Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR
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1 Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000
2 Set of Integers The set of integers, denoted by Z, contains all integral numbers (with no fraction) from negative infinity to positive infinity.
3 Binary Operations A binary operation takes two inputs and creates one output. Three binary operations for the set of integers
4 Example Results of the three binary operations on two integers. Because each input can be either positive or negative, there can be four cases for each operation.
5 Integer Division In integer arithmetic, if we divide a by n, we can get q and r. The relationship between these four integers can be shown as a = q n + r
6 Example Assume that a = 255 and n = 11. We can find q = 23 and R = 2 using the division algorithm. Finding the quotient and the remainder
7 Division algorithm for integers
8 Example When we use a computer or a calculator, r and q are negative when a is negative. How can we apply the restriction that r needs to be positive? The solution is simple, we decrement the value of q by 1 and we add the value of n to r to make it positive.
9 Graph of division alogorithm
10 Divisbility If a is not zero and we let r = 0 in the division relation, we get a = q n If the remainder is zero, If the remainder is not zero,
11 Example a. The integer 4 divides the integer 32 because 32 = 8 4. b. The number 8 does not divide the number 42 because 42 = There is a remainder, the number 2, in the equation.
12 Properties Property 1: if a 1, then a = ±1. Property 2: if a b and b a, then a = ±b. Property 3: if a b and b c, then a c. Property 4: if a b and a c, then a (m b + n c), where m and n are arbitrary integers
13 Example
14 Example
15 Note Fact 1: The integer 1 has only one divisor, itself. Fact 2: Any positive integer has at least two divisors, 1 and itself (but it can have more).
16 Common divisors of two integers
17 Note Greatest Common Divisor The greatest common divisor of two positive integers is the largest integer that can divide both integers. Note Euclidean Algorithm Fact 1: gcd (a, 0) = a Fact 2: gcd (a, b) = gcd (b, r), where r is the remainder of dividing a by b
18 Euclidean Algorithm Note When gcd (a, b) = 1, we say that a and b are relatively prime. 2.18
19 Note When gcd (a, b) = 1, we say that a and b are relatively prime.
20 Example Find the greatest common divisor of 2740 and Solution We have gcd (2740, 1760) = 20.
21 Example Find the greatest common divisor of 25 and 60. Solution We have gcd (25, 65) = 5.
22 Extended Euclidean Algorithm Given two integers a and b, we often need to find other two integers, s and t, such that The extended Euclidean algorithm can calculate the gcd (a, b) and at the same time calculate the value of s and t.
23 Extended Euclidean algorithm
24 Extended Euclidean algorithm
25 Example Given a = 161 and b = 28, find gcd (a, b) and the values of s and t. Solution We get gcd (161, 28) = 7, s = 1 and t = 6.
26 Example Given a = 17 and b = 0, find gcd (a, b) and the values of s and t. Solution We get gcd (17, 0) = 17, s = 1, and t = 0.
27 Example Given a = 0 and b = 45, find gcd (a, b) and the values of s and t. Solution We get gcd (0, 45) = 45, s = 0, and t = 1.
28 Linear Diophantine Equation Note A linear Diophantine equation of two variables is ax + by = c. To find the integer values of x and y which satisfy the equation. This type of equation has no solution or infinite number of solutions. If d c, then infinite number of solutions, else no solution.
29 Linear Diophantine Equation Note Particular solution: x 0 = (c/d)s and y 0 = (c/d)t Note General solutions: x = x 0 + k (b/d) and y = y 0 k(a/d) where k is an integer
30 Example Find the particular and general solutions to the equation 21x + 14y = 35. Solution
31 Example For example, imagine we want to cash a $100 check and get some $20 and some $5 bills. We have many choices, which we can find by solving the corresponding Diophantine equation 20x + 5y = 100. Since d = gcd (20, 5) = 5 and 5 100, the equation has an infinite number of solutions, but only a few of them are acceptable in this case. The general solutions with x and y nonnegative are (0, 20), (1, 16), (2, 12), (3, 8), (4, 4), (5, 0).
32 MODULAR ARITHMETIC The division relationship (a = q n + r) has two inputs (a and n) and two outputs (q and r). In modular arithmetic, we are interested in only one of the outputs, the remainder r. 1 Modular Operator 2 Set of Residues 3 Congruence 4 Operations in Z n 5 Addition and Multiplication Tables 6 Different Sets
33 Modulo Operator The modulo operator is shown as mod. The second input (n) is called the modulus. The output r is called the residue. Division algorithm and modulo operator
34 Example Find the result of the following operations: a. 27 mod 5 b. 36 mod 12 c. 18 mod 14 d. 7 mod 10 Solution a. Dividing 27 by 5 results in r = 2 b. Dividing 36 by 12 results in r = 0. c. Dividing 18 by 14 results in r = 4. After adding the modulus r = 10 d. Dividing 7 by 10 results in r = 7. After adding the modulus to 7, r = 3.
35 Set of Residues The modulo operation creates a set, which in modular arithmetic is referred to as the set of least residues modulo n, or Z n. Some Z n sets
36 Congruence To show that two integers are congruent, we use the congruence operator ( ). For example, we write: 2 mod 10 = 2, 12 mod 10 = 2, 22 mod 10 = 2 In modular arithmetic 2, 12, 22 are called congruent mod
37 Concept of congruence
38 Residue Classes A residue class [a] or [a] n is the set of integers congruent modulo n. The set of all integers such that x= a (mod n)
39 Residue Classes In each set, there is one element called the least nonnegative residue. [0] 0, [1] 1.. and so on. The set of all these residues are Zn
40 Comparison of Z and Z n using graphs The circle shows the distribution of integers Zn. All congruent integers modulo n occupy the same point on the circle.
41 Example We use modular arithmetic in our daily life; for example, we use a clock to measure time. Our clock system uses modulo 12 arithmetic. However, instead of a 0 we use the number 12.
42 Operation in Z n The three binary operations that we used for the set Z can also be defined for the set Zn. The result may need to be mapped to Zn using the mod operator. Binary operations in Z n
43 Example Perform the following operations (the inputs come from Zn): a. Add 7 to 14 in Z15. b. Subtract 11 from 7 in Z13. c. Multiply 11 by 7 in Z20. Solution
44 Example Perform the following operations (the inputs come from either Z or Z n ): a. Add 17 to 27 in Z 14. b. Subtract 43 from 12 in Z 13. c. Multiply 123 by 10 in Z 19. Solution
45 Properties
46 Properties of mod operator The first two operands are reduced.
47 Example The following shows the application of the above properties: 1. (1,723, ,124,945) mod 11 = (8 + 9) mod 11 = 6 2. (1,723,345 2,124,945) mod 16 = (8 9) mod 11 = (1,723,345 2,124,945) mod 16 = (8 9) mod 11 = 6
48 Example In arithmetic, we often need to find the remainder of powers of 10 when divided by an integer.
49 Inverses To find the inverse of a number relative to an operation. - an additive inverse (relative to an addition operation) or - a multiplicative inverse (relative to a multiplication operation).
50 Additive Inverse In Z n, two numbers a and b are additive inverses of each other if Note In modular arithmetic, each integer has an additive inverse. The sum of an integer and its additive inverse is congruent to 0 modulo n.
51 Example Find all additive inverse pairs in Z10. Solution The six pairs of additive inverses are (0, 0), (1, 9), (2, 8), (3, 7), (4, 6), and (5, 5).
52 Multiplicative Inverse In Z n, two numbers a and b are the multiplicative inverse of each other if Note In modular arithmetic, an integer may or may not have a multiplicative inverse. When it does, the product of the integer and its multiplicative inverse is congruent to 1 modulo n.
53 Example Find the multiplicative inverse of 8 in Z 10. Solution There is no multiplicative inverse because gcd (10, 8) = 2 1. In other words, we cannot find any number between 0 and 9 such that when multiplied by 8, the result is congruent to 1. Example Find all multiplicative inverses in Z 10. Solution There are only three pairs: (1, 1), (3, 7) and (9, 9). The numbers 0, 2, 4, 5, 6, and 8 do not have a multiplicative inverse.
54 Example Find all multiplicative inverse pairs in Z 11. Solution We have seven pairs: (1, 1), (2, 6), (3, 4), (5, 9), (7, 8), (9, 9), and (10, 10).
55 Note The extended Euclidean algorithm finds the multiplicative inverses of b in Z n when n and b are given and gcd (n, b) = 1. The multiplicative inverse of b is the value of t after being mapped to Z n.
56 Using extended Euclidean algorithm to find multiplicative inverse
57 Example Find the multiplicative inverse of 11 in Z 26. Solution The gcd (26, 11) is 1; the inverse of 11 is -7 or 19.
58 Example Find the multiplicative inverse of 23 in Z 100. Solution The gcd (100, 23) is 1; the inverse of 23 is -13 or 87.
59 Example Find the inverse of 12 in Z 26. Solution The gcd (26, 12) is 2; the inverse does not exist.
60 Addition and Multiplication Tables Both tables are symmetric about the diagonal commutative property of addition and multiplication; a+b=b+a, axb=bxa. In Addition table each row and column is a permutation of another row or column.
61 Different Sets Some Z n and Z n * sets In cryptography uses an integer as the encryption key, The inverse of that integer is used as the decryption key. If the encryption / decryption operation is addition, Then Zn is the set of possible keys, as each element of Zn has an inverse. For multiplication, it is not true. Not all elements of Zn has an inverse. Zn* is a new set, a subset of Zn, which includes those members of Zn which has a unique multiplicative inverse.
62 Different Sets Some Z n and Z n * sets Note We need to use Zn when additive inverses are needed; we need to use Zn* when multiplicative inverses are needed.
63 Two More Sets Cryptography often uses two more sets: Z p and Z p *. The modulus in these two sets is a prime number.
64 MATRICES In cryptography we need to handle matrices. 1 Definitions 2 Operations and Relations 3 Determinants 4 Residue Matrices
65 Definition A matrix of size l x m
66 Examples of matrices
67 Operations and Relations Example An example of addition and subtraction. Addition and subtraction of matrices
68 Example Figure shows the product of a row matrix (1 3) by a column matrix (3 1). The result is a matrix of size 1 1. Figure Multiplication of a row matrix by a column matrix
69 Example Figure below shows the product of a 2 3 matrix by a 3 4 matrix. The result is a 2 4 matrix. Figure Multiplication of a 2 3 matrix by a 3 4 matrix
70 Example Figure below shows an example of scalar multiplication. Figure Scalar multiplication
71 Determinant The determinant of a square matrix A of size m m denoted as det (A) is a scalar calculated recursively as shown below: Note The determinant is defined only for a square matrix.
72 Example Figure below shows how we can calculate the determinant of a 2 2 matrix based on the determinant of a 1 1 matrix. Figure Calculating the determinant of a 2 2 matrix
73 Example Figure below shows the calculation of the determinant of a 3 3 matrix. Figure Calculating the determinant of a 3 3 matrix
74 Inverses Additive inverse: A+B=0 = -A Multiplicative inverse: MI of a square matrix A is a square matrix B, such that A x B = B x A = I. Multiplicative inverse exists only if the det(a) has a multiplicative inverse in the corresponding set.
75 Inverses Note Multiplicative inverses are only defined for square matrices.
76 Residue Matrices Cryptography uses residue matrices: matrices where all elements are in Z n. Property: A residue matrix has a multiplicative inverse if gcd (det(a), n) = 1. Example Figure A residue matrix and its multiplicative inverse
77 LINEAR CONGRUENCE Cryptography often involves solving an equation or a set of equations of one or more variables with coefficient in Z n. To show how to solve equations when the power of each variable is 1 (linear equation). 1 Single-Variable Linear Equations 2 Set of Linear Equations
78 Single-Variable Linear Equations Equations of the form ax b (mod n ) might have no solution or a limited number of solutions.
79 Example Solve the equation 10 x 2(mod 15). Solution First we find the gcd (10 and 15) = 5. Since 5 does not divide 2, we have no solution. Example Solve the equation 14 x 12 (mod 18). Solution
80 Example Solve the equation 3x (mod 13). Solution First we change the equation to the form ax b (mod n). We add 4 (the additive inverse of 4) to both sides, which give 3x 2 (mod 13). Because gcd (3, 13) = 1, the equation has only one solution, which is x 0 = (2 3 1 ) mod 13 = 18 mod 13 = 5. We can see that the answer satisfies the original equation: (mod 13).
81 Single-Variable Linear Equations We can also solve a set of linear equations with the same modulus if the matrix formed from the coefficients of the variables is invertible. Figure Set of linear equations
82 Example Solve the set of following three equations: Solution The result is x 15 (mod 16), y 4 (mod 16), and z 14 (mod 16). We can check the answer by inserting these values into the equations.
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