Cipher image damage and decisions in real time

The algorithm defines a bijective function between nonnegative integers and permutation sets.

“To this end, an algorithm that defines a bijective function between the nonnegative integer and permutation sets is built.”

• Also stated in: § 2 Preliminary Concept · plain

  “Then the function I m : N m → Π m is bijective.”

The method builds a pseudorandom permutation of 250,000 position arrays or more.

“which means building a pseudorandom permutation of 250,000 position arrays or more.”

The algorithm can build permutations on the 0 to m-1 array knowing m-1 constants.

“From this algorithm, the way to build permutations on the 0; 1; : : : ; m -1 array, knowing m -1 constants, is presented.”

The method permutes the original image pixel positions before encryption with AES variable permutations.

“This is achieved by permuting the original image pixel positions before being encrypted with AES variable permutations”

The τ value is around 35% when the damage is 40%.

“The τ value is around 35% when the damage is 40%.”

The proposed test uses a goodness-of-fit test with χ2 for each basic color to measure randomness of the basic color distribution of an encrypted image.

“Working with images, a test of randomness based on the way the bits are arranged in an encrypted figure is proposed and the statistical, Chi-square, χ 2 ¼ P k i¼1 ½ðo ie i Þ 2 ∕e i is used for each of the basic colors.”

• AssumptionRests on an assumption: NIST 800-22 tests do not measure the randomness of the basic color distribution of an encrypted image.

The algorithm's implementation complexity is O(m).

“Regarding the complexity to implement this algorithm is OðmÞ because at every step a removal and replacement of an item is made.”

A failure less than or equal to 40% has a ratio τ of at least about 34% for images of about 512×512 or higher.

“It will be noted in Sec.7 that a failure less or equal to 40% has a ratio, as minimum, of about 34%.That is τ ≥ 34% for images of about 512 × 512 or higher.”

When an encrypted figure with damage is decrypted, the decoded image with failure has a higher degree of disorder in its bits than the original image.

“When an encrypted figure with damage is decrypted, such as that illustrated in Fig. 6, the decoded image with failure has a higher degree of disorder in its bits than the original image, that is, the Fig. 8 clause (c) has a higher randomness degree in its bits than the Fig. 8 clause (a).”

For all cases in the proposed test, the null hypothesis is accepted.

“Note that the analysis for each primary color is included; it is also important to mention that for all the cases, the null hypothesis is accepted.”

The χ2 values mentioned above are much higher, about 6 million.

“Furthermore, the χ 2 values mentioned above are much higher, in fact, about 6 million.”

Using the transcendental number pi and this algorithm, it is possible to construct a pseudorandom permutation over 250,000 array positions or more in less than 10 ms.

“Using the transcendental number pi and this algorithm, it is possible to construct a pseudorandom permutation over 250,000 array positions or more in less than 10 ms.”

The proposed image has χ2 values χ2r=106361059.17, χ2g=106647915.51, χ2b=107366956.56 for red, green, and blue.

“In this paper, a figure with the values of χ 2 : χ 2 r ¼ 106361059.17,χ 2 g ¼ 106647915.51and χ 2 b ¼ 107366956.56,for the red, green, and blue colors, respectively, is proposed.”

The algorithm constructs a permutation of the 0 to m-1 array in m-1 steps.

“This procedure is made in m -1 steps.”

The number of elements in N_m and Π_m is equal.

“The test that I m function is surjective is simple, since the number of elements in the sets N m and Π m is equal.”

The encrypted images pass the entire randomness test applied for a particular key proposal, including DFT, entropy, correlation coefficient in horizontal, vertical, and diagonal directions, and the proposal test.

“The encrypted images pass the entire randomness test applied for a particular key proposal and reported in Sec.7.1.These randomness tests are: DFT, entropy, correlation coefficient in: horizontal, vertical, and diagonal directions and the proposal test.”

For the 100,000 images encrypted in Fig. 6 clause (c), the average values of the correlation coefficient in the three directions and for each primary color were close to zero.

“For the 100,000 images encrypted in Fig. 6 clause (c), the average values of the correlation coefficient in the three directions and for each of the primary colors were reported in Table 7.The amounts found were close to zero.”

The algorithm defines a bijective function between the nonnegative integer and permutation sets.

“This paper has built an algorithm which defines a bijective function between the nonnegative integer and permutation sets.”

The images' encryption times of Fig. 6 are about 85 ms.

“Finally, it is reported that the images' encryption times of Fig. 6 are about 85 ms.”

Even in the worst cases, these images have correlation coefficients near to zero.

“These amounts are the furthest from zero, which means that even in the worst cases, these images have correlation coefficients near to zero.”

For entropy, the results are better than other studies.

“Indeed, for entropy, the results are better than other studies.4”

• cites [4]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Figure 5 shows encryption can be ineffective without variable permutations, with χ2 values χ2r=5691872.24, χ2g=5714519.25, χ2b=5706278.92.

“Figure 5 shows when the encryption process can be ineffective in a simple way, since the χ 2 for the primary colors are: χ 2 r ¼ 5691872.24,χ 2 g ¼ 5714519.25, and χ 2 b ¼ 5706278.92 for red, green, and blue, respectively.”

The proposed method allows recovery of the original image when the encrypted image suffers a failure from an attack or otherwise.

“Furthermore, this research offers the possibility of knowing the original image when the encrypted figure suffered a failure from either an attack or not.”

Finding the AES key from the permuted image using brute force has complexity 2^843.

“However, if we want to find the key, taking as a plaintext the permuted image using the brute force attack, then this could be a problem with a complexity of 2 843 ; because in a previous work 34 a strong evidence was given that the DES algorithm with variable permutation has a complexity of 2 56 × ð64!Þ, where the Monte Carlo method was used.”

• cites [34]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For figures with very large χ2 for each basic color (more than 100 million), variable permutations must be applied for encryption to be effective.

“For figures with a very large χ 2 for each different basic color, say more than 100 million, the variable permutations must be applied so that the encryption process is effective.”

Some countries' security areas do not allow compression in encryption images.

“This investigation does not use compression on images because there are some countries whose security areas do not allow compression in the encryption image.23”

• cites [23]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The random variable χ2 can be approximated to the normal distribution according to the central limit theorem.

“Moreover, the random variable χ 2 can be approximated to the normal distribution according to the central limit theorem.36”

• cites [36]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The size of the encrypted images failure is 40%.

“Section 7.2 carried out the analysis of encrypted images with damage, noting that the size of the encrypted images failure is 40%, see Fig. 8.”

At the end of the encryption process the zeroes and ones string will be random.

“Therefore, at the end of the encryption process the zeroes and ones string will be random.”

• AssumptionRests on an assumption: In AES, information is mixed in each round.

Finding the key from the initial image is more complex than from the permuted image.

“Clearly, if the problem is to find the key from the initial image, the solution to the problem would be more complex.”

The permutation on the whole original image before encryption is intended to disperse the pixels, so when the encrypted file is damaged and later decrypted, the result does not present the failure in a focalized way.

“The permutation on the whole original image before the encryption stage is intended to disperse the pixels, thus, when the encrypted file is damaged and later is decrypted, the result does not present the failure in a focalized way as shown in Fig. 7.”

The permutations generated depend on the AES key.

“On the other hand, the permutations generated depend on the AES key in accordance with the following procedure: denoted as l the integer which represents the 128 bits chain AES key.”

Differential and linear attacks are not applicable to AES due to the construction of the substitution box.

“Cases of plaintext are chosen as differential and linear attacks, which are not applicable to the AES cryptosystem due to the way the substitution box is constructed.6”

• cites [6]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The rejection situation can be described using the binomial model.

“Then, taking into account that rejecting an encrypted image happens when at least one of the primary colors fails the randomness test and considering that the probability of rejecting any of them is p ¼ 0.01, it follows that this situation can be described using the binomial model; 36”

• cites [36]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The constants used to build the permutations can be obtained from pi multiplied by l.

“Thus, using this last number it is possible to get the constants to build the permutations.”

If an image with low randomness is encrypted using AES without modification, the encrypted image could provide information because the distribution of shades follows a pattern.

“However, if a figure with a low randomness degree in its bits distribution is encrypted using the AES algorithm without any modification, perhaps the image encrypted could provide information, i.e., the distribution of the different shades of basic colors follow a certain pattern.”

Applying the permutation in the first round gives more opportunity to mix the information.

“Well, it is understood that the information is mixed in each round, so any change carried out by the permutation at the first round will have more opportunity to mix the information.”

• AssumptionRests on an assumption: In AES, information is mixed in each round.

If the tone distribution in each basic color was completely random, then χ2 = 0.

“if the tone distribution in each basic color was completely random, then χ 2 ¼ 0.”

Applying the permutation after the x-or operation allows changing some string bits.

“If the permutation is used after the x-or operation, then it allows for changing some string bits.”

If χ2 has a large value for each basic color, it means they have a defined order.

“However, if χ 2 has a large value for each basic color, it means they have a defined order.”