Appendix A Symmetry Groups A mathematical group G is a set of objects (the group's elements) with a binary op- eration denoted by "+" or by "*" defined on the elements that satisfies the following requirements. 1. Closure: for any a, b G G, the sum (a + b) is an element of G. 2. Associativity: any a,b,c G G satisfies (a + b) + c = a + (b + c). 3. Identity: there exists e G G such that for all a G (7 (a + e) = (e + a) = a. 4. Inverses: for each a G G, there exists a unique element a~l G G such that a-fa"1 — a"1 + a = e. 5. If the group operation is commutative, i.e., if a + b = b + a for any a,b £ G, the group is called Abelian. Question: What's purple and commutes? Answer: An Abelian grape. Examples of groups: 1. The set of all the integers with integer addition. The identity element is the integer 0. This is an infinite group. 2. The (finite) set of the integers 0, 1, 2,... ,m — 1 with modulo-m addition. 3. The integers 1, 2,. .. ,g — 1 for a prime q with modulo-g multiplication. 4. The set of all rotations in two dimensions under the operation: The sum of the two rotations by a and (3 degrees is a rotation by a + (3 degrees. The set (0,1, 2,3) with modulo-4 addition is a group denoted by G(4). It obeys the 448 Appendix A Symmetry Groups addition table + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 The order of a group (its cardinality) is the number of elements. It is denoted by ord(G). The order of G(4) is 4. A subgroup is a subset of the elements of a group that is closed under the group's operation. A theorem by Lagrange states that if S is a subgroup of G, then ord(5) divides ord(G). For example, if S is the subgroup (0,1) of G(4), then ord(5) = 2 divides ord(G(4)) = 4 and G(4) can be partitioned into the cosets S and S + 2. This appendix deals with symmetry groups. The elements of such a group are sym- metry operations (or transformations) on an object; they are not numbers. Figure A.la shows four symmetric objects: a rhombus, a rectangle, a square, and a pentagon. The term symmetric means an object that retains its shape and location under certain trans- formations. A square, for example, is highly symmetric, because it preserves its shape and position when rotated by a multiple of 90° or when reflected about the four axes shown by dashed lines in Figure A.lb. A rectangle is less symmetric because a rotation of 90° changes its shape from horizontal to vertical or vice versa. b a b a b c d c (a) (b) c) Figure A.I: Symmetries of Rhombus, Square, and Pentagon. For simple geometric objects, it is possible to express rotations and reflections by listing the new position of each vertex of the object. When the square is rotated 90° clockwise, for example, vertex a moves to b, b moves to c, and so on, which can be expressed as the permutation ab c dN b c d a Symmetry Groups 449 Reflections of the square about a vertical axis and about the main diagonal are expressed by a b c d \ /abed' bade/' Vadcb The connection between symmetry transformations and groups becomes clear when we consider combinations of transformations. The rectangle is transformed to itself after (1) a 0° rotation, (2) a reflection about a central horizontal axis, (3) a reflection about a central vertical axis, and (4) a 180° rotation. Examining the diagram, the following properties become clear: 1. Transformation 1 followed by transformation i (or i followed by 1) is equivalent to just transformation i for any i. 2. Any of the four transformations followed by itself returns the rectangle to its original shape, so it is identical to transformation 1. 3. Transformation 3 followed by 2 is equivalent to 4. An analysis of all the combinations of two transformations of the rectangle yields Table A.2a. The table can be considered the definition of a symmetry group of four ele- ments, because it specifies the group operation for the elements. A direct check verifies that element 1 (the null transformation) is the group's identity, that the operation is closed, and that it is noncommutative. This symmetry group is denoted by D4 (D for dihedral, meaning bending the arms up; anhedral means the opposite). (A dihedral group is a group whose elements correspond to a closed set of rotations and reflections in the plane. The dihedral group with 2n elements is denoted by either Dn or D2n- The group consists of n reflections, n — 1 rotations, and the identity transformation.) * 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 * 0 1 2 3 4 5 6 7 1 1 2 3 4 0 6 7 8 9 5 0 0 1 2 3 4 5 6 7 2 2 3 4 0 1 7 8 9 5 6 1 1 2 3 0 6 7 5 4 3 3 4 0 1 2 8 9 5 6 7 2 2 3 0 1 5 4 7 6 4 4 0 1 2 3 9 5 6 7 8 * 1 2 3 4 3 3 0 1 2 7 6 4 5 5 5 9 8 7 6 0 4 3 2 1 1 1 2 3 4 4 4 7 5 6 0 2 3 1 6 6 5 9 8 7 1 0 4 3 2 2 2 1 4 3 5 5 6 4 7 2 0 1 3 7 7 6 5 9 8 2 1 0 4 3 3 3 4 1 2 6 6 4 7 5 1 3 0 2 8 8 7 6 5 9 3 2 1 0 4 4 4 3 2 1 7 7 5 6 4 3 1 2 0 9 9 8 7 6 5 4 3 2 1 0 (a) (b) (c) Table A.2: The D4, D%, and D10 Symmetry Groups. Similarly, the rhombus has limited symmetry. Its four symmetry transformations are (1) the null transformation, (2) a reflection about the line bd, (3) a reflection about 450 Appendix A Symmetry Groups line ac, and (4) a 180° rotation. An analysis of all the combinations of two of these transformations, however, results in the same symmetry group. Thus, even though the rhombus and rectangle are different objects and their symmetry transformations are different, we can say that they have the same symmetries and we call them isometric. Intuitively, a square is more symmetric than a rectangle or a rhombus. There are more transformations that leave it unchanged. It is easy to see that these are the four rotations by multiples of 90° and the four reflections about the vertical, horizontal, and two diagonal axes. These eight transformations can be written as the permutations _/abcd\ _ /abcd\ _ /ab c d\ _ / a b c d ~ \abcdj' "ybcday5 \cdaby' ""\dabc __/abcd\ /abcd\ /ab cd\ 7 __ /^ a ^ c d ybadcy' ydcbay yadcby ycbad which can immediately be used to construct the symmetry dihedral group Dg listed in Table A.2b. Finally, the pentagon is used to create the larger symmetry group Dio, because it has 10 symmetry transformations. Figure A.la shows that the pentagon is transformed to itself by any rotation through a multiple of 60°, while Figure A.lc shows that it can be symmetrically reflected about five different axes. These ten transformations give rise to the DIQ symmetry group of Table A.2c (identical to Table 2.14a), and it is this group that is used by the Verhoeff check digit method of Section 2.11. The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful. —Aristotle, Metaphysica Appendix B Galois Fields This appendix is an introduction to finite fields for those who need to brush up on this topic. Finite fields are used in cryptography in the Rijndael (AES) algorithm and in stream ciphers. In the field of error-control codes, they are used extensively. The Reed-Solomon codes of Section 1.14 operate on the elements of such a field. B.I Field Definitions and Operations The mathematical concept of a field is based on that of a group, which has been intro- duced at the start of Appendix A. A field F is a set with two operations—addition "+" and multiplication ax"—that satisfies the following conditions. 1. F is an Abelian group under the 4- operation. 2. F is closed under the x operation. 3. The nonzero elements of F form an Abelian group under x. 4. The elements obey the distributive law (a -{- b)xc ~ axc-{- bxc. Examples of fields are 1. The real numbers under the normal addition and multiplication; 2. The complex numbers; and 3. The rational numbers. Notice that the integers do not form a field under addition and multiplication because the multiplicative inverse (reciprocal) of an integer a is I/a, which is generally a noninteger. Also, a finite set of real numbers is not a field under normal addition and multiplication because these operations can create a result outside the set.