Cryptology - I: Symbols and Notation

Instructors: R.E. Newman-Wolfe and M.S. Schmalz

In this class, we must express cryptologic transformations and operations in a rigorous fashion, in order to (a) understand their structure and function, (b) predict consequences of changing transform parameters, and (c) have a unified basis for comparing cryptographic algorithms. Currently, the only rigorous, concise notation that unifies image and signal processing (of which cryptology is a part) is called image algebra. Some of you may have heard of this notation, which was invented here at UF by Gerhard Ritter. Image algebra is a rigorous, concise notation that unifies linear and nonlinear mathematics in the image domain. Image algebra was developed at UF under DARPA and Air Force support since the early 1980s to provide a simple, powerful notation for describing image and signal processing algorithms in a unified, high- level manner.

The following theory is a small subset of image algebra, since the aim of this class is not to teach the details of image algebra. Rather, we are teaching basic cryptology within the context of image algebra, for the above-mentioned reasons. Thus, we provide the following brief theoretical introduction.

NOTE: Bold-italic-face titles or names denote cryptologic terms that we will use in class.

1. Sets and Set Operations

Sets are the basic entities in image algebra and are the building blocks of mappings that we will subsequently use to define images. A few concepts and conventions are noteworthy:

Sets are customarily denoted by upper-case letters in roman (regular) or italic face (e.g., A, G), or by the empty set Ø.

Value Sets or Alphabets are denoted by bold upper-case letters from the head of the alphabet (e.g., A, B, ...). Additionally, we call F the generalized value set. Commonly-used value sets include the
- Natural numbers: N
- Integers: Z
- Real numbers: R
- Complex numbers: C
Furthermore, we have the integers modulo n, which are denoted by Z_n. In the special case of the binary numbers, we write B Z₂ for purposes of brevity.
Intervals. Occasionally, we use a concise specification of a contiguous subset of a given set, called an interval. For example, [1,4]Z denotes the set {1,2,3,4}, while the half-open interval (1,4]Z denotes the set {2,3,4}. Similarly, (2,4]R = {xR: 2 < x 4}.

Point Sets, Domain Sets, or Domains are denoted by bold upper-case letters from the tail of the alphabet (e.g., U, V, W, X, and Y). Additionally, we call X the generalized point set. Customary point sets include
- Euclidean n-space: Rⁿ
- Discrete n-space: Zⁿ

Set Operations. The customary set operations of union (), intersection (), subtraction (\), complement (´), cardinality (|A|), choice (choice), infimum (inf), and supremum (sup) are supported.
Unary arithmetic functions are applicable to subsets of the real numbers (e.g., max or , min or , and abs). Note that abs denotes absolute value of a set instead of |A|, which is reserved for cardinality.

2. Images or Messages

One usually thinks of an image as a two-dimensional array of reals or integers. Unfortunately, that definition is too restrictive for modern image and signal processing, of which cryptography and cryptanalysis are a part. Thus, we use the image algebra definition of an image, which generalizes all images.

An image (or a message in cryptology) is a mapping from a point set (domain) to a value set (alphabet).
Example. The mapping a : X -> F denotes a generalized image a F^X, which is a concise way of denoting the set of all mappings from X to F.

If a F^X, then X = domain(a) and F = range(a).

The graph of a F^X is denoted by:
G(a) = {(x, a(x)) : a(x) F, x X} .

We also write a G(a), where denotes equivalence. Note that p₁(G(a)) = X = domain(a) and p₂(G(a)) = F = range(a), where p_k denotes projection onto the k^th coordinate.

Although a point set or value set can be any set, one must take care to specify images using only those value or point sets for which operations are defined.

A special class of images are called constant-valued images. Given a constant value
k F, we denote a constant image k {k}^X as
k {(x, a(x)) : a(x) = k, x X} = {(x,k): x X} .

Example. The unitary image on X is denoted as 1_X, or simply 1, with X understood.

3. Image Operations.

In order to manipulate images or messages, we must specify a few simple operations. Fortunately, cryptology is not as complex notationally as image processing, so we can concentrate on the pointwise unary and binary operations that are frequently used in cryptography, as well as simple image-template operations (Section 6).

Let a denote an image in F^X. A unary pointwise operation g:F^X -> F^X is induced over X by the corresponding scalar operation g:F -> F.
Example. We have the following cases of real and Boolean negation:
-a {(x, -[a(x)]) : a(x) R, x X}
¬a {(x, ¬[a(x)]) : a(x) B, x X}

where B = {0,1}, as before.

A special type of unary operation is called the global reduce operation O: F^X -> F, which maps an image a F^X to an element of its value set.
Example. If a B^X, then the operation
a = a(x)
determines the number of unitarily-valued pixels in a. Alternatively, replacing with would tell us if there are any unitary values in a.

A binary pointwise image operation g:F^X × F^X -> F^X is induced over X by the corresponding scalar operation g:F × F -> F. Note that the operands (input images) must have the same domain, which the result (output image) also has.
Example. Given the images a,b F^X, we have the pointwise arithmetic operations
c = a + b {(x, c(x)) : c(x) = a(x) + b(x), where a(x),b(x) R, x X}
c = a · b {(x, c(x)) : c(x) = a(x) · b(x), where a(x),b(x) R, x X} .

Example. If a and b are Boolean images on X, then we have the following definition of the exclusive-or function:
c = (a xor b) {(x, c(x)) : c(x) = |a(x) - b(x)|, x X} .

Observation. A wide variety of binary operations can be described (e.g., a^b and log_ab), provided that value sets are properly defined. However, in the case of the logarithm, we must set range(a), range(b) R⁺, the positive reals. Otherwise, the logarithm is not defined.

A special type of binary operation is called the restriction of an image a F^X to a subset W of its domain X, which is denoted by:
b = a|_W {(x, b(x)) : b(x) = a(x), xWX} .
The converse of restriction is extension. For example, the extension of image b F^W to a k-valued constant image on X, the domain of a, is denoted by:
c = b |^k {(x, c(x)) : c(x) = b(x) if x W or c(x) = k(x) if x X\W} .

An image can also be restricted to a subset of its range space. Assuming that a F^X and A F, we have the following notation:
b = a ||_A {(x, b(x)) : b(x) = a(x) if a(x)AF, xX} .

Implementationally, image restriction can be thought of as a cut operation, while extension can be thought of as a paste operation.

Another binary operation is called the characteristic function of an image a F^X. Let S denote a property of a's range space (for example, "A F", such that
b = _S(a) {(x, b(x)) : b(x) = 1 if a(x) satisfies S, or b(x) = 0 otherwise}.
The resultant image b is a Boolean image on X.
Thus, the characteristic function can be thought of as a logical test with predicate S that returns a Boolean image. Wherever the test condition S is satisfied, a unitary value occurs, with zeroes elsewhere.
Example. The characteristic function can be employed in thresholding an image a R^X using a real threshold T, as follows:
b = _>T(a) {(x, b(x)) : b(x) = 1 if a(x) > T, or b(x) = 0 otherwise}.

4. Domain Transformations.

An important class of encryptions are called transpostions, in which the domain of a plaintext message is manipulated so that the symbol order is rearranged. Let a denote an image in F^X, and define a spatial transformation f: Y -> X. For purposes of simplicity, assume that f is a one-to-one and onto mapping.

The composition of a with f is denoted by:

b = a o f {(y, b(y)) : b(y) = a(f(y)), y Y} .

The combination of pointwise operations and domain transformations can produce powerful encryptions, as we shall see when we consider the DES transformation. It is interesting to note that in image processing, spatial transformations are used for image magnification, rotation, and warping.

5. Templates.

The most powerful construct in image algebra is called a template, which is defined as a mapping from a point set to a set of images. Thus, a template is an image whose pixel values are images.

Templates are denoted by bold, lower-case letters from the tail of the alphabet
(e.g., r, s, ...).

We say that t (F^X)^Y is an F-valued template from Y to F^X. This means that a template value t(y) F^X, where yY, is an image on X. For purposes of notational brevity, we write t_y t(y). Note that t_y = {(x,t_y(x)): x X} . Thus, a value of t's weight matrix t_y, also called a weight of template t, is expressed as t_y(x) F. Additionally, we denote the support of t_y as S(t_y) X. Implementationally, we usually think of the support as the set of domain points corresponding to the template's nonzero weights. Although this definition is not
general, it will be sufficient for this class.

Templates generalize the concepts of convolution masks, filter domains, sampling windows, etc., and thus provide a unifying construct for describing neighborhood operations on images. We rarely use templates in cryptography, but employ them extensively in cryptanalysis, particularly for pattern matching and correlation-based attack. These strategies will be discussed at length when we consider substitutional and transpositional ciphers.

6. Image-Template Operations.

Images and templates are combined using the generalized image-template left product

: F^X × (F^X)^Y -> F^Y, which accepts an image and a template and returns an image.

Given the associative, commutative operations ,: F × F -> F, an image a F^X, and a template t (F^X)^X, the left product is defined as
(a t)_y = a(x) t_y(x) , y Y .

The right product t a is symmetrically defined.
Example. When = + and = ·, we have the linear convolution , which is defined in terms of the left product as
(a t)_y = a(x) · t_y(x) , y Y .

If t is a template from X to X and a is an image on X, then a t = t a.
Example. When = and = ·, we have the multiplicative maximum operation , and when = and = +, we have the additive maximum operation . The operations of multiplicative minimum and additive minimum are symmetrically defined.

Although we will not use the nonlinear operations or extensively in this class, such operations can be employed in cryptanalysis, especially for nonlinear analysis attack on compressed or encrypted signals. The topic of nonlinear cryptanalysis will be mentioned at the end of the semester, if time permits.

This concludes our discussion of basic notation for this class. Other notational conventions will be defined as they are introduced in theory development.