AEAD (Authenticated Encryption with Associated Data) block cipher mode. A detailed description of ARIA-GCM is defined similarly as AES-GCM found in [RFC5116][RFC5282].

SM4 Block Cipher (China)

See sm4_encryption_spec 15 May 2008 Version 1.03

SMS4 is a Chinese block cipher standard, mandated for use in protecting wireless networks and issued in January 2006. The input, output, and key of SMS4 are each 128 bits. The algorithm has 32 rounds, each of which modifies one of the four 32-bit words that make up the block by xoring it with a keyed function of the other three words. Encryption and decryption have the same structure except that the round key schedule for decryption is the reverse of the round key schedule for encryption

1. Terminology (Definitions)

1.1 Zi and ZiJie (Word and Byte)

${Z_2}^{e}$ is the set of e-bit vectors. Specifically, the elements of ${Z_2}^{32}$ are called $ Zi$ (32-bit words), and the elements of ${Z_2}^{8}$ are called $ ZiJie$ (8-bit characters, or bytes)

1.2 S box

The S (substitution) box takes in 8 bits and outputs 8 bits. It is written Sbox(.)

1.3 Fundamental Operations

The two fundamental operations used by this algorithm are:

  • the bitwise XOR of two 32-bit vectors
  • $ \ll$ i the circular shift of a 32-bit word, with i bits shifted left

1.4 Input and output blocks and key

The 128-bit input block consists of four 32-bit words $ MK = (MK_1,MK_2,MK_3,MK_4) or MK_i(i = 0, 1, 2, 3)$

The round key schedule, derived from the encryption key, is represented by $ (rk_0, rk_1,...,rk_{31})$ where each $ rk_i, i=0,...,31)$ is 32 bits long

The 128-bit output block consists of four 32-bit words $ FK = (FK_0, FK_1, FK_2, FK_3)$

For decryption, the round key schedule is represented by $ CK = (CK_0,CK_1,...,CK_{31})$ or

  • $ FK_i, (i = 0,...,3), CK_i, (i = 0,...,31)$

2. The round function F

This algorithm uses a nonlinear substitution structure, encrypting 32 bits at a time. This is called a one-round exchange. To illustrate, consider a one-round-substitution:

Let the 128-bit input block be the four 32-bit elements

  • $(X_0,X_1,X_2,X_3) \in {({Z_2}^{32})}^{4}$, with $rk \in {Z_2}^{32}$, then F is given by
  • $ F(X_0,X_1,X_2,X_3,rk) = X_0 \oplus T(X_1 \oplus X_2 \oplus X_3 \oplus rk)$

2.1 Mixer-substitution T

T is a substitution that generates 32 bits from 32 bits $T : {Z_2}^{32} \mapsto {Z_2}^{32}$. This substitution is a reversible process. It consists of a non-linear substitution, $tau$, and a linear substitution L, i.e., $T(.) = L(\tau(.))$

2.1.1 Non-linear substitution $\tau$

$\tau$ applies 4 S-boxes in parallel Let a 32-bit input word be $A = (a_0, a_1, a_2, a_3) \in {({Z_2}^{8})}^{4}$, where each ai is an 8-bit character Let the 32-bit output word be $B = (b_0, b_1, b_2, b_3) \in {({Z_2}^{8})}^{4}$, given by $(b_0, b_1, b_2, b_3) = \tau(A) = (Sbox(a_0), Sbox(a_1), Sbox(a_2), Sbox(a_3))$

2.1.2 Linear substitution L

$B \in {Z_2}^{32}$, the 32-bit output word of the non-linear substitution $tau$ will be the input word of the linear substitution L. Let $C \in {Z_2}^{32}$ be the 32-bit output word generated by L. Then

  • $C = L(B) = B \oplus (B <<< 2) \oplus (B <<< 10) \oplus (B <<< 18) \oplus (B <<< 24)$

2.2 S box

All Sbox numbers are in hexadecimal notation

$ S Box$
0123456789101112131415
d6 90 e9 fe cc e1 3d b7 16 b6 14 c2 28 fb 2c 05
2b 67 9a 76 2a be 04 c3 aa 44 13 26 49 86 06 99
9c 42 50 f4 91 ef 98 7a 33 54 0b 43 ed cf ac 62
e4 b3 1c a9 c9 08 e8 95 80 df 94 fa 75 8f 3f a6
47 07 a7 fc f3 73 17 ba 83 59 3c 19 e6 85 4f a8
68 6b 81 b2 71 64 da 8b f8 eb 0f 4b 70 56 9d 35
1e 24 0e 5e 63 58 d1 a2 25 22 7c 3b 01 21 78 87
d4 00 46 57 9f d3 27 52 4c 36 02 e7 a0 c4 c8 9e
ea bf 8a d2 40 c7 38 b5 a3 f7 f2 ce f9 61 15 a1
e0 ae 5d a4 9b 34 1a 55 ad 93 32 30 f5 8c b1 e3
1d f6 e2 2e 82 66 ca 60 c0 29 23 ab 0d 53 4e 6f
d5 db 37 45 de fd 8e 2f 03 ff 6a 72 6d 6c 5b 51
8d 1b af 92 bb dd bc 7f 11 d9 5c 41 1f 10 5a d8
0a c1 31 88 a5 cd 7b bd 2d 74 d0 12 b8 e5 b4 b0
89 69 97 4a 0c 96 77 7e 65 b9 f1 09 c5 6e c6 84
18 f0 7d ec 3a dc 4d 20 79 ee 5f 3e d7 cb 39 48

For example, if the input to the Sbox is 'ef', then go to e-th row and f-th column, to find $ Sbox('ef')='84'$

3. Encryption and decryption

Let the reverse substitution $ R$ be:

  • $R(A_0,A_1,A_2,A_3) = (A_3,A_2,A_1,A_0),A_i \in {Z_2}^{32}, i = 0,1,2,3$

Let the plaintext input be $(X_0,X_1,X_2,X_3) \in {({Z_2}^{32})}^{4}$, the ciphertext output be $(Y_0,Y_1,Y_2,Y_3) \in $ ${({Z_2}^{32})}^{4}$, and the encrypting key be $rk_i \in {Z_2}^{32}, i = 0,1,2,...,31$

Then encryption proceeds as follows:

  • $ X_{i+4} = F(X_i,X_{i+1},X_{i+2},X_{i+3}, rk_i) = Xi \oplus T(X_{i+1} \oplus X_{i+2} \oplus X_{i+3} \oplus rk_i)$ where $ i = 0,1,...,31$
  • $(Y_0, Y_1, Y_2, Y_3) = R(X_{32},X_{33},X_{34},X_{35}) = (X_{35},X_{34},X_{33},X_{32})$

This algorithm's encryption and decryption methods have the same structure, except the order in which the round keys are used is reversed

The key order for encryption is: $(rk_0, rk_1,...,rk_{31})$. The key order for decryption is:

  • $(rk_{31}, rk_{30},...,rk_0)$

4. Key expansion when encrypting

The $rk_i$ round key used for encrypting in this algorithm is derived from the encryption key MK.

Let $MK = (MK_0,MK_1,MK_2,MK_3), MK_i \in {Z_2}^{32}, i = 0,1,2,3;$

  • $K_i \in {Z_2}^{32}, i = 0,1,...,31;$
  • $rk_i \in {Z_2}^{32}, i = 0,1,...,31$;

the derivation proceeds as follows:

First,

  • $(K_0,K_1,K_2,K_3) = (MK_0 \oplus FK_0; MK_1 \oplus FK_1; MK_2 \oplus FK_2; MK_3 \oplus FK_3)$

Then for i = 0,1,2,...,31:

  • $rk_i = K_{i+4} = K_i \oplus T\prime(K_{i+1} \oplus K_{i+2} \oplus K_{i+3} \oplus CK_i)$

Notes:

  1. T&prime substitution uses the same T as in encryption, except the linear substitution L is changed to L&prime: L&prime $(B) = B \oplus (B <<< 13) \oplus (B <<< 23)$;
  2. The system parameter $ FK$ given in hexadecimal notation is
  • $FK_0 = (a3b1bac6), FK_1 = (56aa3350), FK_2 = (677d9197), FK_3 = (b27022dc)$
  1. The constant parameter $ CK$ is calculated as follows:

Let $ck_{i,j}$ be the j-th byte of $CK_{i,j}(i = 0,1,...,31; j = 0,1,2,3)$, i.e., $CK_i = (ck_{i,0}, ck_{i,1}; ck_{i,2}; ck_{i,3}) \in {({Z_2}^{8})}^{4}, then ck_{i,j} = (4i + j) \times (mod 256)$

The 32 constants $CK_i$ are represented in hexadecimal as tabulated below:

$ CK_i$
0123
00070e15 1c232a31 383f464d 545b6269
70777e85 8c939aa1 a8afb6bd c4cbd2d9
e0e7eef5 fc030a11 181f262d 343b4249
50575e65 6c737a81 888f969d a4abb2b9
c0c7ced5 dce3eaf1 f8ff060d 141b2229
30373e45 4c535a61 686f767d 848b9299
a0a7aeb5 bcc3cad1 d8dfe6ed f4fb0209
10171e25 2c333a41 484f565d 646b7279

Camellia Block Cipher (Japan)

See https://en.wikipedia.org/wiki/Camellia_(cipher)
See https://info.isl.ntt.co.jp/crypt/eng/camellia/dl/01espec.pdf

In cryptography, Camellia is a symmetric key block cipher with a block size of 128 bits and key sizes of 128, 192 and 256 bits. It was jointly developed by Mitsubishi Electric and NTT of Japan. The cipher has been approved for use by the ISO/IEC, the European Union's NESSIE project and the Japanese CRYPTREC project. The cipher has security levels and processing abilities comparable to the Advanced Encryption Standard

The cipher was designed to be suitable for both software and hardware implementations, from low-cost smart cards to high-speed network systems. It is part of the Transport Layer Security (TLS) cryptographic protocol designed to provide communications security over a computer network such as the Internet

The cipher was named for the flower Camellia japonica, which is known for being long-lived as well as because the cipher was developed in Japan

Design

Camellia is a Feistel cipher with either 18 rounds (when using 128-bit keys) or 24 rounds (when using 192- or 256-bit keys). Every six rounds, a logical transformation layer is applied: the so-called "FL-function" or its inverse. Camellia uses four 8×8-bit S-boxes with input and output affine transformations and logical operations. The cipher also uses input and output key whitening. The diffusion layer uses a linear transformation based on a matrix with a branch number of 5

Security analysis

Camellia is considered a modern, safe cipher. Even using the smaller key size option (128 bits), it's considered infeasible to break it by brute-force attack on the keys with current technology. There are no known successful attacks that weaken the cipher considerably. The cipher has been approved for use by the ISO/IEC, the European Union's NESSIE project and the Japanese CRYPTREC project. The Japanese cipher has security levels and processing abilities comparable to the AES/Rijndael cipher

Camellia is a block cipher which can be completely defined by minimal systems of multivariate polynomials:

  • The Camellia (as well as AES) S-boxes can be described by a system of 23 quadratic equations in 80 terms
  • The key schedule can be described by 1,120 equations in 768 variables using 3,328 linear and quadratic terms
  • The entire block cipher can be described by 5,104 equations in 2,816 variables using 14,592 linear and quadratic terms
  • In total, 6,224 equations in 3,584 variables using 17,920 linear and quadratic terms are required
  • The number of free terms is 11,696, which is approximately the same number as for AES

Theoretically, such properties might make it possible to break Camellia (and AES) using an algebraic attack, such as extended sparse linearisation, in the future, provided that the attack becomes feasible

ARC4 Stream Cipher

See https://en.wikipedia.org/wiki/RC4
See http://cypherpunks.venona.com/archive/1994/09/msg00304.html

In cryptography, RC4 (Rivest Cipher 4 also known as ARC4 or ARCFOUR meaning Alleged RC4, see below) is a stream cipher. While it is remarkable for its simplicity and speed in software, multiple vulnerabilities have been discovered in RC4, rendering it insecure. It is especially vulnerable when the beginning of the output keystream is not discarded, or when nonrandom or related keys are used. Particularly problematic uses of RC4 have led to very insecure protocols such as WEP

As of 2015, there is speculation that some state cryptologic agencies may possess the capability to break RC4 when used in the TLS protocol. IETF has published RFC 7465 to prohibit the use of RC4 in TLS; Mozilla and Microsoft have issued similar recommendations

A number of attempts have been made to strengthen RC4, notably Spritz, RC4A, VMPC, and RC4+

History

RC4 was designed by Ron Rivest of RSA Security in 1987. While it is officially termed "Rivest Cipher 4", the RC acronym is alternatively understood to stand for "Ron's Code" (see also RC2, RC5 and RC6)

RC4 was initially a trade secret, but in September 1994 a description of it was anonymously posted to the Cypherpunks mailing list. It was soon posted on the sci.crypt newsgroup, where it was broken within days by Bob Jenkins. From there it spread to many sites on the Internet. The leaked code was confirmed to be genuine as its output was found to match that of proprietary software using licensed RC4. Because the algorithm is known, it is no longer a trade secret. The name RC4 is trademarked, so RC4 is often referred to as ARCFOUR or ARC4 (meaning alleged RC4) to avoid trademark problems. RSA Security has never officially released the algorithm; Rivest has, however, linked to the English Wikipedia article on RC4 in his own course notes in 2008 and confirmed the history of RC4 and its code in a 2014 paper by him

RC4 became part of some commonly used encryption protocols and standards, such as WEP in 1997 and WPA in 2003/2004 for wireless cards; and SSL in 1995 and its successor TLS in 1999, until it was prohibited for all versions of TLS by RFC 7465 in 2015, due to the RC4 attacks weakening or breaking RC4 used in SSL/TLS. The main factors in RC4's success over such a wide range of applications have been its speed and simplicity: efficient implementations in both software and hardware were very easy to develop

Description

RC4 generates a pseudorandom stream of bits (a keystream). As with any stream cipher, these can be used for encryption by combining it with the plaintext using bit-wise exclusive-or; decryption is performed the same way (since exclusive-or with given data is an involution). This is similar to the one-time pad except that generated pseudorandom bits, rather than a prepared stream, are used

To generate the keystream, the cipher makes use of a secret internal state which consists of two parts:

  1. A permutation of all 256 possible bytes (denoted "S" below)
  2. Two 8-bit index-pointers (denoted "i" and "j")

The permutation is initialized with a variable length key, typically between 40 and 2048 bits, using the key- scheduling algorithm (KSA). Once this has been completed, the stream of bits is generated using the pseudo-random generation algorithm (PRGA).

Key-scheduling algorithm (KSA)

The key-scheduling algorithm is used to initialize the permutation in the array "S". "keylength" is defined as the number of bytes in the key and can be in the range 1 ≤ keylength ≤ 256, typically between 5 and 16, corresponding to a key length of 40–128 bits. First, the array "S" is initialized to the identity permutation. S is then processed for 256 iterations in a similar way to the main PRGA, but also mixes in bytes of the key at the same time

for i from 0 to 255
S[i] := i
endfor
j := 0
for i from 0 to 255
j := (j + S[i] + key[i mod keylength]) mod 256
swap values of S[i] and S[j]
endfor
ProtocolPP::swap
static void swap(w128_t *array, int size)
Definition: SFMT-alti.h:148

In the above lookup stage of ARC4, the output byte is selected by looking up the values of S[i] and S[j], adding them together modulo 256, and then using the sum as an index into S; S(S[i] + S[j]) is used as a byte of the key stream, K

Pseudo-random generation algorithm (PRGA)

The lookup stage of RC4. The output byte is selected by looking up the values of S[i] and S[j], adding them together modulo 256, and then using the sum as an index into S; S(S[i] + S[j]) is used as a byte of the key stream, K

For as many iterations as are needed, the PRGA modifies the state and outputs a byte of the keystream. In each iteration, the PRGA:

  • increments i
  • looks up the ith element of S, S[i], and adds that to j
  • exchanges the values of S[i] and S[j] then uses the sum S[i] + S[j] (modulo 256) as an index to fetch a third element of S (the keystream value K below)
  • then bitwise exclusive ORed (XORed) with the next byte of the message to produce the next byte of either ciphertext or plaintext.

Each element of S is swapped with another element at least once every 256 iterations

i := 0
j := 0
while GeneratingOutput:
i := (i + 1) mod 256
j := (j + S[i]) mod 256
swap values of S[i] and S[j]
K := S[(S[i] + S[j]) mod 256]
output K
endwhile

Security

Unlike a modern stream cipher (such as those in eSTREAM), RC4 does not take a separate nonce alongside the key. This means that if a single long-term key is to be used to securely encrypt multiple streams, the protocol must specify how to combine the nonce and the long-term key to generate the stream key for RC4. One approach to addressing this is to generate a "fresh" RC4 key by hashing a long-term key with a nonce. However, many applications that use RC4 simply concatenate key and nonce; RC4's weak key schedule then gives rise to related key attacks, like the Fluhrer, Mantin and Shamir attack (which is famous for breaking the WEP standard)

Because RC4 is a stream cipher, it is more malleable than common block ciphers. If not used together with a strong message authentication code (MAC), then encryption is vulnerable to a bit-flipping attack. The cipher is also vulnerable to a stream cipher attack if not implemented correctly

It is noteworthy, however, that RC4, being a stream cipher, was for a period of time the only common cipher that was immune to the 2011 BEAST attack on TLS 1.0. The attack exploits a known weakness in the way cipher block chaining mode is used with all of the other ciphers supported by TLS 1.0, which are all block ciphers

In March 2013, there were new attack scenarios proposed by Isobe, Ohigashi, Watanabe and Morii, as well as AlFardan, Bernstein, Paterson, Poettering and Schuldt that use new statistical biases in RC4 key table to recover plaintext with large number of TLS encryptions

The use of RC4 in TLS is prohibited by RFC 7465 published in February 2015.

Triple DES

See https://en.wikipedia.org/wiki/Triple_DES

In cryptography, Triple DES (3DES or TDES), officially the Triple Data Encryption Algorithm (TDEA or Triple DEA), is a symmetric-key block cipher, which applies the DES cipher algorithm three times to each data block. The Data Encryption Standard’s(DES) 56-bit key is no longer considered adequate in the face of modern cryptanalytic techniques and supercomputing power. However, an adapted version of DES, Triple DES (3DES), uses the same algorithm to produce a more secure encryption

While the government and industry standards abbreviate the algorithm's name as TDES (Triple DES) and TDEA (Triple Data Encryption Algorithm),[1] RFC 1851 referred to it as 3DES from the time it first promulgated the idea, and this namesake has since come into wide use by most vendors, users, and cryptographers

Algorithm

The original DES cipher's key size of 56 bits was generally sufficient when that algorithm was designed, but the availability of increasing computational power made brute-force attacks feasible. Triple DES provides a relatively simple method of increasing the key size of DES to protect against such attacks, without the need to design a completely new block cipher algorithm

A naive approach to increase strength of a block encryption algorithm with short key length (like DES) would be to use two keys (K1, K2) instead of one, and encrypt each block twice: $ E_K2(E_K1(plaintext)) $. If the original key length is n bits, one would hope this scheme provides security equivalent to using key 2n bits long. Unfortunately, this approach is vulnerable to meet-in-the-middle attack: given a known plaintext pair (x, y), such that $ y = E_K2(E_K1(x))$, one can recover the key pair (K1, K2) in $ {2}^{n}$ steps, instead of $ {2}^{2n}$ steps one would expect from an ideally secure algorithm with 2n bits of key

Therefore, Triple DES uses a "key bundle" that comprises three DES keys, K1, K2 and K3, each of 56 bits (excluding parity bits). The encryption algorithm is:

$ ciphertext = E_K3 (D_K2(E_K1(plaintext)))$

That is, DES encrypt with K1, DES decrypt with K2, then DES encrypt with K3. Decryption is the reverse:

$ plaintext = D_K1(E_K2(D_K3(ciphertext)))$

That is, decrypt with K3, encrypt with K2, then decrypt with K1

Each triple encryption encrypts one block of 64 bits of data

In each case the middle operation is the reverse of the first and last. This improves the strength of the algorithm when using keying option 2 and provides backward compatibility with DES with keying option 3

Keying options

The standards define three keying options:

  • Keying option 1
    • All three keys are independent. Sometimes known as 3TDEA or triple-length keys. This is the strongest, with 3 × 56 = 168 independent key bits. It is still vulnerable to meet-in-the-middle attack, but the attack requires $ {2}^{2 × 56} steps$
  • Keying option 2
    • K1 and K2 are independent, and K3 = K1. Sometimes known as 2TDEA or double-length keys. This provides a shorter key length of 112 bits and a reasonable compromise between DES and Kehttps://en.wikipedia.org/wiki/Triple_DESying option 1, with the same caveat as above. This is an improvement over "double DES" which only requires 256 steps to attack. NIST has deprecated this option
  • Keying option 3
    • All three keys are identical, i.e. K1 = K2 = K3. This is backward compatible with DES, since two operations cancel out. ISO/IEC 18033-3 never allowed this option, and NIST no longer allows K1 = K2 or K2 = K3

Each DES key is 8 odd-parity bytes, with 56 bits of key and 8 bits of error-detection. A key bundle requires 24 bytes for option 1, 16 for option 2, or 8 for option 3

NIST (and the current TCG specifications version 2.0 of approved algorithms for Trusted Platform Module) also disallows using any one of the 64 following 64-bit values in any keys (note that 32 of them are the binary complement of the 32 others; and that 32 of these keys are also the reverse permutation of bytes of the 32 others), listed here in hexadecimal (in each byte, the least significant bit is a odd-parity generated bit, it is discarded when forming the effective 56-bit keys):

01.01.01.01.01.01.01.01, FE.FE.FE.FE.FE.FE.FE.FE, E0.FE.FE.E0.F1.FE.FE.F1, 1F.01.01.1F.0E.01.01.0E,
01.01.FE.FE.01.01.FE.FE, FE.FE.01.01.FE.FE.01.01, E0.FE.01.1F.F1.FE.01.0E, 1F.01.FE.E0.0E.01.FE.F1,
01.01.E0.E0.01.01.F1.F1, FE.FE.1F.1F.FE.FE.0E.0E, E0.FE.1F.01.F1.FE.0E.01, 1F.01.E0.FE.0E.01.F1.FE,
01.01.1F.1F.01.01.0E.0E, FE.FE.E0.E0.FE.FE.F1.F1, E0.FE.E0.FE.F1.FE.F1.FE, 1F.01.1F.01.0E.01.0E.01,
01.FE.01.FE.01.FE.01.FE, FE.01.FE.01.FE.01.FE.01, E0.01.FE.1F.F1.01.FE.0E, 1F.FE.01.E0.0E.FE.01.F1,
01.FE.FE.01.01.FE.FE.01, FE.01.01.FE.FE.01.01.FE, E0.01.01.E0.F1.01.01.F1, 1F.FE.FE.1F.0E.FE.FE.0E,
01.FE.E0.1F.01.FE.F1.0E, FE.01.1F.E0.FE.01.0E.F1, E0.01.1F.FE.F1.01.0E.FE, 1F.FE.E0.01.0E.FE.F1.01,
01.FE.1F.E0.01.FE.0E.F1, FE.01.E0.1F.FE.01.F1.0E, E0.01.E0.01.F1.01.F1.01, 1F.FE.1F.FE.0E.FE.0E.FE,
01.E0.01.E0.01.F1.01.F1, FE.1F.FE.1F.FE.0E.FE.0E, E0.1F.FE.01.F1.0E.FE.01, 1F.E0.01.FE.0E.F1.01.FE,
01.E0.FE.1F.01.F1.FE.0E, FE.1F.01.E0.FE.0E.01.F1, E0.1F.01.FE.F1.0E.01.FE, 1F.E0.FE.01.0E.F1.FE.01,
01.E0.E0.01.01.F1.F1.01, FE.1F.1F.FE.FE.0E.0E.FE, E0.1F.1F.E0.F1.0E.0E.F1, 1F.E0.E0.1F.0E.F1.F1.0E,
01.E0.1F.FE.01.F1.0E.FE, FE.1F.E0.01.FE.0E.F1.01, E0.1F.E0.1F.F1.0E.F1.0E, 1F.E0.1F.E0.0E.F1.0E.F1,
01.1F.01.1F.01.0E.01.0E, FE.E0.FE.E0.FE.F1.FE.F1, E0.E0.FE.FE.F1.F1.FE.FE, 1F.1F.01.01.0E.0E.01.01,
01.1F.FE.E0.01.0E.FE.F1, FE.E0.01.1F.FE.F1.01.0E, E0.E0.01.01.F1.F1.01.01, 1F.1F.FE.FE.0E.0E.FE.FE,
01.1F.E0.FE.01.0E.F1.FE, FE.E0.1F.01.FE.F1.0E.01, E0.E0.1F.1F.F1.F1.0E.0E, 1F.1F.E0.E0.0E.0E.F1.F1,
01.1F.1F.01.01.0E.0E.01, FE.E0.E0.FE.FE.F1.F1.FE, E0.E0.E0.E0.F1.F1.F1.F1, 1F.1F.1F.1F.0E.0E.0E.0E,

With these restrictions on allowed keys, Triple DES has been reapproved with keying options 1 and 2 only. Generally the three keys are generated by taking 24 bytes from a strong random generator and only keying option 1 should be used (option 2 needs only 16 random bytes, but strong random generators are hard to assert and it's considered best practice to use only option 1)

Usage

The electronic payment industry uses Triple DES and continues to develop and promulgate standards based upon it, such as EMV

Earlier versions of Microsoft OneNote, Microsoft Outlook 2007 and Microsoft System Center Configuration Manager 2012 use Triple DES to password-protect user content and system data. However, in December 2018, Microsoft announced the retirement of 3DES throughout their Office 365 service

Firefox and Mozilla Thunderbird use Triple DES in CBC mode to encrypt website authentication login credentials when using a master password

SEED Block Cipher (Korea)

SEED is a block cipher developed by the Korea Internet & Security Agency (KISA). It is used broadly throughout South Korean industry, but seldom found elsewhere. It gained popularity in Korea because 40-bit encryption was not considered strong enough, so the Korea Information Security Agency developed its own standard. However, this decision has historically limited the competition of web browsers in Korea, as no major SSL libraries or web browsers supported the SEED algorithm, requiring users to use an ActiveX control in Internet Explorer for secure web sites

On April 1, 2015 the Ministry of Science, ICT and Future Planning (MSIP) announced its plan to remove the ActiveX dependency from at least 90 percent of the country's top 100 websites by 2017. Instead, HTML5-based technologies will be employed as they operate on many platforms, including mobile devices. Starting with the private sector, the ministry plans to expand this further to ultimately remove this dependency from public websites as well

Design

SEED is a 16-round Feistel network with 128-bit blocks and a 128-bit key. It uses two 8 × 8 S-boxes which, like those of SAFER, are derived from discrete exponentiation (in this case, x247 and x251 – plus some "incompatible operations"). It also has some resemblance to MISTY1 in the recursiveness of its structure: the 128-bit full cipher is a Feistel network with an F-function operating on 64-bit halves, while the F-function itself is a Feistel network composed of a G-function operating on 32-bit halves. However the recursion does not extend further because the G-function is not a Feistel network. In the G-function, the 32-bit word is considered as four 8-bit bytes, each of which is passed through one or the other of the S-boxes, then combined in a moderately complex set of boolean functions such that each output bit depends on 3 of the 4 input bytes

SEED has a fairly complex key schedule, generating its thirty-two 32-bit subkeys through application of its G-function on a series of rotations of the raw key, combined with round constants derived (as in TEA) from the Golden ratio

ZUC Encrytpion Algorithm (128-EEA3 and 128-EIA3)

See also
Specification of the 3GPP Confidentiality and Integrity Algorithms 128-EEA3 and 128-EIA3 Document 2: ZUC Specification

General structure of the algorithm

ZUC has three logical layers, see the figure below. The top layer is a linear feedback shift register (LFSR) of 16 stages, the middle layer is for bit-reorganization ( BR), and the bottom layer is a nonlinear function F

General Structure of ZUC

The linear feedback shift register (LFSR)

The linear feedback shift register (LFSR) has 16 of 31-bit cells $ (s_0, s_1,\cdots,s_{15})$. Each cell $ s_i (0 \leq i \leq 15)$ is restricted to take values from the following set $ \{1,2,3,\cdots,{2}^{31} – 1\}$. The LFSR has 2 modes of operations: the initialization mode and the working mode

In the initialization mode, the LFSR receives a 31-bit input word u, which is obtained by removing the rightmost bit from the 32-bit output W of the nonlinear function F, i.e., u=W>>1. More specifically, the initialization mode works as follows:

LFSRWithInitialisationMode(u) {

  1. $ v={2}^{15}s_{15}+{2}^{17}s_{13}+{2}_{21}s_{10}+{2}^{20}s_4+(1+{2}^{8})s_0$ mod $ ({2}^{31}-1)$
  2. $ s_{16}=(v+u)$ mod $ ({2}^{31}-1)$
  3. $ If s_{16}=0$, then set $s_{16}={2}^{31}-1$
  4. $ (s_1,s_2,\cdots,s_{15},s_{16})\rightarrow(s_0,s_1,\cdots,s_{14},s_{15})$

}

In the working mode, the LFSR does not receive any input, and it works as follows:

LFSRWithWorkMode() {

  1. $ s_{16}={2}^{15}s_15+{2}^{17}s_13+{2}^{21}s_10+{2}^{20}s_4+(1+{2}^{8})s_0$ mod $ ({2}^{31}-1)$
  2. $ If s_{16}=0$, then set $ s_{16}={2}^{31}-1$
  3. $ (s_1,s_2,\cdots,s_{15},s_{16})\rightarrow(s_0,s_1,\cdots,s_{14},s_{15})$

}

Informative note: Since the multiplication of a 31-bit string s by $ {2}^{i}$ over $ GF({2}^{31}-1)$ can be implemented by a cyclic shift of s to the left by i bits, only addition modulo $ {2}^{31}-1$ is needed in step 1 of the above functions. More precisely, step 1 of the function LFSRWithInitialisationMode can be implemented by

$ v=(s_{15} <<<_{31} 15)+(s_{13} <<<_{31} 17)+(s_{10} <<<_{31} 21)+(s_4 <<<_{31} 20)+(s_0 <<<_{31} 8)+s_0$ mod $ ({2}^{31}-1)$

and the same implementation is needed for step 1 of the function LFSRWithWorkMode

Informative note: For two elements $ a,b$ over $ GF({2}^{31}-1)$, the computation of $ v=a+b$ mod $ ({2}^{31}-1)$ can be done by

(1) compute v=a+b and

(2) if the carry bit is 1, then set v=v+1

Alternatively (and better if the implementation should resist possible timing attacks):

(1) compute w=a+b, where w is a 32-bit value; and

(2) set v = (least significant 31 bits of w) + (most significant bit of w)

The bit-reorganization

The middle layer of the algorithm is the bit-reorganization. It extracts 128 bits from the cells of the LFSR and forms 4 of 32-bit words, where the first three words will be used by the nonlinear function F in the bottom layer, and the last word will be involved in producing the keystream

Let $ s_0, s_2, s_5, s_7, s_9, s_{11}, s_{14}, s_{15}$ be 8 cells of LFSR as in section 3.2. Then the bitreorganization forms 4 of 32-bit words $ X_0, X_1, X_2, X_3$ from the above cells as follows:

Bitreorganization() {

  1. $ X_0=s_{15H} \parallel s_{14L}$
  2. $ X_1=s_{11L} \parallel s_{9H}$
  3. $ X_2=s_{7L} \parallel s_{5H}$
  4. $ X_3=s_{2L} \parallel s_{0H}$

}

Note: The $ s_i$ are 31-bit integers, so $ s_{iH}$ means bits 30...15 and not 31...16 of $ s_i, for 0 \leq i \leq 15$

The nonlinear function F

The nonlinear function F has 2 of 32-bit memory cells $ R_1$ and $ R_2$. Let the inputs to F be $ X_0, X_1$ and $ X_2$, which come from the outputs of the bit-reorganization (see section 3.3), then the function F outputs a 32-bit word W. The detailed process of F is as follows:

F ( $ X_0, X_1, X_2$) {

  1. $ W=(X_0 \oplus R_1) \oplus R_2$
  2. $ W_1=R_1 \oplus X_1$
  3. $ W_2=R_2 \oplus X_2$
  4. $ R_1=S(L_1(W_{1L} \parallel W_{2H}))$
  5. $ R_2=S(L_2(W_{2L} \parallel W_{1H}))$

}

where S is a 32×32 S-box, see section 3.4.1, $ L_1$ and $ L_2$ are linear transforms as defined in section 3.4.2

The S-box S

The 32×32 S-box S is composed of 4 juxtaposed 8×8 S-boxes, i.e., S=( $ S_0,S_1,S_2,S_3$), where $ S_0=S_2$, $ S_1=S_3$. The definitions of $ S_0$ and $ S_1$ can be found in table 3.1 and table 3.2 respectively

Let x be an 8-bit input to $ S_0$ (or $ S_1$). Write x into two hexadecimal digits as $ x=h \parallel l$, then the entry at the intersection of the h-th row and the l-th column in table 3.1 (or table 3.2) is the output of $ S_0$ (or $ S_1$)

The S-Box S0
The S-Box S1

The linear transforms L1 and L2

Both L1 and L2 are linear transforms from 32-bit words to 32-bit words, and are defined as follows:

$ L_1(X)=X \oplus (X <<<_{32} 2) \oplus (X <<<_{32} 10) \oplus (X <<<_{32} 18) \oplus (X <<<_{32} 24)$

$ L_2(X)=X \oplus (X <<<_{32} 8) \oplus (X <<<_{32} 14) \oplus (X <<<_{32} 22) \oplus (X <<<_{32} 30)$

Key loading

The key loading procedure will expand the initial key and the initial vector into 16 of 31-bit integers as the initial state of the LFSR. Let the 128-bit initial key k and the 128-bit initial vector iv be

$ k=k_0 \parallel k_1 \parallel k_2 \parallel \cdots \parallel k_{15}$

and

$ iv=iv_0 \parallel iv_1 \parallel iv_2 \parallel \cdots \parallel iv_{15}$

respectively, where $ k_i$ and $ iv_i, 0\leq i\leq 15$, are all bytes. Then k and iv are loaded to the cells $ s_0, s_1,\cdots, s_{15}$ of LFSR as follows:

  1. Let D be a 240-bit long constant string composed of 16 substrings of 15 bits:

$ D=d_0 \parallel d_1 \parallel \cdots \parallel d_{15}$

where

$ d00 = 100010011010111_2 $

$ d01 = 010011010111100_2 $

$ d02 = 110001001101011_2 $

$ d03 = 001001101011110_2 $

$ d04 = 101011110001001_2 $

$ d05 = 011010111100010_2 $

$ d06 = 111000100110101_2 $

$ d07 = 000100110101111_2 $

$ d08 = 100110101111000_2 $

$ d09 = 010111100010011_2 $

$ d10 = 110101111000100_2 $

$ d11 = 001101011110001_2 $

$ d12 = 101111000100110_2 $

$ d13 = 011110001001101_2 $

$ d14 = 111100010011010_2 $

$ d15 = 100011110101100_2 $

  1. $ For 0 \leq i \leq 15, let s_i=k_i \parallel d_i \parallel iv_i$

The execution of ZUC

The execution of ZUC has two stages: the initialization stage and the working stage

The initialization stage

During the initialization stage, the algorithm calls the key loading procedure (see section 3.5) to load the 128-bit initial key k and the 128-bit initial vector iv into the LFSR, and set the 32bit memory cells R1 and R2 to be all 0. Then the cipher runs the following operations 32 times:

  1. Bitreorganization();
  2. w=F( $ X_0, X_1, X_2$);
  3. LFSRWithInitialisationMode(w >> 1)

The working stage

After the initialization stage, the algorithm moves into the working stage. At the working stage, the algorithm executes the following operations once, and discards the output W of F:

  1. Bitreorganization();
  2. F( $ X_0, X_1, X_2$);
  3. LFSRWithWorkMode()

Then the algorithm goes into the stage of producing keystream, i.e., for each iteration, the following operations are executed once, and a 32-bit word Z is produced as an output:

  1. Bitreorganization();
  2. Z=F( $ X_0, X_1, X_2) \oplus X3$;
  3. LFSRWithWorkMode()

Snow3g Stream Cipher (UEA2 and UIA2)

See also
Specification of the 3GPP Confidentiality and Integrity Algorithms UEA2 and UIA2 Document 2: SNOW 3G Specification

SNOW 3G is a word-oriented stream cipher that generates a sequence of 32-bit words under the control of a 128-bit key and a 128-bit initialisation variable. These words can be used to mask the plaintext. First a key initialisation is performed, i.e. the cipher is clocked without producing output, see 4.1. Then with every clock tick it produces a 32-bit word of output

Functions used in different Components of SNOW 3G

MULx maps 16 bits to 8 bits. Let V and c be 8-bit input values. Then MULx is defined:

If the leftmost (i.e. the most significant) bit of V equals 1, then

$ MULx(V, c) = (V <<_8 1) \oplus c, else MULx(V, c) = V <<_8 1$

MULxPOW maps 16 bits and an positive integer i to 8 bit. Let V and c be 8-bit input values, then MULxPOW(V, i, c) is recursively defined:

If i equals 0, then $ MULxPOW(V, i, c) = V$, else $ MULxPOW(V, i, c) = MULx(MULxPOW(V, i-1, c), c)$

Linear Feedback Shift Register (LFSR)

The Linear Feedback Shift Register (LFSR) consists of sixteen stages $ s_0, s_1, s_2,\cdots, s_{15}$ each holding 32 bits

Finite State Machine (FSM)

The Finite State Machine (FSM) has three 32-bit registers R1, R2 and R3. The S-boxes S1 and S2 are used to update the registers R2 and R3

The 32x32-bit S-Box S1

The S-Box $ S_1$ maps a 32-bit input to a 32-bit output. Let $ w = w_0 || w_1 || w_2 || w_3$ the 32-bit input with $ w_0$ the most and $ w3$ the least significant byte

Let $ S_1(w)= r_0 || r_1 || r_2 || r_3$ with $ r_0$ the most and $ r_3$ the least significant byte. We use the 8 to 8 bit Rijndael S-Box $ S_R$ defined in 5.1

Then $ r_0, r_1, r_2, r_3$ are defined as

$ r_0= MULx(S_R(w_0), 0x1B) \oplus S_R(w_1) \oplus S_R(w_2) \oplus MULx(S_R(w_3), 0x1B) \oplus S_R(w_3)$

$ r_1= MULx(S_R(w_0), 0x1B) \oplus S_R(w_0) \oplus MULx(S_R(w_1), 0x1B) \oplus S_R(w_2) \oplus S_R(w_3)$

$ r_2= S_R(w_0) \oplus MULx(S_R(w_1), 0x1B) \oplus S_R(w_1) \oplus MULx(S_R(w_2), 0x1B) \oplus S_R(w_3)$

$ r_3= S_R(w_0) \oplus S_R(w_1) \oplus MULx(S_R(w_2), 0x1B) \oplus S_R(w_2) \oplus MULx(S_R(w_3), 0x1B)$

The 32x32-bit S-Box S2

The S-Box $ S_2$ maps a 32-bit input to a 32-bit output. Let $ w = w_0 || w_1 || w_2 || w_3$ the 32-bit input with $ w_0$ the most and $ w_3$ the least significant byte. Let $ S_2(w)= r_0 || r_1 || r_2 || r_3$ with $ r0$ the most and $ r3$ the least significant byte. We use the 8 to 8 bit S-Box $ S_Q$ defined in 5.2

Then $ r_0, r_1, r_2, r_3$ are defined as

$ r_0= MULx(S_Q(w_0), 0x69) \oplus S_Q(w_1) \oplus S_Q(w_2) \oplus MULx(S_Q(w_3), 0x69) \oplus SQ(w3)$

$ r_1= MULx(S_Q(w_0), 0x69) \oplus S_Q(w_0) \oplus MULx(S_Q(w_1), 0x69) \oplus S_Q(w_2) \oplus S_Q(w_3)$

$ r_2= S_Q(w_0) \oplus MULx(S_Q(w_1), 0x69) \oplus S_Q(w_1) \oplus MULx(S_Q(w_2), 0x69) \oplus S_Q(w_3)$

$ r_3= S_Q(w_0) \oplus S_Q(w_1) \oplus MULx(S_Q(w_2), 0x69) \oplus S_Q(w_2) \oplus MULx(S_Q(w_3), 0x69)$

The Clocking Operations

Clocking the LFSR

The clocking of the LFSR has two different modes of operation, the Initialisation Mode 3.4.4 and the Keystream Mode 3.4.5. In both modes the functions $ MUL_α$ and $ DIV_α$ are used which are defined in 3.4.2 resp. 3.4.3

The function $ MUL_α$

The function $ MUL_α$ maps 8 bits to 32 bits. Let c be the 8-bit input, then $ MUL_α$ is defined as

$ MUL_α(c) = (MULxPOW(c, 23, 0xA9) || MULxPOW(c, 245, 0xA9) || MULxPOW(c, 48, 0xA9) || MULxPOW(c, 239, 0xA9))$

The function $ DIV_α$

The function $ DIV_α$ maps 8 bits to 32 bits. Let c be the 8-bit input, then $ DIV_α$ is defined as

$ DIV_α(c) = (MULxPOW(c, 16, 0xA9) || MULxPOW(c, 39, 0xA9) || MULxPOW(c, 6, 0xA9) || MULxPOW(c, 64, 0xA9))$

Initialisation Mode

In the Initialisation Mode the LFSR receives a 32-bit input word F, which is the output of the FSM

Let $ s_0 = s_{0,0} || s_{0,1} || s_{0,2} || s_{0,3}$ with $ s_{0,0}$ being the most and $ s_{0,3}$ being the least significant byte of $ s_0$.

Let $ s_{11} = s_{11,0} || s_{11,1} || s_{11,2} || s_{11,3}$ with $ s_{11,0}$ being the most and $ s_{11,3}$ being the least significant byte of $ s_{11}$

Compute the intermediate value v as

$ v = (s_{0,1} || s_{0,2} || s_{0,3} || 0x00) \oplus MUL_α(s_{0,0}) \oplus s_2 \oplus (0x00 || s_{11,0} || s_{11,1} || s_{11,2}) \oplus DIV_α(s_{11,3}) \oplus F$

Set

$ s_0=s_1, s_1=s_2, s_2=s_3, s_3=s_4, s_4=s_5, s_5=s_6, s_6=s_7, s_7=s_8$

$ s_8=s_9, s_9=s_{10}, s_{10}=s_{11}, s_{11}=s_{12}, s_{12}=s_{13}, s_{13}=s_{14}, s_{14}=s_{15}, s_{15} = v$

Keystream Mode

In the Keystream Mode the LFSR does not receive any input

Let $ s_0 = s_{0,0} || s_{0,1} || s_{0,2} || s_{0,3}$ with $ s_{0,0}$ being the most and $ s_{0,3}$ being the least significant byte of $ s_0$

Let $ s_11 = s_{11,0} || s_{11,1} || s_{11,2} || s_{11,3}$ with $ s_{11,0}$ being the most and $ s_{11,3}$ being the least significant byte of $ s_11$

Compute the intermediate value v as

v = $ (s_{0,1} || s_{0,2} || s_{0,3} || 0x00) \oplus MUL_α(s_{0,0}) \oplus s_2 \oplus (0x00 || s_{11,0} || s_{11,1} || s_{11,2}) \oplus DIV_α(s_{11,3})$

Set

$ s_0=s_1, s_1=s_2, s_2=s_3, s_3=s_4, s_4=s_5, s_5=s_6, s_6=s_7, s_7=s_8$

$ s_8=s_9, s_9=s_{10}, s_{10}=s_{11}, s_{11}=s_{12}, s_{12}=s_{13}, s_{13}=s_{14}, s_{14}=s_{15}, s_{15}=v$

Clocking the FSM

The FSM has two input words $ s_{15}$ and $ s_5$ from the LFSR. It produces a 32-bit output word F:

$ F = (s_{15} \oplus R1) \oplus R2$

Then the registers are updated. Compute the intermediate value r as

$ r = R2 \oplus (R3 \oplus s_5)$

Set

$ R3 = S_2(R2)$

$ R2 = S_1(R1)$

$ R1 = r$

Operation of SNOW 3G

Initialisation

SNOW 3G is initialized with a 128-bit key consisting of four 32-bit words $ k_0, k_1, k_2, k_3$ and an 128-bit initialisation variable consisting of four 32-bit words $ IV_0, IV_1, IV_2, IV_3$ as follows

Let 1 be the all-ones word (0xffffffff)

$ s_{15}=k_3 \oplus IV_0$

$ s_{14}=k_2$

$ s_{13}=k_1$

$ s_{12}=k_0 \oplus IV_1$

$ s_{11}=k_3 \oplus 1$

$ s_{10}=k_2 \oplus 1 \oplus IV_2 s_9 = k_1 \oplus 1 \oplus IV_3 s_8 = k_0 \oplus 1$

$ s_7=k_3$

$ s_6=k_2$

$ s_5=k_1$

$ s_4=k_0$

$ s_3=k_3 \oplus 1$

$ s_2=k_2 \oplus 1$

$ s_1=k_1 \oplus 1$

$ s_0=k_0 \oplus 1$

The FSM is initialised with $ R1 = R2 = R3 = 0$

Then the cipher runs in a special mode without producing output:

repeat 32-times {

STEP 1: The FSM is clocked (see 3.4.6) producing the 32-bit word F

STEP 2: Then the LFSR is clocked in Initialisation Mode (see 3.4.4) consuming F

}

Snow 3G Algorithm During Initialization

Generation of Keystream

First the FSM is clocked once, see 3.4.6. The output word of the FSM is discarded. Then the LFSR is clocked once in Keystream Mode, see 3.4.4

After that n 32-bit words of keystream are produced:

for t = 1 to n {

STEP 1: The FSM is clocked (see 3.4.6) and produces a 32-bit output word F

STEP 2: The next keystream word is computed as $ z_t = F \oplus s_0$

STEP 3: Then the LFSR is clocked in Keystream Mode, see 3.4.4

}

Snow 3G Algorithm During Key-Stream Generation

The Rijndael S-box $ S_R$

The S-box SR maps 8 bit to 8 bit. Here the input and output is presented in hexadecimal form

Let $ x_0, x_1, y_0, y_1$ be hexadecimal digits with $ S_R(x_0 {2}^{4}+x_1) = y_0 {2}^{4} + y_1$, then the cell at the intersection of the $ {x_0}^{th}$ row and the $ {x_1}^{th}$ column contains the values for $ y_0 || y_1$ in hexadecimal form

For example $ S_R(42) = S_R(0x2A) = 0xE5 = 229$

Rijndael S-Box

The S-box $ S_Q$

The S-box $ S_Q$ maps 8 bit to 8 bit. Here the input is presented in hexadecimal form

Let $ x_0, x_1, y_0, y_1$ be hexadecimal digits with $ S_Q(x_0 {2}^{4}+x_1) = y_0 {2}^{4} + y_1$, then the cell at the intersection of the $ {x_0}^{th}$ row and the $ {x_1}^{th}$ column contains the values for $ y_0 || y_1$ in hexadecimal form

For example $ S_Q(42) = S_Q(0x2A) = 0xAC = 172$

S-Box SQ

For API Documentation:

See also
ProtocolPP::jconfident
ProtocolPP::jconfidentsa
ProtocolPP::jsnow3g
ProtocolPP::jmodes
ProtocolPP::jzuc

For Additional Documentation:

See also
jconfident
jconfidentsa
jsnow3g
jmodes
jzuc

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