Reconstruction of images encrypted in Nostradamus's "Prophecies".

Razumov Il'ya Kimovich ORCID: 0000-0002-7277-2638 Doctor of Physics and Mathematics Senior Researcher, Institute of Metal Physics, Ural Branch of the Russian Academy of Sciences 620108, Russia, Yekaterinburg, S.Kovalevskaya str., 18 ilya.k.razumoff@gmail.com Other publications by this author

1. The "Prophecies" of Nostradamus as a carrier of a hidden message.

Despite the extensive bibliographic material accumulated over the centuries, including hundreds of titles ^[1], the scientific study of the life and work of Nostradamus began relatively recently. In addition to a detailed study of the predictor's biography ^[2,3], historical and philological comments on the quatrains ^[4-8] were compiled, which led to several important conclusions regarding the content of the "Prophecies". Firstly, from comparison with other astrological sources of the first half of the XVI century, such as the treatises of Trithemius and Russ, it was concluded that the end point of Nostradamus's "Prophecies" seems to be the year 2242, close to the end of the 6,000 years of the Jewish calendar. Secondly, it was found that many quatrains describe events preceding 1555, so that in the absence of reference to real-time dates, they are difficult to consider predictive. Moreover, for example, in quatrain 10-67 ("A strong earthquake in May, / Saturn in Capricorn, Jupiter and Mercury in Taurus, / Venus also, [in] Cancer Mars, in [An]nonai / Hail larger than an egg will fall") – a unique planetary configuration is indicated, which took place only on May 1, 1549, when an earthquake really occurred in the vicinity of the town of Annonai and large hail fell ^[7]. It remains unclear how such quatrains are consistent with the predictive motivation of the text. Thirdly, many quatrains seem too vague and insufficiently informative, so, for example, Prof. Denis Kruse declares: "Nostradamus makes it clear to his reader that he must boldly look beyond the words" ^[9]. Meanwhile, the first half of the XVI century was marked by increased interest in the development of methods of covert transmission of information (Trithemia steganography, Cardano lattice, Belazo cipher, etc.) ^[10]. In this context, the vagueness and lack of information content of the "Prophecies" can serve as an indirect sign of the presence of an encrypted message. At least, the need to arrange the quatrains in accordance with some algorithm is unequivocally stated by Nostradamus himself in the preface to the "Prophecies" addressed to King Henry II.

In previous publications of the author ^[11-13] it was shown that the "Prophecies" of Nostradamus, although published in parts (1555, 1557, 1568), are a complete work, subject to a strict plan, which contains two types of cipher. Firstly, there are serious reasons to believe that the order of the quatrains and their corresponding real-time dates can be restored using a cipher such as a modified scytale, using the biblical chronologies from the Letter to Henry as key sequences, which should allow a better understanding of Nostradamus's ideas about the distant future. However, the procedure itself for restoring the order of quatrains requires painstaking work, and not everything is clear at the moment. Secondly, certain words or letters of the text, applied in the form of spots on the plane in the coordinates "century number – quatrain number" lead to images of human faces, which presumably serve as illustrations for predictive texts. However, the quality of these images turns out to be low, so the characters present in them can hardly be identified. The purpose of this article is to propose a method for processing the source data, which, compared with previous results, leads to clearer images, as well as to discuss the graphic cipher of Nostradamus in a historical context.

The first signs of a graphic cipher in "Prophecies" are detected if some letters or words from the text are displayed with dots on the plane, according to the rule (for certainty, a 200x200 pixel area is used):

X=2Nk , Y=20(Nc-1)+5(Ns-1)+Int(1+5q), (1)

where N c , N k, N s are the numbers of the century, quatrain and quatrain lines, respectively, 0<q<1 is the relative position of the symbol in the string, X and Y are the coordinates of the symbol on the plane. Using formula (1) means that the numbers of the centuries are laid down vertically, the numbers of the quatrains horizontally, and the symbols inside the quatrain are in the form of a column within the allotted position (20 pixels).

We will display the word with the coordinates of its first letter. Since most words occur several times in the text, such words will correspond to a set of points on the plane. Figure 1 shows some examples. You can see that the pattern corresponding to the word "bras" has almost perfect symmetry. In the pattern corresponding to the word "Arabe", in addition to the axis of symmetry, there is an abnormal clustering of points. Equidistant points are found in the pattern corresponding to the word "Qui". These features of the patterns clearly indicate the presence of a graphical cipher. Further observations led the author to the conclusion that such patterns are components of more complex images, which can be composed, for example, of all words containing a given letter or starting with a given letter. However, the pictures obtained when displaying a large number of points on a plane are difficult for visual perception, and apparently require special processing to transition from the original discrete images to continuous ones. For example, Figure 2 shows the initial patterns obtained from all words containing the letters "A(a)", "O(o)", as well as from all words starting with neighboring letters of the alphabet R(r) or S(s). The images obtained by processing these patterns are given in Section 3.

Fig.1. Patterns corresponding to the words: "bras" (a), "Arabe" (b), "Qui" (c).

Fig.2. Patterns corresponding to words: containing the letter "A(a)" (a), containing the letter "O(o)" (with a rotation of 90 ⁰) (b); starting with neighboring letters of the alphabet R(r) or S(s) (c).

2. Reconstruction of encrypted images using the "smooth assembly" method.

As a result of displaying certain words or letters of the text as dots on a plane, we get an irregular, highly sparse image, the visual perception of which is difficult. Modern approaches to processing sparse images (including those arising in steganography) demonstrate impressive results ^[14-17], allowing even images that are practically unobservable in their original state to be restored. However, these approaches are reduced, one way or another, to algorithms for interpolating brightness in voids, while in our case the sparse image is given by points of the same brightness, which makes interpolation impossible. Rather, a preliminary reconstruction of the image by points is needed, rather than improving its quality. However, some general principles of image processing can be used.

In previous publications of the author ^[12,13], two approaches were proposed. The first approach relies on physical assumptions about the encryption technique used by Nostradamus (see discussion in Section 4). Based on the ideas about the propagation of light passing through holes in the screen, it is proposed to replace the points of the pattern with light spots, summing up the brightness of the spots when they are superimposed. The task of image optimization is reduced to finding the shape of a light spot (choosing a smoothing filter), including the rule of dimming brightness at the edges of the spot. Unfortunately, the resulting images in this case turn out to be too blurry, requiring additional processing by standard image enhancement methods. The second approach proceeds from the idea that the main property of an image on a plane is the presence of lines that can be restored by connecting the points with segments. Since, however, the rule for choosing such points is unknown, and most likely absent, it was proposed to connect each point with segments to all points within the neighborhood of a given radius, and at the intersections of the segments to sum up their brightness. Although this approach can, in principle, lead to clearer images, it is not sufficiently justified (it allows the appearance of unnecessary lines), and requires careful parameter settings. In this paper, a method is proposed that combines the prerequisites of previous approaches, i.e. the idea of superimposing light spots and searching for an image in the form of a set of brightness isolines.

To reconstruct the image, a set of elementary shapes (assembly elements) is used: a square and rectangles with an aspect ratio of 1:2, oriented vertically, horizontally and diagonals of the screen. All assembly elements contain the same number of pixels. The starting picture is made up of squares placed on the points of the initial pattern; when the squares are superimposed, their brightness is summed up. The size of the square is chosen in such a way as to optimize the visual perception of the image. Next, two points of the pattern are randomly selected, located at a distance of no more than the side of the square from each other. The assembly elements at these points are randomly assigned, the brightness distribution pattern is recalculated, after which the standard deviation of the brightness is calculated in the vicinity of each of these points (the radius of the neighborhood is equal to half the side of the square). The resulting picture is accepted if the sum of the specified standard deviations turned out to be lower than the corresponding amount calculated with the previous assembly elements; otherwise, the previous view of the picture is restored. The procedure is repeated iteratively until it goes into stationary mode. The image is a smoothed picture obtained by averaging over a number of iterations in a stationary mode. It is additionally smoothed by averaging the brightness over the isolines. For each pixel, an array of pixels is determined, the brightness of which differs by no more than a predetermined amount, which can be achieved by iterating through the nearest neighbors, starting from the original pixel. The brightness of the original pixel is recalculated as the average value in this array.

The proposed procedure implements the smoothest possible picture, as far as the set of assembly elements allows; in this case, the brightness isolines are detected in the form of stripes, the width of which is determined by the characteristic size of the assembly element. The search for an image in the form of a smooth picture is consistent with the ideas of the sparsity of natural images in spectral space ^[18], while the search procedure itself, in our opinion, has certain similarities with Monte Carlo simulation of spin dynamics to search for an equilibrium configuration ^[19], where the root-mean-square deviation of the brightness acts as the energy of the interaction of spins pixels on the elementary site.

3. Examples of the images obtained and their specifics.

Images can be applied as light spots on a dark background, or vice versa, dark spots on a light background; in the second case, the picture is displayed in a dark frame. To preserve the uniformity of the picture, the section of missing quatrains of the 7th century (7-43 : 7-100) was filled with randomly generated dots. In some cases, the resulting image requires the rotation of the painting, which is stipulated in the captions to the drawings. The images turn out to be clearer compared to those previously obtained using other computational procedures ^[12,13].

Figure 3 shows an image obtained from all the words containing the letter "O(o)". A fragment of a face can be seen, apparently of a young man. It is unlikely that this image should be classified as a portrait. Rather, the artist demonstrates here the general features of the cipher: images are applied in shades of gray to convey a three-dimensional shape, the subject of the image here and further are faces, the purpose of transmitting a hidden image is not necessarily predictive, i.e. the meaning is already contained in the method itself.

In Figure 4, dots have been added to the previous pattern corresponding to all words containing the letters "P(p)" or "Q(q)", i.e. letters adjacent to the letter "O(o)" in the alphabet. The image transforms somewhat: it seems that the character's face becomes more masculine. This may indicate that the cipher method is capable of transmitting dynamics.

In Figure 5, from the previous sample, only the dots corresponding to words starting with neighboring letters of the alphabet "O(o)" or "P(p)" are left, that is, the corresponding letters in the middle or end of the words are omitted. This results in a completely new image. Fig.6 and Fig.7 also show images obtained from all words containing the letter "A(a)" and from all words starting with neighboring letters of the alphabet "C(c)" or "D(d)", respectively. The images in these three drawings are related in execution, since they show twin faces, in particular the characters in Figure 5 have a common eye. The cipher's penchant for surrealism is again not entirely consistent with the idea of transmitting predictive information, however, the duality of persons may indicate the connectedness of destinies.

Figure 8 shows an image obtained from all words starting with neighboring letters of the alphabet R(r) or S(s), and Figure 9 shows the result of color inversion, resulting in a change in the image. Thus, the artist clearly shows that color inversion can be used as one of the ways of image steganography.

In most cases, the author of this article observed images that are collected either from all words containing the specified letters of the alphabet, or from all words starting with the specified letters of the alphabet ^[12]. Both of these rules allow you to use almost the entire alphabet, which indicates a fairly high system organization of the cipher. The complexity of the implementation of such a cipher allows us to assert that it is a significant part of the content transmitted in the carrier text. However, the low quality of the images and the artist's penchant for surrealism make it impossible to identify the characters and, therefore, raise doubts that the images were embedded in the text for a predictive purpose.

Fig.3. The image obtained from all words containing the letter O(o) (the pattern is rotated by 90 ⁰). On the right is a manual markup.

Fig.4. The image obtained from all words containing adjacent letters of the alphabet O(o),P(p) or Q(q) (the pattern is rotated by 90 ⁰). On the right is a manual markup.

Fig.5. The image obtained from all words starting with neighboring letters of the alphabet O(o) or P(p). On the right is a manual markup.

Fig.6. The image obtained from all words containing the letter A(a). On the right is a manual markup.

Fig.7. The image obtained from all words starting with neighboring letters of the alphabet C(c) or D(d) (the pattern is rotated by 1800). On the right is a manual markup.

Figure 8. An image obtained from all words starting with adjacent letters of the alphabet R(r) or S(s). On the right is a manual markup.

Fig.9. The image obtained from all words starting with neighboring letters of the alphabet R(r) or S(s) (the result of color inversion in Fig.8). On the right is a manual markup.

4. Steganography of Nostradamus: physical foundations in a historical context.

To reconstruct encrypted images, we used resource-intensive computational procedures that were not available in the XVI century. It is logical to assume that the creator of the cipher used rather physical methods, reducing computational work to a minimum. Let's discuss possible ways to implement such a cipher by technical means known and available in that historical period.

Since ancient times and the early Middle Ages, it has been known that light rays passing through a small hole in the screen create an inverted image of objects on the wall of a dark room behind the screen ^[20]. In the Renaissance, an optical device was created based on this principle, called the camera obscura, which was actively used by astronomers and artists. In particular, it was described in Leonardo da Vinci's "Treatise on Painting" (1452-1519). The first published illustration of a pinhole camera is given in Gemma Frisius's book De Radio Astronomica et Geometrica (1545) (see Fig.10). An improved device using a biconvex lens was described in 1550 by the Italian mathematician, astrologer and cryptographer J.Cardano ^[21].

Fig.10. The image of the pinhole camera from Gemma Frisius's book "De Radio Astronomica et Geometrica" (1545).

Our hypothesis is that Nostradamus applied pattern points as holes on the screen, using the latter by analogy with the pinhole camera screen. The light from a non-point source (for example, the sun or candle flame), passing through the holes in the screen, falls on another screen located in a dark room. As a result, a discrete image defined by the points of the pattern and difficult for visual perception is transformed into a continuous image created by superimposing projections of the light source. Figure 11 shows a diagram of such a device. The light of the candle flame passes through two point holes and is projected onto another screen in the form of two inverted flames. With a sufficient distance between the screens, the projections of the flames have an overlap area, within which the brightness of these projections naturally adds up.

Fig. 11. The image of a candle flame on the screen in a pinhole camera with two close holes: the projections of the flame tongue are superimposed on each other.

Note that if the real hole on the screen is not a point, it can be represented as a collection of infinitely small elementary sites, then the projection of the light source corresponding to such a hole is obtained as a result of summing the projections corresponding to elementary sites, which leads to a blurring of the light spot. In the case where the light source can be considered small compared to the holes on the screen, the light spots take the form of holes. If, moreover, a dark room is filled with smoke or fog, it is necessary to take into account the laws of light scattering in such an environment. As a result, it can be stated that the shape of the light spot corresponding to the point of the pattern in this scheme depends on the shape of the light source, on the shape of the holes on the screen, as well as on the properties of the medium between the screen with holes and the projection. A consistent solution to the problem of converting a discrete image into a continuous one requires the determination of many unknown parameters, however, ideas about the general properties of images (for example, the sparsity of natural images in spectral space) allow us to propose approximate reconstruction methods.

It is noteworthy that a number of the images we have obtained require light inversion, i.e. images are obtained by applying dark spots on a light background. Moreover, in some cases, the main and inverse images are visually perceived as different paintings. The association with the well-known positives and negatives in photography is not entirely correct. A negative image in a photograph appears as a result of a chemical reaction stimulated by light. The photosensitivity of silver nitrate was discovered by Wilhelm Gomberg in 1694, and the first negative image on paper impregnated with silver chloride was obtained using a pinhole camera by J.N. Neps only in 1816 ^[22], that is, two and a half centuries after the "Prophecies" of Nostradamus. Despite this, the author of the XVI century was still technically able to create manually and consider positive and negative images together. Let's say the light falls on a translucent dark screen, and a positive image is applied with opaque white paint on the side of the screen facing the light source. To see a negative image, just look at the screen from the opposite side: the areas covered with opaque paint are now darker. At the same time, however, the processing of steganographic patterns corresponding to the positive and negative should be carried out independently. Figure 12 shows a possible scheme for the reconstruction of a negative image, which arises as a result of the superimposition of penumbra created by opaque spots applied on a transparent screen.

12. Shadows and penumbra cast by two opaque objects fixed on a transparent screen illuminated by a non-point light source.

Most surprisingly, although the idea of image steganography is an obvious development of methods of covert transmission of information, examples of hiding images in text, such as the graphic cipher of Nostradamus, are unknown in the history of the Middle Ages. Perhaps the closest example in terms of the nature of the execution is Trithemy's "Steganography", written around 1499 and included in the index of forbidden books, in which cryptographic techniques were outlined under the guise of magical practices of summoning spirits, some of which were only recently understood ^[23]. However, the possibility of a graphical cipher by Trithemy was not considered. Thus, Nostradamus finds himself at the forefront of scientific thought of his time, participating in the development of new promising methods of covert transmission of information. Meanwhile, modern steganography is developing much more complex and qualitative approaches based on digital technologies, when an image or text is imperceptibly embedded in another image ^[17,24,25]. At the same time, embedding hidden images in texts is not practiced, apparently due to the high labor intensity and low efficiency of this approach. Thus, the graphic cipher in the "Prophecies" of Nostradamus still remains a unique experiment of its kind.

5. Conclusions.

(1) The text of the "Prophecies" of Nostradamus acts as a container for the hidden transmission of images. Fitting words to image parameters could be one of the reasons for the vagueness of the text content. Uninformative quatrains serve as building blocks in a graphic cipher.

(2) A method of reconstruction of encrypted images in the form of a "smooth assembly" is proposed, which leads to clearer pictures compared to previous results.

(3) The resulting images show mostly human faces, executed in a formal or surreal manner. Character identification is problematic, which, generally speaking, makes it possible to doubt the existence of a predictive purpose of these images.

(4) The method of embedding hidden images in the text is unique for the XVI century, it had a significant scientific value for this historical period. This suggests that the author of the "Prophecies", among other things, had a completely scientific motivation – the creation of a new promising method of covert transmission of information.