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98-519: The GCM algorithm provides both data authenticity (integrity) and confidentiality and belongs to the class of authenticated encryption with associated data (AEAD) methods. This means that as input it takes a key K, some plaintext P, and some associated data AD; it then encrypts the plaintext using the key to produce ciphertext C, and computes an authentication tag T from the ciphertext and the associated data (which remains unencrypted). A recipient with knowledge of K, upon reception of AD, C and T, can decrypt

196-422: A 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ + a n x n = a 0 + x ( a 1 + x ( a 2 + x ( a 3 + ⋯ + x ( a n − 1 + x

294-410: A 0 + x 0 ( a 1 + x 0 ( a 2 + ⋯ + x 0 ( a n − 1 + b n x 0 ) ⋯ ) ) = a 0 + x 0 ( a 1 + x 0 (

392-450: A 0 , … , a n {\displaystyle a_{0},\ldots ,a_{n}} are constant coefficients, the problem is to evaluate the polynomial at a specific value x 0 {\displaystyle x_{0}} of x . {\displaystyle x.} For this, a new sequence of constants is defined recursively as follows: Then b 0 {\displaystyle b_{0}}

490-1068: A 1 + b 2 x , d 1 = b 1 + d 2 y , b 0 = a 0 + b 1 x . {\displaystyle {\begin{aligned}b_{n}&=a_{n},&\quad d_{n}&=b_{n},\\b_{n-1}&=a_{n-1}+b_{n}x,&\quad d_{n-1}&=b_{n-1}+d_{n}y,\\&{}\ \ \vdots &\quad &{}\ \ \vdots \\b_{1}&=a_{1}+b_{2}x,&\quad d_{1}&=b_{1}+d_{2}y,\\b_{0}&=a_{0}+b_{1}x.\end{aligned}}} At completion, we have p ( x ) = b 0 , p ( y ) − p ( x ) y − x = d 1 , p ( y ) = b 0 + ( y − x ) d 1 . {\displaystyle {\begin{aligned}p(x)&=b_{0},\\{\frac {p(y)-p(x)}{y-x}}&=d_{1},\\p(y)&=b_{0}+(y-x)d_{1}.\end{aligned}}} This computation of

588-709: A 2 + ⋯ + x 0 b n − 1 ) )     ⋮ = a 0 + x 0 b 1 = b 0 . {\displaystyle {\begin{aligned}p(x_{0})&=a_{0}+x_{0}{\Big (}a_{1}+x_{0}{\big (}a_{2}+\cdots +x_{0}(a_{n-1}+b_{n}x_{0})\cdots {\big )}{\Big )}\\&=a_{0}+x_{0}{\Big (}a_{1}+x_{0}{\big (}a_{2}+\cdots +x_{0}b_{n-1}{\big )}{\Big )}\\&~~\vdots \\&=a_{0}+x_{0}b_{1}\\&=b_{0}.\end{aligned}}} Now, it can be proven that; This expression constitutes Horner's practical application, as it offers

686-734: A 2 i + 1 x 2 i = p 0 ( x 2 ) + x p 1 ( x 2 ) . {\displaystyle {\begin{aligned}p(x)&=\sum _{i=0}^{n}a_{i}x^{i}\\[1ex]&=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}\\[1ex]&=\left(a_{0}+a_{2}x^{2}+a_{4}x^{4}+\cdots \right)+\left(a_{1}x+a_{3}x^{3}+a_{5}x^{5}+\cdots \right)\\[1ex]&=\left(a_{0}+a_{2}x^{2}+a_{4}x^{4}+\cdots \right)+x\left(a_{1}+a_{3}x^{2}+a_{5}x^{4}+\cdots \right)\\[1ex]&=\sum _{i=0}^{\lfloor n/2\rfloor }a_{2i}x^{2i}+x\sum _{i=0}^{\lfloor n/2\rfloor }a_{2i+1}x^{2i}\\[1ex]&=p_{0}(x^{2})+xp_{1}(x^{2}).\end{aligned}}} More generally,

784-443: A 4 x 4 + ⋯ ) + x ( a 1 + a 3 x 2 + a 5 x 4 + ⋯ ) = ∑ i = 0 ⌊ n / 2 ⌋ a 2 i x 2 i + x ∑ i = 0 ⌊ n / 2 ⌋

882-410: A i x i = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ + a n x n , {\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n},} proceed as follows b n =

980-423: A n x n = ( a 0 + a 2 x 2 + a 4 x 4 + ⋯ ) + ( a 1 x + a 3 x 3 + a 5 x 5 + ⋯ ) = ( a 0 + a 2 x 2 +

1078-420: A n ) ⋯ ) ) )   . {\displaystyle p(x)=a_{0}+x{\bigg (}a_{1}+x{\Big (}a_{2}+x{\big (}a_{3}+\cdots +x(a_{n-1}+x\,a_{n})\cdots {\big )}{\Big )}{\bigg )}\ .} Thus, by iteratively substituting the b i {\displaystyle b_{i}} into the expression, p ( x 0 ) =

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1176-523: A n ) ⋯ ) ) ) . {\displaystyle {\begin{aligned}&a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}\\={}&a_{0}+x{\bigg (}a_{1}+x{\Big (}a_{2}+x{\big (}a_{3}+\cdots +x(a_{n-1}+x\,a_{n})\cdots {\big )}{\Big )}{\bigg )}.\end{aligned}}} This allows the evaluation of a polynomial of degree n with only n {\displaystyle n} multiplications and n {\displaystyle n} additions. This

1274-460: A n , d n = b n , b n − 1 = a n − 1 + b n x , d n − 1 = b n − 1 + d n y ,     ⋮     ⋮ b 1 =

1372-614: A Chinese origin, but the lapse of time sufficiently makes it not altogether impossible that the Europeans could have known of the Chinese method in a direct or indirect way." Ulrich Libbrecht concluded: It is obvious that this procedure is a Chinese invention ... the method was not known in India . He said, Fibonacci probably learned of it from Arabs, who perhaps borrowed from the Chinese. The extraction of square and cube roots along similar lines

1470-459: A GCM variant Sophie Germain Counter Mode (SGCM) based on Sophie Germain primes . Authenticated encryption Authenticated Encryption (AE) is an encryption scheme which simultaneously assures the data confidentiality (also known as privacy: the encrypted message is impossible to understand without the knowledge of a secret key ) and authenticity (in other words, it is unforgeable:

1568-489: A cryptographic algorithm and generates code that runs well on the target processor. GCM has been criticized in the embedded world (for example by Silicon Labs) because the parallel processing is not suited for performant use of cryptographic hardware engines. As a result, GCM reduces the performance of encryption for some of the most performance-sensitive devices. Specialized hardware accelerators for ChaCha20-Poly1305 are less complex compared to AES accelerators. According to

1666-404: A hardware pipeline. By contrast, the cipher block chaining (CBC) mode of operation incurs pipeline stalls that hamper its efficiency and performance. Like in normal counter mode , blocks are numbered sequentially, and then this block number is combined with an initialization vector (IV) and encrypted with a block cipher E , usually AES . The result of this encryption is then XORed with

1764-507: A message and subsequently applying a MAC to the ciphertext (the Encrypt-then-MAC approach) implies security against an adaptive chosen ciphertext attack , provided that both functions meet minimum required properties. Katz and Yung investigated the notion under the name "unforgeable encryption" and proved it implies security against chosen ciphertext attacks. In 2013, the CAESAR competition

1862-476: A particular way of forging a GCM message, given a valid GCM message, that works with probability of about n ⋅2 for messages that are n × 128 bits long. However, this work does not show a more effective attack than was previously known; the success probability in observation 1 of this paper matches that of lemma 2 from the INDOCRYPT 2004 analysis (setting w = 128 and l = n × 128 ). Saarinen also described

1960-474: A poor protocol design or implementation turning Alice's side into an oracle . Naturally, this attack cannot be mounted at all when the keys are generated randomly. Key commitment was originally studied in the 2010s by Abdalla et al. and Farshim et al. under the name "robust encryption". To mitigate the attack described above without removing the "oracle", a key-committing AEAD that does not allow this type of crafted messages to exist can be used. AEGIS

2058-444: A power of 2 is merely a register shift operation. Thus, multiplying by 2 is calculated in base-2 by an arithmetic shift . The factor (2 ) is a right arithmetic shift , a (0) results in no operation (since 2 = 1 is the multiplicative identity element ), and a (2 ) results in a left arithmetic shift. The multiplication product can now be quickly calculated using only arithmetic shift operations, addition and subtraction. The method

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2156-419: A second (wrong) key K M will be incorrect, the authentication tag would still match. Since crafting a message with such property requires Mallory to already possess both K A and K M , the issue might appear to be one of a purely academic interest. However, under special circumstances, practical attacks can be mounted against vulnerable implementations. For example, if an identity authentication protocol

2254-437: A single key should be limited by 2. The authentication strength depends on the length of the authentication tag, like with all symmetric message authentication codes. The use of shorter authentication tags with GCM is discouraged. The bit-length of the tag, denoted t , is a security parameter. In general, t may be any one of the following five values: 128, 120, 112, 104, or 96. For certain applications, t may be 64 or 32, but

2352-447: A single polynomial evaluation. If, however, one is evaluating a single polynomial of very high order, it may be useful to break it up as follows: p ( x ) = ∑ i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ +

2450-476: A very quick way of determining the outcome of; p ( x ) / ( x − x 0 ) {\displaystyle p(x)/(x-x_{0})} with b 0 {\displaystyle b_{0}} (which is equal to p ( x 0 ) {\displaystyle p(x_{0})} ) being the division's remainder, as is demonstrated by the examples below. If x 0 {\displaystyle x_{0}}

2548-693: Is 5 . But by the polynomial remainder theorem , we know that the remainder is f ( 3 ) {\displaystyle f(3)} . Thus, f ( 3 ) = 5 {\displaystyle f(3)=5} . In this example, if a 3 = 2 , a 2 = − 6 , a 1 = 2 , a 0 = − 1 {\displaystyle a_{3}=2,a_{2}=-6,a_{1}=2,a_{0}=-1} we can see that b 3 = 2 , b 2 = 0 , b 1 = 2 , b 0 = 5 {\displaystyle b_{3}=2,b_{2}=0,b_{1}=2,b_{0}=5} ,

2646-960: Is 5 . This makes Horner's method useful for polynomial long division . Divide x 3 − 6 x 2 + 11 x − 6 {\displaystyle x^{3}-6x^{2}+11x-6} by x − 2 {\displaystyle x-2} : The quotient is x 2 − 4 x + 3 {\displaystyle x^{2}-4x+3} . Let f 1 ( x ) = 4 x 4 − 6 x 3 + 3 x − 5 {\displaystyle f_{1}(x)=4x^{4}-6x^{3}+3x-5} and f 2 ( x ) = 2 x − 1 {\displaystyle f_{2}(x)=2x-1} . Divide f 1 ( x ) {\displaystyle f_{1}(x)} by f 2 ( x ) {\displaystyle f_{2}\,(x)} using Horner's method. The third row

2744-426: Is a fast, code-efficient method for multiplication and division of binary numbers on a microcontroller with no hardware multiplier . One of the binary numbers to be multiplied is represented as a trivial polynomial, where (using the above notation) a i = 1 {\displaystyle a_{i}=1} , and x = 2 {\displaystyle x=2} . Then, x (or x to some power)

2842-423: Is a root of p ( x ) {\displaystyle p(x)} , then b 0 = 0 {\displaystyle b_{0}=0} (meaning the remainder is 0 {\displaystyle 0} ), which means you can factor p ( x ) {\displaystyle p(x)} as x − x 0 {\displaystyle x-x_{0}} . To finding

2940-519: Is an authentication-only variant of the GCM which can form an incremental message authentication code . Both GCM and GMAC can accept initialization vectors of arbitrary length. Different block cipher modes of operation can have significantly different performance and efficiency characteristics, even when used with the same block cipher. GCM can take full advantage of parallel processing and implementing GCM can make efficient use of an instruction pipeline or

3038-556: Is an example fast (if the AES instruction set is present), key-committing AEAD. It is possible to add key-commitment to an existing AEAD scheme. The plaintext is first encrypted, then a MAC is produced based on the resulting ciphertext. The ciphertext and its MAC are sent together. ETM is the standard method according to ISO/IEC 19772:2009. It is the only method which can reach the highest definition of security in AE, but this can only be achieved when

Galois/Counter Mode - Misplaced Pages Continue

3136-483: Is based on successful decryption of a message that uses a password-based key, Mallory's ability to craft a single message that would be successfully decrypted using 1000 different keys associated with weak , and thus known to her, potential passwords, can speed up her search for passwords by a factor of almost 1000. For this dictionary attack to succeed, Mallory also needs an ability to distinguish successful decryption by Alice from an unsuccessful one, due, for example, to

3234-476: Is defined by the polynomial The authentication tag is constructed by feeding blocks of data into the GHASH function and encrypting the result. This GHASH function is defined by where H = E k (0) is the hash key , a string of 128 zero bits encrypted using the block cipher , A is data which is only authenticated (not encrypted), C is the ciphertext , m is the number of 128-bit blocks in A (rounded up), n

3332-655: Is drawn in red in the figure to the right. Newton's method is used to find the largest zero of this polynomial with an initial guess of 7. The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by ( x − 3 ) {\displaystyle (x-3)} to obtain p 4 ( x ) = x 4 + 14 x 3 + 47 x 2 − 38 x − 240 {\displaystyle p_{4}(x)=x^{4}+14x^{3}+47x^{2}-38x-240} which

3430-458: Is further reduced to p 2 ( x ) = x 2 + 13 x + 40 {\displaystyle p_{2}(x)=x^{2}+13x+40} which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing p 2 ( x ) {\displaystyle p_{2}(x)} and solving

3528-611: Is included in the NSA Suite B Cryptography and its latest replacement in 2018 Commercial National Security Algorithm (CNSA) suite. GCM mode is used in the SoftEther VPN server and client, as well as OpenVPN since version 2.4. GCM requires one block cipher operation and one 128-bit multiplication in the Galois field per each block (128 bit) of encrypted and authenticated data. The block cipher operations are easily pipelined or parallelized;

3626-401: Is increased by exploiting instruction-level parallelism by interleaving operations. This process is called function stitching, and while in principle it can be applied to any combination of cryptographic algorithms, GCM is especially suitable. Manley and Gregg show the ease of optimizing when using function stitching with GCM. They present a program generator that takes an annotated C version of

3724-411: Is limited to encrypting 2 − 256 bits of plain text (64 GiB). NIST Special Publication 800-38D includes guidelines for initialization vector selection and limits the number of possible initialization vector values for a single key. As the security assurance of GCM degrades with more data being processed using the same key, the total number of blocks of plaintext and AD protected during the lifetime of

3822-383: Is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials. The algorithm is based on Horner's rule , in which a polynomial is written in nested form :

3920-408: Is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations. Alexander Ostrowski proved in 1954 that the number of additions required is minimal. Victor Pan proved in 1966 that the number of multiplications is minimal. However, when x {\displaystyle x} is a matrix, Horner's method is not optimal . This assumes that

4018-457: Is optimal, since there are polynomials of degree n that cannot be evaluated with fewer arithmetic operations. Alternatively, Horner's method and Horner–Ruffini method also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the Newton–Raphson method made more efficient for hand calculation by application of Horner's rule. It

Galois/Counter Mode - Misplaced Pages Continue

4116-438: Is particularly fast on processors supporting a single-instruction shift-and-addition-accumulate. Compared to a C floating-point library, Horner's method sacrifices some accuracy, however it is nominally 13 times faster (16 times faster when the " canonical signed digit " (CSD) form is used) and uses only 20% of the code space. Horner's method can be used to convert between different positional numeral systems – in which case x

4214-1033: Is repeatedly factored out. In this binary numeral system (base 2), x = 2 {\displaystyle x=2} , so powers of 2 are repeatedly factored out. For example, to find the product of two numbers (0.15625) and m : ( 0.15625 ) m = ( 0.00101 b ) m = ( 2 − 3 + 2 − 5 ) m = ( 2 − 3 ) m + ( 2 − 5 ) m = 2 − 3 ( m + ( 2 − 2 ) m ) = 2 − 3 ( m + 2 − 2 ( m ) ) . {\displaystyle {\begin{aligned}(0.15625)m&=(0.00101_{b})m=\left(2^{-3}+2^{-5}\right)m=\left(2^{-3})m+(2^{-5}\right)m\\&=2^{-3}\left(m+\left(2^{-2}\right)m\right)=2^{-3}\left(m+2^{-2}(m)\right).\end{aligned}}} To find

4312-419: Is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow. Horner's method is now used to obtain p 3 ( x ) = x 3 + 16 x 2 + 79 x + 120 {\displaystyle p_{3}(x)=x^{3}+16x^{2}+79x+120} which is shown in green and found to have a zero at −3. This polynomial

4410-439: Is that all of the operations are sequentially dependent , so it is not possible to take advantage of instruction level parallelism on modern computers. In most applications where the efficiency of polynomial evaluation matters, many low-order polynomials are evaluated simultaneously (for each pixel or polygon in computer graphics, or for each grid square in a numerical simulation), so it is not necessary to find parallelism within

4508-417: Is the header of a network packet that contains its destination address. To properly route the packet, all intermediate nodes in the message path need to know the destination, but for security reasons they cannot possess the secret key. Schemes that allow associated data provide authenticated encryption with associated data , or AEAD . A typical programming interface for an AE implementation provides

4606-400: Is the base of the number system, and the a i coefficients are the digits of the base- x representation of a given number – and can also be used if x is a matrix , in which case the gain in computational efficiency is even greater. However, for such cases faster methods are known. Using the long division algorithm in combination with Newton's method , it is possible to approximate

4704-404: Is the bit length of the final block of A , u  = len( C ) mod 128 is the bit length of the final block of C , and ∥ {\displaystyle \parallel } denotes concatenation of bit strings. Then X i is defined as: The second form is an efficient iterative algorithm (each X i depends on X i −1 ) produced by applying Horner's method to

4802-432: Is the number of 128-bit blocks in C (rounded up), and the variable X i for i = 0, ..., m + n + 1 is defined below. First, the authenticated text and the cipher text are separately zero-padded to multiples of 128 bits and combined into a single message S i : where len( A ) and len( C ) are the 64-bit representations of the bit lengths of A and C , respectively, v  = len( A ) mod 128

4900-512: Is the sum of the first two rows, divided by 2 . Each entry in the second row is the product of 1 with the third-row entry to the left. The answer is f 1 ( x ) f 2 ( x ) = 2 x 3 − 2 x 2 − x + 1 − 4 2 x − 1 . {\displaystyle {\frac {f_{1}(x)}{f_{2}(x)}}=2x^{3}-2x^{2}-x+1-{\frac {4}{2x-1}}.} Evaluation using

4998-416: Is the value of p ( x 0 ) {\displaystyle p(x_{0})} . To see why this works, the polynomial can be written in the form p ( x ) = a 0 + x ( a 1 + x ( a 2 + x ( a 3 + ⋯ + x ( a n − 1 + x

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5096-456: The linear equation . As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were found. Horner's method can be modified to compute the divided difference ( p ( y ) − p ( x ) ) / ( y − x ) . {\displaystyle (p(y)-p(x))/(y-x).} Given the polynomial (as before) p ( x ) = ∑ i = 0 n

5194-443: The plaintext to produce the ciphertext . Like all counter modes, this is essentially a stream cipher , and so it is essential that a different IV is used for each stream that is encrypted. The ciphertext blocks are considered coefficients of a polynomial which is then evaluated at a key-dependent point H , using finite field arithmetic . The result is then encrypted, producing an authentication tag that can be used to verify

5292-674: The 11th century Song dynasty mathematician Jia Xian ; for example, one method is specifically suited to bi-quintics, of which Qin gives an instance, in keeping with the then Chinese custom of case studies. Yoshio Mikami in Development of Mathematics in China and Japan (Leipzig 1913) wrote: "... who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe ;... We of course don't intend in any way to ascribe Horner's invention to

5390-699: The E&;M approach has not been proved to be strongly unforgeable in itself, it is possible to apply some minor modifications to SSH to make it strongly unforgeable despite the approach. A MAC is produced based on the plaintext, then the plaintext and MAC are together encrypted to produce a ciphertext based on both. The ciphertext (containing an encrypted MAC) is sent. Until TLS 1.2, all available SSL/TLS cipher suites were MtE. MtE has not been proven to be strongly unforgeable in itself. The SSL/TLS implementation has been proven to be strongly unforgeable by Krawczyk who showed that SSL/TLS was, in fact, secure because of

5488-475: The MAC used is "strongly unforgeable". IPSec adopted EtM in 2005. In November 2014, TLS and DTLS received extensions for EtM with RFC   7366 . Various EtM ciphersuites exist for SSHv2 as well (e.g., hmac-sha1-etm@openssh.com ). A MAC is produced based on the plaintext, and the plaintext is encrypted without the MAC. The plaintext's MAC and the ciphertext are sent together. Used in, e.g., SSH . Even though

5586-596: The XMPMUL instruction, which performs XOR multiplication of much larger values, up to 2048 × 2048 bit input values producing a 4096-bit result. These instructions enable fast multiplication over GF(2), and can be used with any field representation. Impressive performance results are published for GCM on a number of platforms. Käsper and Schwabe described a "Faster and Timing-Attack Resistant AES-GCM" that achieves 10.68 cycles per byte AES-GCM authenticated encryption on 64-bit Intel processors. Dai et al. report 3.5 cycles per byte for

5684-657: The above we know that the largest root of this polynomial is 7 so we are able to make an initial guess of 8. Using Newton's method the first zero of 7 is found as shown in black in the figure to the right. Next p ( x ) {\displaystyle p(x)} is divided by ( x − 7 ) {\displaystyle (x-7)} to obtain p 5 ( x ) = x 5 + 11 x 4 + 5 x 3 − 179 x 2 − 126 x + 720 {\displaystyle p_{5}(x)=x^{5}+11x^{4}+5x^{3}-179x^{2}-126x+720} which

5782-433: The adversary chooses a t -bit tag at random, it is expected to be correct for given data with probability measure 2. With GCM, however, an adversary can increase their likelihood of success by choosing tags with n words – the total length of the ciphertext plus any additional authenticated data (AAD) – with probability measure 2 by a factor of n . Although, one must bear in mind that these optimal tags are still dominated by

5880-399: The algorithm's survival measure 1 − n ⋅2 for arbitrarily large t . Moreover, GCM is neither well-suited for use with very short tag-lengths nor very long messages. Ferguson and Saarinen independently described how an attacker can perform optimal attacks against GCM authentication, which meet the lower bound on its security. Ferguson showed that, if n denotes the total number of blocks in

5978-741: The algorithm, it is required that terms with zero-valued coefficients are dropped, so that only binary coefficients equal to one are counted, thus the problem of multiplication or division by zero is not an issue, despite this implication in the factored equation: = d 0 ( m + 2 d 1 d 0 ( m + 2 d 2 d 1 ( m + 2 d 3 d 2 ( m ) ) ) ) . {\displaystyle =d_{0}\left(m+2{\frac {d_{1}}{d_{0}}}\left(m+2{\frac {d_{2}}{d_{1}}}\left(m+2{\frac {d_{3}}{d_{2}}}(m)\right)\right)\right).} The denominators all equal one (or

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6076-625: The authentication assurance is completely lost. Independent of this attack, an adversary may attempt to systematically guess many different tags for a given input to authenticated decryption and thereby increase the probability that one (or more) of them, eventually, will be considered valid. For this reason, the system or protocol that implements GCM should monitor and, if necessary, limit the number of unsuccessful verification attempts for each key. Saarinen described GCM weak keys . This work gives some valuable insights into how polynomial hash-based authentication works. More precisely, this work describes

6174-421: The authors' statement, GCM is unencumbered by patents. GCM is proven secure in the concrete security model . It is secure when it is used with a block cipher that is indistinguishable from a random permutation; however, security depends on choosing a unique initialization vector for every encryption performed with the same key ( see stream cipher attack ). For any given key and initialization vector value, GCM

6272-408: The block size of the encryption function. Padding errors often result in the detectable errors on the recipient's side, which in turn lead to padding oracle attacks, such as Lucky Thirteen . Horner%27s method In mathematics and computer science , Horner's method (or Horner's scheme ) is an algorithm for polynomial evaluation . Although named after William George Horner , this method

6370-401: The ciphertext to recover the plaintext P and can check the tag T to ensure that neither ciphertext nor associated data were tampered with. GCM uses a block cipher with block size 128 bits (commonly AES-128 ) operated in counter mode for encryption, and uses arithmetic in the Galois field GF(2) to compute the authentication tag; hence the name. Galois Message Authentication Code ( GMAC )

6468-879: The consecutive b {\displaystyle b} -values, you start with determining b n {\displaystyle b_{n}} , which is simply equal to a n {\displaystyle a_{n}} . Then you then work recursively using the formula: b n − 1 = a n − 1 + b n x 0 {\displaystyle b_{n-1}=a_{n-1}+b_{n}x_{0}} till you arrive at b 0 {\displaystyle b_{0}} . Evaluate f ( x ) = 2 x 3 − 6 x 2 + 2 x − 1 {\displaystyle f(x)=2x^{3}-6x^{2}+2x-1} for x = 3 {\displaystyle x=3} . We use synthetic division as follows: The entries in

6566-506: The derivative of p ( x ) {\displaystyle p(x)} . Horner's paper, titled "A new method of solving numerical equations of all orders, by continuous approximation", was read before the Royal Society of London, at its meeting on July 1, 1819, with a sequel in 1823. Horner's paper in Part II of Philosophical Transactions of the Royal Society of London for 1819

6664-513: The divided difference is subject to less round-off error than evaluating p ( x ) {\displaystyle p(x)} and p ( y ) {\displaystyle p(y)} separately, particularly when x ≈ y {\displaystyle x\approx y} . Substituting y = x {\displaystyle y=x} in this method gives d 1 = p ′ ( x ) {\displaystyle d_{1}=p'(x)} ,

6762-452: The encoding (the input to the GHASH function), then there is a method of constructing a targeted ciphertext forgery that is expected to succeed with a probability of approximately n ⋅2. If the tag length t is shorter than 128, then each successful forgery in this attack increases the probability that subsequent targeted forgeries will succeed, and leaks information about the hash subkey,  H . Eventually, H may be compromised entirely and

6860-494: The encoding used alongside the MtE mechanism. However, Krawczyk's proof contains flawed assumptions about the randomness of the initialization vector (IV). The 2011 BEAST attack exploited the non-random chained IV and broke all CBC algorithms in TLS 1.0 and under. In addition, deeper analysis of SSL/TLS modeled the protection as MAC-then-pad-then-encrypt, i.e. the plaintext is first padded to

6958-419: The encrypted message includes an authentication tag that the sender can calculate only while possessing the secret key ). Examples of encryption modes that provide AE are GCM , CCM . Many (but not all) AE schemes allow the message to contain "associated data" (AD) which is not made confidential, but its integrity is protected (i.e., it is readable, but tampering with it will be detected). A typical example

7056-532: The entries in the third row. So, synthetic division (which was actually invented and published by Ruffini 10 years before Horner's publication) is easier to use; it can be shown to be equivalent to Horner's method. As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of f ( x ) {\displaystyle f(x)} on division by x − 3 {\displaystyle x-3} . The remainder

7154-455: The first. Only the final X m + n +1 remains an output. If it is necessary to parallelize the hash computation, this can be done by interleaving k times: If the length of the IV is not 96, the GHASH function is used to calculate Counter 0 : GCM was designed by John Viega and David A. McGrew to be an improvement to Carter–Wegman counter mode (CWC mode). In November 2007, NIST announced

7252-424: The following functions: The header part is intended to provide authenticity and integrity protection for networking or storage metadata for which confidentiality is unnecessary, but authenticity is desired. The need for authenticated encryption emerged from the observation that securely combining separate confidentiality and authentication block cipher operation modes could be error prone and difficult. This

7350-972: The following two steps: These two steps are repeated until all real zeros are found for the polynomial. If the approximated zeros are not precise enough, the obtained values can be used as initial guesses for Newton's method but using the full polynomial rather than the reduced polynomials. Consider the polynomial p 6 ( x ) = ( x + 8 ) ( x + 5 ) ( x + 3 ) ( x − 2 ) ( x − 3 ) ( x − 7 ) {\displaystyle p_{6}(x)=(x+8)(x+5)(x+3)(x-2)(x-3)(x-7)} which can be expanded to p 6 ( x ) = x 6 + 4 x 5 − 72 x 4 − 214 x 3 + 1127 x 2 + 1602 x − 5040. {\displaystyle p_{6}(x)=x^{6}+4x^{5}-72x^{4}-214x^{3}+1127x^{2}+1602x-5040.} From

7448-406: The inner summations may be evaluated using separate parallel instances of Horner's method. This requires slightly more operations than the basic Horner's method, but allows k -way SIMD execution of most of them. Modern compilers generally evaluate polynomials this way when advantageous, although for floating-point calculations this requires enabling (unsafe) reassociative math . Horner's method

7546-512: The integrity of both the associated data and the confidential information in a message. AD is useful, for example, in network packets where the header should be visible for routing , but the payload needs to be confidential, and both need integrity and authenticity . The notion of AEAD was formalized by Rogaway (2002). AE was originally designed primarily to provide the ciphertext integrity: successful validation of an authentication tag by Alice using her symmetric key K A indicates that

7644-446: The integrity of the data. The encrypted text then contains the IV, ciphertext, and authentication tag. GCM combines the well-known counter mode of encryption with the new Galois mode of authentication. The key feature is the ease of parallel computation of the Galois field multiplication used for authentication. This feature permits higher throughput than encryption algorithms, like CBC , which use chaining modes. The GF(2) field used

7742-416: The message was not tampered with by an adversary Mallory that does not possess the K A . The AE schemes usually do not provide the key commitment , a guarantee that the decryption would fail for any other key. As of 2021, most existing AE schemes (including the very popular GCM) allow some messages to be decoded without an error using more than just the (correct) K A ; while their plaintext decoded using

7840-503: The method in Horner's 1819 paper differs from what afterwards became known as "Horner's method" and that in consequence the priority for this method should go to Holdred (1820). Unlike his English contemporaries, Horner drew on the Continental literature, notably the work of Arbogast . Horner is also known to have made a close reading of John Bonneycastle's book on algebra, though he neglected

7938-570: The monomial form of a degree n {\displaystyle n} polynomial requires at most n {\displaystyle n} additions and ( n 2 + n ) / 2 {\displaystyle (n^{2}+n)/2} multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. The cost can be reduced to n {\displaystyle n} additions and 2 n − 1 {\displaystyle 2n-1} multiplications by evaluating

8036-533: The multiplication operations are easily pipelined and can be parallelized with some modest effort (either by parallelizing the actual operation, by adapting Horner's method per the original NIST submission, or both). Intel has added the PCLMULQDQ instruction, highlighting its use for GCM. In 2011, SPARC added the XMULX and XMULXHI instructions, which also perform 64 × 64 bit carry-less multiplication . In 2015, SPARC added

8134-432: The number of bits of x {\displaystyle x} . Alternatively, Horner's method can be computed with n {\displaystyle n} fused multiply–adds . Horner's method can also be extended to evaluate the first k {\displaystyle k} derivatives of the polynomial with k n {\displaystyle kn} additions and multiplications. Horner's method

8232-594: The polynomial is evaluated in monomial form and no preconditioning of the representation is allowed, which makes sense if the polynomial is evaluated only once. However, if preconditioning is allowed and the polynomial is to be evaluated many times, then faster algorithms are possible . They involve a transformation of the representation of the polynomial. In general, a degree- n {\displaystyle n} polynomial can be evaluated using only ⌊ n /2 ⌋ +2 multiplications and n {\displaystyle n} additions. A disadvantage of Horner's rule

8330-795: The powers of x {\displaystyle x} by iteration. If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately 2 n {\displaystyle 2n} times the number of bits of x {\displaystyle x} : the evaluated polynomial has approximate magnitude x n {\displaystyle x^{n}} , and one must also store x n {\displaystyle x^{n}} itself. By contrast, Horner's method requires only n {\displaystyle n} additions and n {\displaystyle n} multiplications, and its storage requirements are only n {\displaystyle n} times

8428-759: The product of two binary numbers d and m : In general, for a binary number with bit values ( d 3 d 2 d 1 d 0 {\displaystyle d_{3}d_{2}d_{1}d_{0}} ) the product is ( d 3 2 3 + d 2 2 2 + d 1 2 1 + d 0 2 0 ) m = d 3 2 3 m + d 2 2 2 m + d 1 2 1 m + d 0 2 0 m . {\displaystyle (d_{3}2^{3}+d_{2}2^{2}+d_{1}2^{1}+d_{0}2^{0})m=d_{3}2^{3}m+d_{2}2^{2}m+d_{1}2^{1}m+d_{0}2^{0}m.} At this stage in

8526-629: The real roots of a polynomial. The algorithm works as follows. Given a polynomial p n ( x ) {\displaystyle p_{n}(x)} of degree n {\displaystyle n} with zeros z n < z n − 1 < ⋯ < z 1 , {\displaystyle z_{n}<z_{n-1}<\cdots <z_{1},} make some initial guess x 0 {\displaystyle x_{0}} such that z 1 < x 0 {\displaystyle z_{1}<x_{0}} . Now iterate

8624-525: The release of NIST Special Publication 800-38D Recommendation for Block Cipher Modes of Operation: Galois/Counter Mode (GCM) and GMAC making GCM and GMAC official standards. GCM mode is used in the IEEE 802.1AE (MACsec) Ethernet security, WPA3-Enterprise Wifi security protocol, IEEE 802.11ad (also dubbed WiGig ), ANSI ( INCITS ) Fibre Channel Security Protocols (FC-SP), IEEE P1619 .1 tape storage, IETF IPsec standards, SSH , TLS 1.2 and TLS 1.3. AES-GCM

8722-486: The same algorithm when using Intel's AES-NI and PCLMULQDQ instructions. Shay Gueron and Vlad Krasnov achieved 2.47 cycles per byte on the 3rd generation Intel processors. Appropriate patches were prepared for the OpenSSL and NSS libraries. When both authentication and encryption need to be performed on a message, a software implementation can achieve speed gains by overlapping the execution of those operations. Performance

8820-688: The summation can be broken into k parts: p ( x ) = ∑ i = 0 n a i x i = ∑ j = 0 k − 1 x j ∑ i = 0 ⌊ n / k ⌋ a k i + j x k i = ∑ j = 0 k − 1 x j p j ( x k ) {\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}=\sum _{j=0}^{k-1}x^{j}\sum _{i=0}^{\lfloor n/k\rfloor }a_{ki+j}x^{ki}=\sum _{j=0}^{k-1}x^{j}p_{j}(x^{k})} where

8918-744: The term is absent), so this reduces to = d 0 ( m + 2 d 1 ( m + 2 d 2 ( m + 2 d 3 ( m ) ) ) ) , {\displaystyle =d_{0}(m+2{d_{1}}(m+2{d_{2}}(m+2{d_{3}}(m)))),} or equivalently (as consistent with the "method" described above) = d 3 ( m + 2 − 1 d 2 ( m + 2 − 1 d 1 ( m + d 0 ( m ) ) ) ) . {\displaystyle =d_{3}(m+2^{-1}{d_{2}}(m+2^{-1}{d_{1}}(m+{d_{0}}(m)))).} In binary (base-2) math, multiplication by

9016-429: The third row are the sum of those in the first two. Each entry in the second row is the product of the x -value ( 3 in this example) with the third-row entry immediately to the left. The entries in the first row are the coefficients of the polynomial to be evaluated. Then the remainder of f ( x ) {\displaystyle f(x)} on division by x − 3 {\displaystyle x-3}

9114-567: The use of these two tag lengths constrains the length of the input data and the lifetime of the key. Appendix C in NIST ;SP 800-38D provides guidance for these constraints (for example, if t = 32 and the maximal packet size is 2 bytes, the authentication decryption function should be invoked no more than 2 times; if t = 64 and the maximal packet size is 2 bytes, the authentication decryption function should be invoked no more than 2 times). Like with any message authentication code, if

9212-529: The work of Paolo Ruffini . Although Horner is credited with making the method accessible and practical, it was known long before Horner. In reverse chronological order, Horner's method was already known to: Qin Jiushao , in his Shu Shu Jiu Zhang ( Mathematical Treatise in Nine Sections ; 1247), presents a portfolio of methods of Horner-type for solving polynomial equations, which was based on earlier works of

9310-460: Was announced to encourage design of authenticated encryption modes. In 2015, ChaCha20-Poly1305 is added as an alternative AE construction to GCM in IETF protocols. Authenticated encryption with associated data (AEAD) is a variant of AE that allows the message to include "associated data" (AD, additional non-confidential information, a.k.a. "additional authenticated data", AAD). A recipient can check

9408-1147: Was confirmed by a number of practical attacks introduced into production protocols and applications by incorrect implementation, or lack of authentication. Around the year 2000, a number of efforts evolved around the notion of standardizing modes that ensured correct implementation. In particular, strong interest in possibly secure modes was sparked by the publication of Charanjit Jutla 's integrity-aware CBC and integrity-aware parallelizable , IAPM, modes in 2000 (see OCB and chronology ). Six different authenticated encryption modes (namely offset codebook mode 2.0 , OCB   2.0; Key Wrap ; counter with CBC-MAC , CCM; encrypt then authenticate then translate , EAX; encrypt-then-MAC , EtM; and Galois/counter mode , GCM) have been standardized in ISO/IEC 19772:2009. More authenticated encryption methods were developed in response to NIST solicitation. Sponge functions can be used in duplex mode to provide authenticated encryption. Bellare and Namprempre (2000) analyzed three compositions of encryption and MAC primitives, and demonstrated that encrypting

9506-487: Was warmly and expansively welcomed by a reviewer in the issue of The Monthly Review: or, Literary Journal for April, 1820; in comparison, a technical paper by Charles Babbage is dismissed curtly in this review. The sequence of reviews in The Monthly Review for September, 1821, concludes that Holdred was the first person to discover a direct and general practical solution of numerical equations. Fuller showed that

9604-522: Was widely used until computers came into general use around 1970. Given the polynomial p ( x ) = ∑ i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ⋯ + a n x n , {\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n},} where

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