Richard E. Chandler

The Hill cryptographic system with unknown cipher alphabet but known plaintext

The cryptographic equations relating plaintext, ciphertext, and key-matrix elements in the Hill system are nonlinear equations if the cipher alphabet is unknown. In the case where plaintext is known it is possible to reduce these equations to linear equations by the introduction of a larger set of unknowns. These latter equations are analyzed by a method of successive eliminations of unknowns from a series of related row-reduced echelon forms.

A tracking algorithm for implicitly defined curves

A tracking algorithm is given for curves that have equations of the form f(x,y)=0. It produces the next approximating pixel by looking for a sign difference in function evaluations at midpoints between the eight nearest neighboring pixels. The search proceeds in order of decreasing likelihood, examining the most probable candidates first.<<ETX>>

Singular sets and remainders

This paper characterizes the singular sets of several traditional classes of continuous mappings associated with compactifications. By relating remainders of compactifications to singular sets of mappings with compact range, new results are obtained about each. All spaces to be considered here are locally compact and Hausdorff and all functions are continuous. The compact subsets of a space Z will be denoted by %z. For/? G Z, 9l(/?) (9tK(/?)) denotes the family of open neighborhoods (open neighborhoods with compact closure) of /?. For a mapping /: X -* Y, Cain [2] defined the singular set off to be S(/) = {/? G r|V£/ G 9L(/»), 3F G %Y,p G F c U and f~l(F) <2 %x}. In a later paper [3], he further explored the nature of S (/). Whyburn [14] gave an alternate definition. Various authors have seen methods of constructing a compactification of X given a map /: X —> Y where Y is compact. Perhaps the earliest (in the context of this paper) was Loeb [9]. Others who should be mentioned are Steiner and Steiner [11], [12], Magill [10], Blakley, Gerlits and Magill [1], and Choo [7]. Chandler and Tzung [6] defined the remainder induced by f to be £(/)= n{clYf(X\F)\FG%x} and proved that (whenever Y is compact) there is a compactification aX with aX \ X homeomorphic to £(/). We show here that §(/) and £(/) are the same set. We can then use what was previously known about singular sets to obtain results about remainders and vice versa. We will also explore this set for three types of mappings commonly encountered in studying compactifications: (i) the composition of a mapping of X into Y and the inclusion of Y into a compactification of Y; (ii) the composition of a mapping of X into Y and a quotient mapping of Y; and (iii) the evaluation mapping into a product generated by the inclusion of X into a compactification and by a mapping of X into a compact space K. Received by the editors April 9, 1980 and, in revised form, October 1, 1980. AMS (MOS) subject classifications (1970). Primary 54D35, 54D40, 54C10.

A recursive technique for rendering parametric curves

Abstract A method is developed for recursively drawing parametric curves. The advantage of this method is that it avoids the problem of having to choose the subdivision of the parametric interval.

Some further cryptographic applications of permutation polynomials

A cryptographic system is described based on the use of systems of permutation polynomials over finite fields or rings. The method involves two steps of encipherment, but more than two steps can be used if desired.

A new elusive otodontid shark (Lamniformes: Otodontidae) from the lower Miocene, and comments on the taxonomy of otodontid genera, including the ‘megatoothed’ clade

Abstract We describe a new large otodontid lamniform shark, Megalolamna paradoxodon gen. nov. et sp. nov., chronostratigraphically restricted to the early Miocene (Aquitanian–Burdigalian). This new species is based on isolated teeth found from five globally distributed localities: the Jewett Sand in southern California, USA; the Pungo River Formation of North Carolina, USA; the Chilcatay Formation of Peru; the Oi Formation in Mie Prefecture, Japan; and the O’oshimojo Formation in Nagano Prefecture, Japan. Extrapolations based on available published data on modern macrophagous lamniforms suggest that the largest specimen of M. paradoxodon gen. nov. et sp. nov. possibly came from an individual that measured at least 3.7 m in total length. All specimens came from deposits in the mid-latitudinal zones representing shallow-water, shelf-type, coastal environments. Its dentition likely exhibited monognathic heterodonty suited for capturing and cutting relatively large prey (e.g. medium-sized fishes). We recommend the genus Otodus to include sharks of the ‘megatoothed’ (e.g. megalodon) lineage in order to avoid Otodus paraphyly. We also propose the following phylogenetic hypothesis: [Kenolamna + [Cretalamna + [Megalolamna + Otodus]]]. ZooBank LSID for the genus Megalolamna is: urn: lsid:zoobank.org:act:B4791DEF-4D96-4FEB-9B7B-0EF816B96079 ZooBank LSID for the species Megalolamna paradoxodon is: urn:lsid:zoobank.org:act:7D3D7442-53C6-43A2-9E8D-6339729565B6

The calculations of stability constants using a high speed digital computer

Abstract A least squares technique for calculating stability constants from potentiometric titration data is described for use with a high speed digital computer such as the I.B.M. 650.

Continua as remainders, revisited

Abstract We give a sufficient condition for a general class of continua to be remainders of a space X in a compactification α X . We then see that (for these X ) the smallest compactification to which a function f ϵ C ∗ ( X ) extends has a closed, bounded interval for a remainder.

On classes of compactifications which are always singular

Abstract A compactification αX of a locally compact Hausdorff space X is said to be singular if αX β X is a retract of αX. Suppose that S is a class of locally compact, noncompact Hausdorff spaces, and that K is a collection of compact Hausdorff spaces. A general question about the existence of singular compactifications is the following: For what classes S and K is it true that each compactification of X ϵ S having a remainder αX β X ϵ K is singular? In this paper we consider a collection S , which contains the zero-dimensional spaces, and prove, among other things, that in this case K can be taken to be all products of compact metric spaces. In the process we have a variant of the well known result of Sierpinski that in a separable metric space X, a closed subset A having a zero-dimensional complement is a retract of X.

The two-message problem in the Hill cryptographic system with unknown cipher alphabet

The Hill two-message Problem assumes given two ciphertexts obtained by the encipherment of a single plaintext by two different key matrices K 1, K 2. The problem is to determine the (unknown) K 1, K 2 and the cipher alphabet if unknown. Levine and Brawley (Journal fur die reine und angewandte Mathematik 224 (1966): 20–43; 227 (1967): 1–24) and Nance (Doctoral dissertation, N. C. State University) analyzed this problem for the case in which the cipher alphabet is known. In this paper we continue with the problem and consider the (more difficult) case in which the cipher alphabet is unknown. The method used depends in part on some results obtained by Levine and Chandler (Cryptologia 13: 1–28) which determine the matrix A=K 1 K 2. The basic equation AXA=X is then solved for matrix X by use of the row-reduced echelon form of the system of linear equations in the unknown elements of X, the key matrices being among these solutions.