ISSN: 2641-3086
Trends in Computer Science and Information Technology
Review Article       Open Access      Peer-Reviewed

Redefining GCM’s resistance to cryptanalysis with offset mechanisms

Muhammad Faisal Nawaz1 and Yasir Nawaz2*

1University of Lahore, Pakistan
2Shanghai Jiao Tong University, China
*Corresponding author: Yasir Nawaz, Shanghai Jiao Tong University, China, E-mail:,
Received: 19 February, 2024 | Accepted: 29 March, 2024 | Published: 30 March, 2024
Keywords: Authenticated encryption; Galois/Counter Mode (GCM); Differential cryptanalysis offset mechanism; Cryptographic security

Cite this as

Nawaz MF, Nawaz Y (2024) Redefining GCM’s resistance to cryptanalysis with offset mechanisms. Trends Comput Sci Inf Technol 9(1): 042-051. DOI: 10.17352/tcsit.000079

Copyright License

© 2024 Nawaz MF, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

The research paper proposes an enhancement to the Galois/Counter Mode (GCM) of authenticated encryption by introducing an “offset” mechanism. This modification aims to improve privacy and resist differential cryptanalysis without significantly impacting the mode’s efficiency and simplicity. The improved GCM maintains its original features, such as minimal block cipher invocations, the use of a single cryptographic key, and efficient offset computation. It provides a detailed analysis of the operational framework, including the integration and calculation of offsets in encryption and decryption processes. By complicating the predictability of differential cryptanalysis through unique offsets, the paper asserts this enhancement significantly increases GCM’s security within a concrete security model. The discussion emphasizes the benefits of the offset-enhanced GCM over other modes, highlighting its suitability for high-speed, parallelizable cryptographic applications while bolstering resistance against cryptanalytic attacks.


Authenticated encryption overview

Block cipher mode of operation is a scrutinized cryptographic primitive for secure encryption and decryption that ensures privacy, authenticity, and authenticated encryption [1]. Authenticated encryption is a term that simultaneously provides confidentiality and authenticity to the data. Every cryptosystem requires both forms of security, but until relatively recently confidentiality and authenticity have been designed separately. Now, Authenticated encryption is implemented using a block cipher mode of operation structure. Recently, many authenticated encryption modes have been proposed [2,3]. The first authenticated encryption mode was IAPM (Integrity-Aware Parallelizable) mode proposed by Jutla’s [4]. The OCB (offset Code Book) mode which refine one of IAPM [5,6]. The OCB2, and OCB3 are refine version of OCB mode [7,8]. All of these are parallelized authenticated encryption mode suitable for high-speed cryptosystem [9,10]. There are some others motivated work combined with Counter mode and CBC − MAC is CCM mode uses only one key, however, it is not a suitable for high speed authenticated encryption because CBC − MAC is not parallelizable [11]. The other mode similar to CCM is EAX mode, combined Counter mode with OMAC [2,12]. The OMAC is not parallelized, so, EAX is not high speed authenticated encryption mode, but it refines some properties of CCM mode. Another authentication encryption mode is cwc combined with Counter mode with MAC based on the universal hash function over GF(2127 -1), But it’s relatively expensive to implement [13,14]. There is some authenticated encryption mode’s ability to authenticate with associated which simultaneously assures the confidentiality and authenticity of data. The method is sometimes termed for AHED (authenticated encryption with associated data) [15]. The CCM mode and GCM mode both have facilities for AHED and increase usability [16-18]. The Galois/Counter mode is recommended by the National Institute of Standards and Technology (NIST) and most favorite than CCM due to parallelizability [19,20].

Galois/Counter Mode (GCM) is a block cipher mode of operation designed to meet the need for confidentiality and authenticity of data and use universal hashing over a binary Galois field. It is implemented in many cryptosystems to achieve high speeds with low latency and low cost. Its design is supported by a well-understood theoretical foundation. There is an enthralling need for a mode of operation that can efficiently provide parallel authenticated encryption. The modes of operation must admit pipelined and parallelized construction and have high data rates. The Counter mode meets those requirements and has become a mode for high-speed cryptosystems [21,22]. However, the Counter mode provide only confidentiality not message authentication. So GCM incorporates with Counter mode and builds on it by adding a Message Authentication Code (MAC) based on universal hashing provide message authentication that can keep up with our cipher [23]. It uses Polynomial hashing in the finite field GF(2n) [24]. The multiplication in GF(2n) can be efficiently implemented using XOR and shift operation. Additionally, GCM also has useful properties, it can be used as an incremental MAC and stand-alone MAC. These properties of GCM unique among all of the proposed authenticated encryption modes.

The GCM associated Counter mode, changes the inputs bits of underlying block cipher serially, and it is well known that the successive block of Counter has small hamming difference underlying the block cipher, this led to concern that AdvErsary can obtain many plaintext pairs with a known small plaintext difference, which would facilitate the differential cryptanalysis [25]. It is the responsibility of the mode to compensate for the weak block cipher. Our work refines the privacy property of GCM by using an extra input value that is an offset, such that each offset input is unique.

The principal characteristics of GCM offset retain the same (like: fully parallelizable) only add a small overhead compared to conventional GCM mode. Now, the GCM offset combines the following features:

Arbitrary − message length: The GCM encrypt and authenticate a nonempty any length of string M ∈ {0,1}* using |M|/n + 1 block cipher invocations. The message length (|M|) need to be a multiple of n.

Minimal requirement on counter: Like another authenticated encryption mode GCM require a nonce as counter. The counter value must be non-repeating (each block cipher chooses a new counter value for every message block with restriction no counter value used twice).

offset calculation: We need a sequence of offset as with [26,27]. The offset value generate in a particularly cheap way, and each offset value need just a machine cycle.

Single key: The GCM offset used a single block cipher key. All the block cipher invocations are keyed by this one key.

The paramount contribution of this paper is the introduction of an “offset” mechanism to the GCM of authenticated encryption, aimed at significantly enhancing its resistance to differential cryptanalysis without detracting from its efficiency or simplicity. By integrating unique offsets in the encryption and decryption processes, this enhancement complicates the predictability upon which differential cryptanalysis relies, thereby strengthening GCM’s security posture within a robust security model. Through comprehensive analysis and discussion, we demonstrate the practical application of this offset-enhanced GCM in modern cryptographic systems, emphasizing its minimal overhead and retained efficiency. This advancement not only fortifies GCM against sophisticated cryptanalytic attacks but also underscores the feasibility of such an approach in high-speed, parallelizable cryptographic operations, marking a significant stride forward in the domain of authenticated encryption.



Let there are two integers a and b, if a £ b, then it means {a,a + 1, …,b}. If i > 0 is an integer, then ntz(i) is the trailing 0 - bits in the binary representation of . A string {0,1} represent the set of binary numbers and a string {0,1}* denote the set of all strings. The set {0,1}n denote all the strings of length n. If there is no element in the string, then it’s called the empty string denoted e. A||B represents the concatenation of set A and B where A,B Î {0,1}*. If A ¹ e then firstbit (A) represent the first bit of A, in such a way lastbit (A) denote the last bit of the A. Let I and n be two integers then 0i and 1i represent the string of 0¢s and 1¢s respectively. If A Î {0,1}* then |A| represent the bit length of A while ||A||n = max{1, é|A|/nù} represent the length of A in n - bit block. Let A Î {0,1}* and t Î [0..|A|] then A[firstbit t] denote first t bit of A and A[lastbit t] denote the last tbit of A respectively. If A,B Î {0,1}* then AB is the bitwise XOR of firstbit (A) and firstbit (B), where |A| = |B|. If A = an-1a1a0 Î{0,1}nthen stir2num is the number . If a Î [0.. 2n - 1] then num2strn(a) is n - bit string A such that stir2num(A) = a. lenn(A) = num2strn(|A|). If A = an-1 an-2…a1a0 Î {0,1}n then A ≪ 1 is the n - bit string an-2…a1a0 0 which is the left shift of A by one bit, while A ≫ 1 is the n - bit string 0an-1 an-2…a1 which is the right shift of A by 1 - bit. The plaintext message M partitioned into m1m2…mn and |m1| = n for 1 < I < n. We partition C into c1c2…cn, where C refers to the ciphertext resulting from the encryption process. The partitioning of  into multiple blocks c1c2…cn facilitates the processing of the ciphertext in blocks, aligning with the block cipher mode of operation used by GCM. This is crucial for both encrypting the plaintext message in blocks and subsequently or generating or verifying the authentication tag, which ensures data integrity and authenticity.

The field with 2n points

Lets GF(2n) represent a field with 2n point [28]. We interchangeably think of a point a in GF(2n) in any of the following ways:

  1. As an abstract point in a field
  2. As an n − bit string an-1 …a1a0 Î {0,1}n
  3. As a formal polynomial a(x) = anxn-1 + a1x + a0 with binary coefficients.
  4. As an integer between 0 and 2n − 1, where the string a Î {0,1}n corresponding to the number str2num(a).

We write a(x) instead of a if we wish to emphasize that we are thinking of a as a polynomial. We take XOR to add two points in GF(2n), and for the multiplication of two points, we fix an irreducible polynomial pn (x) having binary coefficients and degree n. For n = 128 the indicated polynomial is p128 (x) = x128 + x7 + x2 + x + 1. A few other pn (x) values are x64 + x4 + x3 + x + 1 and x96 + x10 + x9 + x6 + 1 and x160 + x5 + x3 + x2 + 1 and x192 + x7 + x2 + x + 1 and x224 + x9 + x8 + x3 + 1 and x256 + x10 + x5 + x2 + 1. To multiply a,b GF(2n) represent a and b as polynomial a(x) = an-1 xn-1 + ⋅⋅⋅ a1x + a0+ a1x + a0 and b(x) = bn-1 xn-1 + ⋅⋅ b1x + b0+ b1x + b0, form product c(x) over GF(2). When dividing c(x) by pn (x) it takes a reminder. The multiplication of x and a ∈ {0,1}n computationally simple. When n = 128, in this case multiplying a = an-1 + ⋅⋅⋅ + a1 + a0 by x give an-1 xn + an-2 xn-1 + ⋅⋅ + a1x2 + a0x. If first bit of a is 0, then a. x << 1. When first bit of a is 1, then add x128 to << 1. Since p128 (x) = x128 + x7 + x2 + x + 1 = 0, in such a way x128 + x7 + x2 + x + 1. So, add x128 means to XOR by 0120 10000111. So, when n = 128,

a.x={ a1,  if firstbit( a )=0 (a1) 0 120 10000111,  if firstbit( a )=1        (1) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7FB6@

On the other hand, in the case of divide a ∈ {0,1}n by x, if the last bit of a is 0, then a.x-1 is a >>1. In such a way, if the last bit of a is 1, then XOR to a >>1 the value x-1. Since x128 + x7 + x2 + x + 1 we know that x-1 = x127 + x6 + x + 1 = 10120 1000011. So, when n = 128,

a. x 1 ={ a1,  if lastbit( a )=0 (a1) 10 120 1000011,  if lastbit( a )=1       (2) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8039@

If L ∈ {0,1}n and i ≥ −1, then L(i) = L. xi. so, we can compute from L the values L(−1), L(0), L(1), …, L(µ), where µ is a small number.

Gray codes

Gray code is a sequence of γ l =( γ 0 l γ 1 l γ   2 l 1 l ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiabeo7aNLqba+aadaahaaWcbeqaaKqzGeWdbiaadYgaaaGaeyypa0Jaaiikaiabeo7aNLqba+aadaqhaaWcbaqcLbsapeGaaGimaaWCPAqaaKqzGeWdbiaadYgaaaGaeq4Sdcwcfa4damaaDaaaleaajugib8qacaaIXaaal8aabaqcLbsapeGaamiBaaaacqGHMacVcqaHZoWzjuaGpaWaa0baaSqaaKqzGeWdbiaabckacaaIYaqcfa4damaaCaaameqabaqcLbsapeGaamiBaaaacqGHsislcaaIXaaal8aabaqcLbsapeGaamiBaaaacaGGPaaaaa@5414@ of {0,1}l, where l ≥ 1 and successive points just one bit differ. When n is a fixed number GCM use canonical gray code γ = γl from γl = (0 1). So, for l > 0,

γ l+1 =(0 γ 0  l  0 γ 1 l 0 γ   2 l 2 l  0 γ   2 l 1 l  1 γ   2 l 1 l  1 γ   2 l 2 l  1 γ 1 l  1 γ 0 l )           (3) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@98D5@

Thus, γ is a gray code, for computing successive points,

1i 2 n 1,   γ i = γ i1 ( 0 n1 1ntz(i)) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruqtLjNCPDxzHrhALjharmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaiaabiWacmaadaGabiaaeaGaauaaaOqaaKqzGeaeaaaaaaaaa8qacaaIXaGaeyizImQaamyAaiabgsMiJkaaikdajuaGpaWaaWbaaSqabeaajugib8qacaWGUbaaaiabgkHiTiaaigdacaGGSaGaaeiiaiaabckacqaHZoWzjuaGpaWaaSbaaSqaaKqzGeWdbiaadMgaaSWdaeqaaKqzGeWdbiabg2da9iabeo7aNLqba+aadaWgaaWcbaqcLbsapeGaamyAaiabgkHiTiaaigdaaSWdaeqaaKqzGeWdbiabgwPiflaacIcacaaIWaqcfa4damaaCaaaleqabaqcLbsapeGaamOBaiabgkHiTiaaigdaaaGaaGymaiablQMi9iaad6gacaWG0bGaamOEaiaacIcacaWGPbGaaiykaiaacMcaaaa@5FF8@

Let L ∈ {0,1}n and γ1. L, 2. L, 3. L,…, γm. L are considered the problem of successive forming strings. Thus, γ1. L = 1. L = L. Since γ1 = γn-1 ⨁ (0n-1 1 << ntz(i)) we know that,

γ i .L= γ i1 ( 0 n1 1ntz( i ) ).L       (4) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiabeo7aNLqba+aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqabaqcLbsapeGaaiOlaiaabYeacqGH9aqpcqaHZoWzjuaGpaWaaSbaaSqaaKqzGeWdbiaadMgacqGHsislcaaIXaaal8aabeaajugib8qacqGHvksXjuaGdaqadaGcpaqaaKqzGeWdbiaaicdajuaGpaWaaWbaaSqabeaajugib8qacaWGUbGaeyOeI0IaaGymaaaacaaIXaGaeSOAI0JaamOBaiaadshacaWG6bqcfa4aaeWaaOWdaeaajugib8qacaWGPbaakiaawIcacaGLPaaaaiaawIcacaGLPaaajugibiaac6cacaqGmbGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG0aGaaeykaaaa@5F2B@

γ i .L=( γ i1 .L)( 0 n1 1ntz(i))       (5) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiabeo7aNLqba+aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqabaqcLbsapeGaaiOlaiaabYeacqGH9aqpcaGGOaGaeq4SdCwcfa4damaaBaaaleaajugib8qacaWGPbGaeyOeI0IaaGymaaWcpaqabaqcLbsapeGaaiOlaiaabYeacaGGPaGaeyyLIuSaaiikaiaaicdajuaGpaWaaWbaaSqabeaajugib8qacaWGUbGaeyOeI0IaaGymaaaacaaIXaGaeSOAI0JaamOBaiaadshacaWG6bGaaiikaiaadMgacaGGPaGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeynaiaabMcaaaa@5D00@

γ i .L=( γ i1 .L)(L. x ntz( i ) )      (6) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiabeo7aNLqba+aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqabaqcLbsapeGaaiOlaiaabYeacqGH9aqpcaGGOaGaeq4SdCwcfa4damaaBaaaleaajugib8qacaWGPbGaeyOeI0IaaGymaaWcpaqabaqcLbsapeGaaiOlaiaabYeacaGGPaGaeyyLIuSaaiikaiaabYeacaGGUaGaaeiEaKqba+aadaahaaWcbeqaaKqzGeWdbiaad6gacaWG0bGaamOEaKqbaoaabmaal8aabaqcLbsapeGaamyAaaWccaGLOaGaayzkaaaaaKqzGeGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG2aGaaeykaaaa@5B81@

γ i .L=( γ i1 .L)L(ntz( i ))      (7) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiabeo7aNLqba+aadaWgaaWcbaqcLbsapeGaamyAaaWcpaqabaqcLbsapeGaaiOlaiaabYeacqGH9aqpcaGGOaGaeq4SdCwcfa4damaaBaaaleaajugib8qacaWGPbGaeyOeI0IaaGymaaWcpaqabaqcLbsapeGaaiOlaiaabYeacaGGPaGaeyyLIuSaaeitaiaacIcacaWGUbGaamiDaiaadQhajuaGdaqadaGcpaqaaKqzGeWdbiaadMgaaOGaayjkaiaawMcaaKqzGeGaaiykaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG3aGaaeykaaaa@586A@

The th word can be obtained by xoring L(ntz(i) with previous words. The ith word would be obtained in the same way for I ³ 2 e.i g1. L ⨁ R, g2. L ⨁ R The first word in the sequence is L ⨁ R instead of L.

GCM offset

This section describes the complete definition of GCM with additional input offset for 128 − bit block ciphers. Generally, GCM encryption have the following inputs, each of which is a bit string:

  • A plaintext M, partitioned into m1m2…mn and length of each message block exact multiple of a block cipher.
  • Authenticated data, which is denoted as AD. This data just authenticates but does not encrypt.
  • Secret key K, whose length is multiple of a block cipher.
  • The Counter value, all that is expected of the Counter is that it be as a nonce. it is not required to be random or unpredictable.
  • The offset (zi) for each block cipher, such that each zi is unique.

Each different value of Counter produces a different set of zi. Thus each offset XOR with the corresponding counter value produces an unpredictable value (comparable, to nonrandom and predictable nonce-related counter value) for the underlying block cipher. The calculation of zi is summarized in the following equations.

L 0 =L= E k ( 0 n ) where  0 n  is consist of n zero bits.     (7) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@70C7@

R= E k ( ctr+iL )     (8) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadkfacqGH9aqpcaWGfbqcfa4damaaBaaaleaajugib8qacaWGRbaal8aabeaajuaGpeWaaeWaaOWdaeaajugib8qacaWGJbGaamiDaiaadkhacqGHRaWkcaWGPbGaeyyLIuSaamitaaGccaGLOaGaayzkaaqcfaOaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeioaiaabMcaaaa@4BF3@

L i =2. L i1  1im     (9) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadYeajuaGpaWaaSbaaSqaaKqzGeWdbiaadMgaaSWdaeqaaKqzGeWdbiabg2da9iaaikdacaGGUaGaamitaKqba+aadaWgaaWcbaqcLbsapeGaamyAaiabgkHiTiaaigdaaSWdaeqaaKqzGeWdbiaabckacaaIXaGaeyizImQaamyAaiabgsMiJkaad2gacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG5aGaaeykaaaa@4EE7@

Z( 1 )=RL      (10) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadQfajuaGdaqadaGcpaqaaKqzGeWdbiaaigdaaOGaayjkaiaawMcaaKqzGeGaeyypa0JaamOuaiabgwPiflaadYeacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabcdacaqGPaaaaa@46E9@

z i = z i1 L(ntz(i)) 1im     (11) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadQhajuaGpaWaaSbaaSqaaKqzGeWdbiaadMgaaSWdaeqaaKqzGeWdbiabg2da9iaadQhajuaGpaWaaSbaaSqaaKqzGeWdbiaadMgacqGHsislcaaIXaaal8aabeaajugib8qacqGHvksXcaWGmbGaaiikaiaad6gacaWG0bGaamOEaiaacIcacaWGPbGaaiykaiaacMcacaqGGcGaaGymaiabgsMiJkaadMgacqGHKjYOcaWGTbGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabgdacaqGPaaaaa@57E5@

offset calculation

Initialization of L0: The document describes that the initial value L0 is derived by encrypting a block of n zero bits using the block cipher encryption function Ek under the secret key K. Mathematically, it’s represented as L0 = L = Ek (0n) where 0n denotes a string of n zero bits.

Calculation of R: The value R is computed as R = Ek (ctr + i) ⨁ L, where ctr is the counter value used in the encryption process, and i is an incrementing value for each block to ensure that R is unique for every block of data being processed.

Sequential Calculation of Li: For i ³ 1, each Li ​ is computed by doubling the previous Li-1 ​ in the finite field GF(2n). This operation can be efficiently implemented using shift and conditional XOR operations to account for the field's polynomial representation.

Generation of Zi​: The first offset value Zi ​is simply R L, combining the previously computed R and L values. Subsequent Zi ​ values for i > 1 are derived by XORing Zi-1​ with L shifted by the number of trailing zeros in i (noted as ntz(i)). This is represented as Zi = Zi-1 ⨁ L. (ntz(i)).

Usage in encryption and decryption

Encryption: During the encryption process, the offset values Zi are XORed with the counter values before they are encrypted with the block cipher under the key k. This step generates a unique keystream for each block, which is then XORed with the plaintext blocks to produce the ciphertext. Specifically, if Yi represents the encrypted counter (plus offset) blocks, then C = M (Y1 ||Y2 || Y3 || ×××), where M is the plaintext message.

Decryption: For decryption, the same process is mirrored. The offsets Zi are recalculated in the same manner as during encryption and used to generate the keystream by XORing with the counter values and encrypting the result under k. The ciphertext is then XORed with this keystream to recover the plaintext.

The operator “.” refers to multiplication over the finite field GF(2n), GCM use G(2128) defined in section 2. The operator ntz(i) is the number of trailing 0 - bits in the binary representation of i such as Li can be computed with ntz(i) defined in section 2. Equivalently, ntz(i) is the largest integer z such that 2z divides i. However, authenticated encryption takes these inputs and resulting a ciphertext whose length exactly that of the plaintext and a tag T whose length also be the same. The length of the tag T denoted as t. The authenticated encryption of GCM with extra input Zi shown in Figure 1. The authenticated decryption operation has an extra input than authenticated encryption that is tag  and output, either the plaintext or fail. The symbol fail indicates that the inputs are not authentic. The authenticated data AD is used to protect the message that needs to be authenticated and does not need to be encrypted. When GCM used for secure network protocol, the AD includes protocol version numbers, ports, sequence numbers, addresses, and other fields that indicate how communication should be handled, processed, or forwarded. When the length of M is zero, GCM acts as a MAC on the AD. The mode of operation that uses GCM as a stand-alone MAC is denoted as GMAC. The strength of the authentication is determined by the length of tag t, and t must be fixed for any fixed value of the key. The length of  must be at least 64 bit, whenever possible 128 bit should be used because this length provides the best security.

The plaintext consists of a sequence of n − bit strings (m1, m2, …, mn-1,mn) that is called a data block, and the bit length of each data block is128 bit. Although the bit length of the last data block may not be equal to n bit, so we denoted the bit length of the last block by u, where 1 ≤ u128. Similarly, the corresponding ciphertext block is denoted as c1, c2, …, cn-1, cn, where the bit length of the last block is u. The authenticated data block AD denoted as AD1, AD2,…, ADn-1, ADn, where the bit length of ADn may not be a complete block, so the length of the last block we denote as v, where1 ≤ v128. In the following given equation, we can define the authentication encryption operation.

The successive value of the counter (ctr) generated by using incr () function, which treat 32 lsb (least significant bit) and increment with modulo 232. The authentication decryption process of GCM similar to the authentication encryption process, rather than the hash and encrypt step, both are reversed in authentication decryption.

Generally, additional authenticated data (AD) and blocks of plaintext (m1m2 … mn) is shown. Here Ek denotes the block cipher encryption using the key K, • denotes multiplication in GF(2128) by the hash key H, and + 1(incr) denotes the counter increment function.

Security proof

The block cipher is a function E:K × {0,1}n → {0,1}n and if it is assumed to be a secure pseudorandom permutation (PRP) [29], then GCM stand a secure authentication encryption, where k is a finite set EK(.) = EK(,.) is a permutation on{0,1}n. This requirement is met when E cannot be distinguished from a random permutation (R) by an AdvErsary that can choose its inputs and view its outputs. The block cipher E with a fixed key and permutation oracle, both have the same interface. Let A be AdvErsary given access to permutation oracle to determine whether it is R or E with random selected key. the probability in each case is 1/2. The AdvErsary asks sequence of queries to the oracle and wants to guess whether the response is. Let D be the event that it guesses the E, in such a way, DR denote the guess of R. Then finally we can conclude advantage function of AdvErsary A.

Adv = P[D\E] − P[D\R] (12)

The notation P[X] denotes the probability of event X. The P[X|Y] = P[X ⋂ Y] / P[Y] denotes the probability of event x given event Y equals the probability of event Y and event X divided by the probability of event Y. We make an assumption that advantage Adv > 0, thus the range of Adv between 0 and 1. The AEAD of GCM follows the following security model [15]. It has the following input bit strings: M, counter, AD, and Zi and return C and T. authenticated decryption oracle models take counter, AD, zi, C,T an input and return M or special symbol fail.

According to the definition of privacy (confidentiality), we use the indistinguishability of ciphertext from random under a CPA attack and indistinguishability of plaintext from random under a CCA attack, this definition equivalent to [30]. GCM encryption is secure under these assumptions when AdvErsary presented with these oracles cannot tell if they contain GCM with a randomly selected key (EGCM) or if C and T are a random function of inputs E GCM R MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadweajuaGpaWaa0baaSqaaKqzGeWdbiaadEeacaWGdbGaamytaaWCPAqaaKqzGeWdbiaadkfaaaaaaa@3CCA@ . The probability in each of these cases is 1/2. Generally, the value of H for both (computing the authentication tag and hashing) provides the AdvErsary a potential attack vector against privacy. So, we need to give access to the AdvErsary to the authenticated decryption oracle. GCM takes E as a pseudorandom function PRF. For security of PRF, consider, we give access to the function oracle, and guess whether it contains PRF or a true random function (oracle have the same interface) [31,32]. The advantage of PRF distinguisher are following.

Ad v PRF =P[ D| E PRF ]P[ D| E PRF R ]      (13) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5E51@

Where, EPRF and E PRF R MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaadweajuaGpaWaa0baaSqaaKqzGeWdbiaadcfacaWGsbGaamOraaWCPAqaaKqzGeWdbiaadkfaaaaaaa@3CDB@ denoted corresponding to PRF and random function. Advantage against both PRF and PRP are similar, because having similar properties.

Lemma 1

The advantage AdvPRF of an AdvErsary in distinguishing a n bit PRP E from a random function is bounded by AdvPRFAdvE + q(q-1)2-n-1 where AdvE is the AdvErsary’s advantage in distinguishing E from a

ramdom permutation, and a value q is the number of queries to the function oracle.

Theorem 1

If there is an AdvErsary that can distinguish GCM encryption from a random function with advantage AdvGCM, when the output of that function is limited to q queries to the authenticated encryption and decryption oracles, where the total number of plaintext bits processed is lp and where len(C) + len(Adv) ≤ l and len(ctr) ≤ lenctr for each query, then that AdvErsary can distinguish E from a random permutation with advantage AdvE, where

Ad v E Ad v GCM ( l P n +2q ) 2 2 n1 q(( l P n +2q ) l ctr n + 12 1n + l n + 12 t )      (14) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@87C1@

The formulation for the AdvErsary’s ability to distinguish E (the block cipher encryption function) from a random permutation with an advantage AdvE is derived from a theoretical framework for evaluating the security of cryptographic algorithms. Specifically, in the context of the Galois/Counter Mode (GCM) and its improved versions discussed in the document, the advantage AdvE quantifies the effectiveness of an AdvErsary in distinguishing the encryption function E used within GCM from a perfectly random permutation. This measure is crucial in cryptographic security to assess how well the encryption scheme withstands attempts at cryptanalysis.

The formulation for AdvE is based on several factors, including:

The Number of Queries (q): This represents the number of times the AdvErsary is allowed to interact with the encryption oracle (or the block cipher being analyzed) and observe its outputs. A larger number of queries might increase the AdvErsary’s chances of distinguishing E from a random permutation, up to a certain limit.

The Total Number of Plaintext Bits Processed (lp): This is the cumulative length of all plaintext messages that the AdvErsary encrypts using the oracle. It factors into the AdvErsary’s advantage because processing a large volume of data might reveal patterns or weaknesses in the encryption scheme.

Constraints on the Counter Values and the Size of the Authentication Tag (t): Constraints on the length of the counter and the size of the authentication tag also influence the AdvErsary’s advantage. For instance, a shorter authentication tag might be easier to forge or guess, potentially increasing AdvE​.

Security Bounds of the Underlying Block Cipher: The inherent security of the block cipher itself, against both known and unknown attacks, plays a critical role. The stronger the block cipher, the lower the AdvErsary’s advantage in distinguishing it from a random permutation.

The specific formulation of AdvE provided in the document considers these and potentially other factors, such as the parallelizability of the GCM mode and its resistance to specific types of cryptanalytic attacks (e.g., differential cryptanalysis). The goal of such a formulation is to establish a concrete security proof or bound that quantifies the level of security offered by the encryption scheme against an AdvErsary capable of conducting chosen plaintext attacks (CPA) or ciphertext attacks (CCA).

GCM encryption security also depends on the authentication tag size but it’s relatively weak. In the bound on AdvE contain 2-t not dominate that value as long as t is greater than about nlg(q l n + l ctr n ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaad6gacqGHsislcaqGSbGaae4zaiaacIcacaWGXbqcfa4aaSaaaOWdaeaajugib8qacaWGSbaak8aabaqcLbsapeGaamOBaaaacqGHRaWkjuaGdaWcaaGCPAqaaKqzGeWdbiaadYgajuaGpaWaaSbaaSqaaKqzGeWdbiaadogacaWG0bGaamOCaaWCPAqabaaakeaajugib8qacaWGUbaaaiaacMcaaaa@49BA@ .

In the presence of CPA attack for MAC security we use the standard model and give access to the AdvErsary to tag generation oracle and tag verification oracle. The AdvErsary sends messages to the tag generation oracle tag generation oracle and construct any message pairs and send these to the tag verification oracle. After making the q queries to both of oracles, the probability of the AdvErsary getting verification oracle to accept a message pair other than generated by the tag generation oracle. This is the forgery advantage of GCM (FGCM).

Theorem 2

AdvErsary with FGCM against GCM has AdvE against pseudorandom permutation E used in GCM are.

F GCM ( l P n +2q ) 2 2 n1 q(( l P n +2q+1 ) l ctr n + 12 1n + l n + 12 t )      (15) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibabaaaaaaaaapeGaamOraKqba+aadaWgaaWcbaqcLbsapeGaam4raiaadoeacaWGnbaal8aabeaajugib8qacqGHsisljuaGdaqadaGcpaqaaKqba+qadaWcaaGcpaqaaKqzGeWdbiaadYgajuaGpaWaaSbaaSqaaKqzGeWdbiaadcfaaSWdaeqaaaGcbaqcLbsapeGaamOBaaaacqGHRaWkcaaIYaGaamyCaaGccaGLOaGaayzkaaqcfa4damaaCaaaleqabaqcLbsapeGaaGOmaaaaaOWdaeaajugib8qacaaIYaqcfa4damaaCaaaleqabaqcLbsapeGaeyOeI0IaamOBaiabgkHiTiaaigdaaaGaeyOeI0IaamyCaiaacIcajuaGdaqadaGcpaqaaKqba+qadaWcaaGcpaqaaKqzGeWdbiaadYgajuaGpaWaaSbaaSqaaKqzGeWdbiaadcfaaSWdaeqaaaGcbaqcLbsapeGaamOBaaaacqGHRaWkcaaIYaGaamyCaiabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaalaaak8aabaqcLbsapeGaamiBaKqba+aadaWgaaWcbaqcLbsapeGaam4yaiaadshacaWGYbaal8aabeaaaOqaaKqzGeWdbiaad6gaaaGaey4kaSIaaGymaiaaikdajuaGpaWaaWbaaSqabeaajugib8qacaaIXaGaeyOeI0IaamOBaaaacqGHRaWkjuaGdaWcaaGcpaqaaKqzGeWdbiaadYgaaOWdaeaajugib8qacaWGUbaaaiabgUcaRiaaigdacaaIYaqcfa4damaaCaaaleqabaqcLbsapeGaeyOeI0IaamiDaaaacaGGPaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG1aGaaeykaaaaaa@8025@

Result and discussion

In this section, we discuss the characteristics of GCM offset with respect to another authenticated encryption mode. Our proposal extends by feature to the NIST recommended GCM mode. Counter mode is the obvious choice for the foundation of most of authenticated encryption mode since it is the one simple, efficient and well-known privacy-only mode that is fully parallelizable. It comes with a security proof that guarantees no attacks up to the birthday bound and has been proven secure against CPA attack up to 2 n 2   MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaaaaaaWdbiaaikdajuaGpaWaaWbaaSqabeaajuaGpeWaaSGaaSWdaeaajugib8qacaWGUbaal8aabaqcLbsapeGaaGOmaaaaaaGaaeiOaaaa@3D1B@ encrypted data blocks. We refine the Counter mode with small additional overhead which is known as the offset. The offset is an unpredictable input underlying the block cipher and it led to achieving higher resistance against differential cryptanalysis and improved the security of Counter mode without breaking its important advantages. So that we can improve the security of Counter mode related authenticated encryption mode like XCBC, CCM, EAX, and cwc. In this paper, we refine the GCM with unpredictable offset input. Thus, the desirable characteristics of offset associated GCM are the following:

  • Assuming underlying block cipher is a good PRP and authenticated tag length t equal to the block length n then GCM offset provable secure up to birthday bound.
  • When encrypting the plaintext and getting the corresponding ciphertext, then we have the same length of the plaintext and authentication tag t.
  • We used nonce (each value used at most once in a given session, having the property of counter); it is not required to be random or unpredictable.
  • The offset is unpredictable input underlying the block cipher, which is XOR with corresponding counter value, each input value of offset used at most once in a given session.
  • GCM offset use the forward direction of the block cipher. This saves chip area compared to AEAD constructions. So, for AHED descryption require the block cipher backward direction.
  • GCM offset is fully parallelizable, enabling hardware throughput. Which is not limited by benefiting software embodiments and block cipher latency.
  • The authenticity of the data after decryption can be verified from the recovery of the confidential data. The invalid data cannot be processed without counter mode decrypting them.
  • The confidential portion of GCM is a counter mode with extra input that is offset is a simple and efficient for hardware to construct GF(2128) multiplier. Overall, in hardware, GCM is unrivaled by any authenticated encryption scheme.
  • As well as it can also be efficient for software. GCM is online and no one needs to know the message length in advance of processing it. However, need to know the AD and its length before processing the message. This makes the GCM suitable for networking applications and incremental API (Application Programmers Interface) where, M, C, or AD provided incrementally in chunks.

The characteristics of fully parallelizable authenticated encryption modes of operation are summarized in Table 1. The remaining serial modes are described in Table 2. The modes that come with security proof are based on the assumption that the underlying block cipher is secure. The confidentiality and authenticity of each mode proved together with the fact that no attacker can get a significant advantage to distinguish between a random stream and ciphertext. There are some provably secure modes, and some are not proven both characteristics mentioned in Tables 1,2.

Some characteristics definitions are the following:

  • Patent: Provably secure authentication encryption modes are patent (i.e GCM, OCB, XCBC, IAPM), and some modes trying to e patent aware.
  • Provably Secure: If the underlying block cipher is a secure PRP and modes come with the proof of security and give message privacy and authenticity then modes are known as provably secure.
  • Parallelizability: For a high-speed environment we use parallelize mode, where encryption/decryption can be done parallel (Table 1). In the case of parallelizable authenticated encryption modes, both (encryption and authentication) are parallelizable denoted as A + E.
  • Associated data authentication: The unencrypted data that is used for protection of ciphertext, where authenticated data is denoted as AD. The AD typically used is to encode header information in a networking context.
  • Ciphertext Expansion: Many modes of operation expand message up to authentication tag length, so for the short message this property is important where can overcome a length of the original message.
  • Online message processing: this is an important property for memory memory-restricted environment, where the possibility to encrypt or decrypt a message without obtaining all messages, GCM have this property.
  • Endian dependency: the modes of operation that use the integer multiplication/ addition are endian dependent. All the discussed (in this section) modes are endian dependent other than OCB mode.
  • Incremental MAC: In the application data set frequently changes and must be authenticated remote database or recalculating an authenticator for all data cannot be efficient.
Differential Distribution Table (DDT)

The Differential Distribution Table (DDT) is a crucial tool in differential cryptanalysis, as it maps the difference between two inputs to the difference between the corresponding outputs for each possible input pair. DDT is used to identify differential characteristics with high probabilities that can be exploited in attacks. The probability of a differential characteristic, which can be derived from the DDT, is a measure of how likely it is that a specific input difference will lead to a specific output difference after going through the cipher.

The implementation of an offset in the improved GCM aims to make the prediction of output differences harder by introducing an additional layer of unpredictability into the encryption process. By XORing each block cipher input with a unique offset, the improved scheme aims to disrupt the predictability that differential cryptanalysis relies on. This unpredictability complicates the construction of a DDT with high probability differential characteristics that are useful for an attacker.

Results of attacks and complexity

In terms of the results of attacks against the improved GCM, including the number of rounds of the block cipher attacked and the complexity of these attacks, such specifics would typically result from extensive cryptanalytic research. The document mentions improvements to privacy and a theoretical resistance to differential cryptanalysis but does not provide detailed results of attacks, such as specific numbers of rounds that can be securely encrypted or the exact complexity of potential attacks against the improved mode.

In general, the resistance of a cryptographic algorithm or mode of operation to differential cryptanalysis (or any other form of cryptanalysis) is evaluated based on:

The Number of Rounds: More rounds generally increase security against differential cryptanalysis, as they make it more difficult to find useful differential paths that cover the entire cipher.

Data Complexity: This refers to the amount of plaintext-ciphertext pairs an attacker needs to analyze to successfully exploit a differential characteristic. The introduction of offsets aims to increase the data complexity required for a successful attack.

Attack Complexity: This encompasses both the computational resources and the data required for an attack to be feasible. Ideally, the complexity should be close to or exceed brute-force search complexity, making the attack impractical.

For detailed cryptanalytic results, including specific vulnerabilities and the resistance of the improved GCM to differential cryptanalysis, one would look to specialized cryptographic literature and research that conducts a thorough analysis of the scheme, including practical and theoretical attacks. Without explicit results in the provided document, it’s recommended to consult further cryptographic analysis and peer-reviewed research for a comprehensive understanding of the improved GCM’s resistance to differential cryptanalysis and other cryptographic attacks.

Furthermore, the improved GCMoffset” mechanism is designed to enhance the mode’s privacy and resistance to differential cryptanalysis. This modification is pivotal in the realm of authenticated encryption, where the quest for both robust security and high performance is incessant. To appreciate the value brought by the improved GCM, it’s essential to compare it with other authenticated encryption modes such as CBC-MAC (Cipher Block Chaining Message Authentication Code) and CCM (Counter with Cipher Block Chaining-Message Authentication Code), focusing on their security features and performance metrics.

Starting with the core of its enhancement, the improved GCM integrates an offset into the encryption process, which is a strategic move to complicate the predictability that differential cryptanalysis exploits. This means that for each block encrypted, a unique offset is applied, significantly obstructing the ability of an attacker to use differential techniques to infer key information or plaintext. This addition does not notably impact the operational efficiency of GCM, which is renowned for its parallel processing capabilities. The ability to process multiple encryption and authentication operations in parallel is a crucial determinant of performance in high-speed network environments, making the improved GCM exceptionally well-suited for applications requiring rapid data processing without compromising security.

On the other hand, CBC-MAC, an older mode of authenticated encryption, employs a sequential block cipher operation to provide message integrity and authenticity. While CBC-MAC is fundamentally secure under certain conditions, its security model is contingent upon the proper management of keys and initialization vectors. Specifically, if a key is reused across different sessions or improperly managed, the security of CBC-MAC can be compromised, making it susceptible to forgery attacks. Furthermore, the inherent sequential processing of CBC-MAC limits its throughput and efficiency, especially in comparison to modes like GCM that excel in parallel processing.

CCM mode, another contender in the realm of authenticated encryption, combines the Counter mode of encryption with CBC-MAC for authentication. This dual approach necessitates a unique nonce for each message to ensure security, introducing complexities in nonce management that can be problematic in systems where nonce reuse might occur. Additionally, CCM operates in two passes over the data—one for authentication and one for encryption—which inherently doubles the processing requirement for any given message. This two-pass process significantly affects performance, particularly in systems where latency and throughput are critical factors.

Comparatively, the improved GCM, with its offset mechanism, not only enhances security by thwarting differential cryptanalysis but also maintains high performance through its parallelizable architecture. This unique blend of security and efficiency is not as pronounced in CBC-MAC and CCM. The sequential nature of CBC-MAC’s operation and CCM’s two-pass requirement for encryption and authentication translate into inherent performance bottlenecks. These limitations become increasingly significant in the context of high-speed data transmission and processing, where delays, even milliseconds in length, can be detrimental.

In terms of security, the improved GCM’s offset mechanism offers a tangible advantage by increasing the complexity for attackers attempting to leverage cryptanalysis techniques. Unlike CBC-MAC, where security can be undermined by key management issues, or CCM, which requires stringent nonce management to avoid security pitfalls, the improved GCM provides a robust security model that is less susceptible to such operational hazards. This makes the improved GCM a more resilient choice for environments where the integrity and confidentiality of data are paramount, and where the operational context might not always guarantee perfect key or nonce management.

From a performance standpoint, the improved GCM’s ability to leverage parallel processing stands in stark contrast to the inherently sequential CBC-MAC and the two-pass CCM mode. This architectural advantage enables the improved GCM to achieve higher throughput rates and lower latency, making it exceptionally well-suited for high-performance computing environments, real-time applications, and large-scale data processing scenarios. Furthermore, the minimal overhead introduced by the offset mechanism ensures that the improved GCM maintains its performance advantages without incurring significant computational costs.

In conclusion, while CBC-MAC and CCM have played pivotal roles in the development of authenticated encryption, the AdvEnt of the improved GCM with its offset mechanism signifies a leap forward in both security and performance. The enhanced resistance to differential cryptanalysis, combined with the ability to execute encryption and authentication operations in parallel, positions the improved GCM as a superior choice for modern cryptographic needs. Its design not only addresses the inherent weaknesses observed in CBC-MAC and CCM but also sets a new standard for efficiency, making it a compelling option for securing data in an increasingly interconnected and high-speed digital world.


In conclusion, this research paper introduces a significant enhancement to the GCM mode of authenticated encryption through the incorporation of an “offset” mechanism, aimed at augmenting privacy and bolstering resistance against differential cryptanalysis. The modified GCM mode retains its original advantages, such as high efficiency, simplicity, and the use of a single cryptographic key, while the introduction of unique offsets complicates the predictability that underpins differential cryptanalysis. This innovation ensures that the improved GCM stands as a formidable option for applications requiring authenticated encryption, especially in scenarios where high-speed, parallelizable cryptographic operations are paramount.

The detailed analysis and discussions presented in the paper highlight the practicality of the offset-enhanced GCM in contemporary cryptographic applications. By maintaining the mode’s original features and adding minimal overhead, the paper convincingly argues for the enhanced mode’s suitability in securing high-speed networks and systems against sophisticated cryptanalytic attacks, without compromising on efficiency or security.

Moreover, the paper’s exploration into the operational framework, including the meticulous integration and computation of offsets in both encryption and decryption processes, underscores the thoughtful approach taken to improve GCM. The security proofs and theoretical discussions further solidify the enhanced GCM’s stance as a robust, secure, and efficient mode of operation that can significantly contribute to the field of cryptography.

Future research could potentially explore the practical implications of this enhancement in real-world applications, examining its performance and security in diverse scenarios. Additionally, the adaptability of the offset mechanism in other cryptographic modes and its potential to enhance the security of existing protocols could offer exciting avenues for further exploration. Overall, this paper not only contributes to the cryptographic community by presenting a more secure and efficient version of GCM but also sets the stage for future advancements in the field of authenticated encryption.

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