Cryptography/Mathematical Background

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Introduction[edit]

Modern public-key (asymmetric) cryptography is based upon a branch of mathematics known as number theory, which is concerned solely with the solution of equations that yield only integer results. These type of equations are known as diophantine equations, named after the Greek mathematician Diophantos of Alexandria (ca. 200 CE) from his book Arithmetica that addresses problems requiring such integral solutions.

One of the oldest diophantine problems is known as the Pythagorean problem, which gives the length of one side of a right triangle when supplied with the lengths of the other two side, according to the equation

where is the length of the hypotenuse. While two sides may be known to be integral values, the resultant third side may well be irrational. The solution to the Pythagorean problem is not beyond the scope, but is beyond the purpose of this chapter. Therefore, example integral solutions (known as Pythagorean triplets) will simply be presented here. It is left as an exercise for the reader to find additional solutions, either by brute-force or derivation.

Pythagorean Triplets
3 4 5
5 12 13
7 24 25
8 15 17

Prime Numbers[edit]

Description[edit]

Asymmetric key algorithms rely heavily on the use of prime numbers, usually exceedingly long primes, for their operation. By definition, prime numbers are divisible only by themselves and 1. In other words, letting the symbol | denote divisibility (i.e. - means " divides into "), a prime number strictly adheres to the following mathematical definition

| Where or only

The Fundamental Theorem of Arithmetic states that all integers can be decomposed into a unique prime factorization. Any integer greater than 1 is considered either prime or composite. A composite number is composed of more than one prime factor

| where ultimately

in which is a unique prime number and is the exponent.

Numerical Examples[edit]

543,312 = 24  32  50  73  111
553,696 = 25  30  50  70  113  131

As can be seen, according to this systematic decomposition, each factorization is unique.

In order to deterministically verify whether an integer is prime or composite, only the primes need be examined. This type of systematic, thorough examination is known as a brute-force approach. Primes and composites are noteworthy in the study of cryptography since, in general, a public key is a composite number which is the product of two or more primes. One (or more) of these primes may constitute the private key.

There are several types and categories of prime numbers, three of which are of importance to cryptography and will be discussed here briefly.

Fermat Primes[edit]

Fermat numbers take the following form

If Fn is prime, then it is called a Fermat prime.

Numerical Examples[edit]








The only Fermat numbers known to be prime are . Moreover, the primality of all Fermat numbers was disproven by Euler, who showed that .

Mersenne Primes[edit]

Mersenne primes - another type of formulaic prime generation - follow the form

where is a prime number. The [1] Wolfram Alpha engine reports Mersenne Primes, an example input request being "4th Mersenne Prime".

Numerical Examples[edit]

The first four Mersenne primes are as follows






Numbers of the form Mp = 2p without the primality requirement are called Mersenne numbers. Not all Mersenne numbers are prime, e.g. M11 = 211−1 = 2047 = 23 · 89.

Coprimes (Relatively Prime Numbers)[edit]

Two numbers are said to be coprime if the largest integer that divides evenly into both of them is 1. Mathematically, this is written

where is the greatest common divisor. Two rules can be derived from the above definition

If | and , then |
If with , then both and are squares, i.e. - ,

The Prime Number Theorem[edit]

The Prime Number Theorem estimates the probability that any integer, chosen randomly will be prime. The estimate is given below, with defined as the number of primes

is asymptotic to , that is to say . What this means is that generally, a randomly chosen number is prime with the approximate probability .

The Euclidean Algorithm[edit]

Introduction[edit]

The Euclidean Algorithm is used to discover the greatest common divisor of two integers. In cryptography, it is most often used to determine if two integers are coprime, i.e. - .

In order to find where efficiently when working with very large numbers, as with cryptosystems, a method exists to do so. The Euclidean algorithm operates as follows - First, divide by , writing the quotient , and the remainder . Note this can be written in equation form as . Next perform the same operation using in 's place: . Continue with this pattern until the final remainder is zero. Numerical examples and a formal algorithm follow which should make this inherent pattern clear.

Mathematical Description[edit]









When , stop with .

Numerical Examples[edit]

Example 1 - To find gcd(17,043,12,660)

17,043 = 1  12,660 + 4383
12,660 = 2  4,383 + 3894
 4,383 = 1  3,894 + 489
 3,894 = 7  489 + 471
   489 = 1  471 + 18
   471 = 26  18 + 3
    18 = 6  3 + 0

gcd (17,043,12,660) = 3 \ </math>

Example 2 - To find gcd(2,008,1,963)

2,008 = 1  1,963 + 45
1,963 = 43  45 + 28
   45 = 1  28 + 17
   28 = 1  17 + 11
   17 = 1  11 + 6
   11 = 1  6 + 5
    6 = 1  5 + 1
    5 = 5  1 + 0

gcd (2,008,1963) = 1 Note: the two number are coprime.

Algorithmic Representation[edit]

Euclidean Algorithm(a,b)
Input:     Two integers a and b such that a > b
Output:    An integer r = gcd(a,b)
  1.   Set a0 = a, r1 = r
  2.   r = a0 mod r1
  3.   While(r1 mod r  0) do:
  4.      a0 = r1
  5.      r1 = r
  6.      r = a0 mod r1
  7.   Output r and halt

The Extended Euclidean Algorithm[edit]

In order to solve the type of equations represented by Bézout's identity, as shown below

where , , , and are integers, it is often useful to use the extended Euclidean algorithm. Equations of the form above occur in public key encryption algorithms such as RSA (Rivest-Shamir-Adleman) in the form where . There are two methods in which to implement the extended Euclidean algorithm; the iterative method and the recursive method.

As an example, we shall solve an RSA key generation problem with e = 216 + 1, p = 3,217, q = 1,279. Thus, 62,537d + 51,456w = 1.

Methods[edit]

The Iterative Method[edit]

This method computes expressions of the form for the remainder in each step of the Euclidean algorithm. Each modulus can be written in terms of the previous two remainders and their whole quotient as follows:

By substitution, this gives:

The first two values are the initial arguments to the algorithm:

The expression for the last non-zero remainder gives the desired results since this method computes every remainder in terms of a and b, as desired.

Example[edit]
Step Quotient Remainder Substitute Combine terms
1 4,110,048 = a 4,110,048 = 1a + 0b
2 65,537 = b 65,537 = 0a + 1b
3 62 46,754 = 4,110,048 - 65,537 62 46,754 = (1a + 0b) - (0a + 1b) 62 46,754 = 1a - 62b
4 1 18,783 = 65,537 - 46,754 1 18,783 = (0a + 1b) - (1a - 62b) 1 18,783 = -1a + 63b
5 2 9,188 = 46,754 - 18,783 2 9,188 = (1a - 62b) - (-1a + 62b) 2 9,188 = 3a - 188b
6 2 407 = 18,783 - 9,188 2 407 = (-1a + 63b) - (3a - 188b) 2 407 = -7a + 439b
7 22 234 = 9,188 - 407 22 234 = (3a - 188b) - (-7a + 439b) 22 234 = 157a - 9,846b
8 1 173 = 407 - 234 1 173 = (-7a + 439b) - (157a - 9,846b) 1 173 = -164a + 10,285b
9 1 61 = 234 - 173 1 61 = (157a - 9,846b) - (-164a + 10,285b) 1 61 = 321a + 20,131b
10 2 51 = 173 - 61 2 51 = (-164a + 10,285b) - (321a +20,131b) 2 51 = -806a + 50,547b
11 1 10 = 61 - 51 1 61 = (321a +20,131b) - (-806a + 50,547b) 1 10 = 1,127a - 70,678b
12 5 1 = 51 -10 5 1 = (-806a + 50,547b) - (1,127a - 70,678b) 5 1 = -6,441a + 403,937b
13 10 0 End of algorithm

Putting the equation in its original form yields , it is shown that and . During the process of key generation for RSA encryption, the value for w is discarded, and d is retained as the value of the private key In this case

d = 0x629e1 = 01100010100111100001

The Recursive Method[edit]

This is a direct method for solving Diophantine equations of the form . Using this method, the dividend and the divisor are reduced over a series of steps. At the last step, a trivial value is substituted into the equation, and is then worked backward until the solution is obtained.

Example[edit]

Using the previous RSA vales of and

Euclidean Expansion Collect Terms Substitute Retrograde Substitution Solve For dx
4,110,048 w0 + 65,537d0 = 1
(62 65,537 + 46,754) w0 + 65,537d0 = 1
65,537 (62w0 + d0) + 46,754w0 = 1 w1 = 62w0 + d0 4,595 = (62)(-6441) + d0 d0 = 403,937
65,537 w1 + 46,754d1 = 1 d1 = w0 w1 = -6,441
(1 46,754 + 18,783) w1 + 46,754d1 = 1
46,754 (w1 + d1) + 18,783w1 = 1 w2 = w1 + d1 -1,846 = 4,595 + d1 d1 = -6,441
46,754 w2 + 18,783d2 = 1 d2 = w1
(2 18,783 + 9,188) w2 + 18,783d2 = 1
18,783 (2w2 + d2) + 9,188w2 = 1 w3 = 2w2 + d2 903 = (2)(-1,846) + d2 d2 = 4,595
18,783 w3 + 9,188d3 = 1 d3 = w2
(2 9,188 + 407) w3 + 9,188d3 = 1
9,188 (2w3 + d3) + 407w3 = 1 w4 = 2w3 + d3 -40 = (2)(903) + d3 d3 = -1846
9,188 w4 + 407d4 = 1 d4 = w3
(22 407 + 234) w4 + 407d4 = 1
407 (22w4 + d4) + 234w4 = 1 w5 = 22w4 +d4 23 = (22)(-40) + d4 d4 = 903
407 w5 + 234d5 = 1 d5 = w4
(1 234 + 173) w5 + 234d5 = 1
234 (w5 + d5) + 173w5 = 1 w6 = w5 +d5 -17 = 23 + d5 d5 = -40
234 w6 + 173d6 = 1 d6 = w5
(1 173 + 61) w6 + 173d6 = 1
173 (w6 + d6) + 61w6 = 1 w7 = w6 +d6 6 = -17 + d6 d6 = 23
173 w7 + 61d7 = 1 d7 = w6
(2 61 + 51) w7 + 61d7 = 1
61 (2w7 + d7) + 51w7 = 1 w8 = 2w7 +d7 -5 = (2)(6) + d7 d7 = -17
61 w8 + 51d8 = 1 d8 = w7
(1 51 + 10) w8 + 51d8 = 1
51 (w8 + d8) + 10w8 = 1 w9 = w8 +d8 1 = -5 + d8 d8 = 6
51 w9 + 10d9 = 1 d9 = w8
(5 10 + 1) w9 + 10d9 = 1
10 (5w9 + d9) + 1w9 = 1 w10 = 5w9 +d9 0 = (5)(1) + d9 d9 = -5
10 w10 + 1d10 = 1 d10 = w9
(1 10 + 0) w10 + 1d10 = 1
1 (10w10 + d10) + 0w10 = 1 w11 = 10w10 +d10 1 = (10)(0) + d10 d10 = 1
1 w11 + 0d11 = 1 d11 = w10 w11 = 1, d11 = 0

Euler's Totient Function[edit]

Significant in cryptography, the totient function (sometimes known as the phi function) is defined as the number of nonnegative integers less than that are coprime to . Mathematically, this is represented as

Which immediately suggests that for any prime

The totient function for any exponentiated prime is calculated as follows

The Euler totient function is also multiplicative

where

Finite Fields and Generators[edit]

A field is simply a set which contains numerical elements that are subject to the familiar addition and multiplication operations. Several different types of fields exist; for example, , the field of real numbers, and , the field of rational numbers, or , the field of complex numbers. A generic field is usually denoted .

Finite Fields[edit]

Cryptography utilizes primarily finite fields, nearly exclusively composed of integers. The most notable exception to this are the Gaussian numbers of the form which are complex numbers with integer real and imaginary parts. Finite fields are defined as follows

The set of integers modulo
The set of integers modulo a prime

Since cryptography is concerned with the solution of diophantine equations, the finite fields utilized are primarily integer based, and are denoted by the symbol for the field of integers, .

A finite field contains exactly elements, of which there are nonzero elements. An extension of is the multiplicative group of , written , and consisting of the following elements

such that

in other words, contains the elements coprime to

Finite fields form an abelian group with respect to multiplication, defined by the following properties

 The product of two nonzero elements is nonzero 
 The associative law holds 
 The commutative law holds 
 There is an identity element 
 Any nonzero element has an inverse 

A subscript following the symbol for the field represents the set of integers modulo , and these integers run from to as represented by the example below

The multiplicative order of is represented and consists of all elements such that . An example for is given below

If is prime, the set consists of all integers such that . For example

Composite n Prime p

Generators[edit]

Every finite field has a generator. A generator is capable of generating all of the elements in the set by exponentiating the generator . Assuming is a generator of , then contains the elements for the range . If has a generator, then is said to be cyclic.

The total number of generators is given by

Examples[edit]

For  (Prime)




Total number of generators  generators

Let , then ,  is a generator

Since  is a generator, check if 
, and , , therefore,  is not a generator
, and , , therefore,  is not a generator

Let , then ,  is a generator
Let , then ,  is a generator
Let , then ,  is a generator

There are a total of  generators,  as predicted by the formula 
For  (Composite)




Total number of generators  generators

Let , then ,  is a generator
Let , then ,  is a generator

There are a total of  generators  as predicted by the formula 

Congruences[edit]

Description[edit]

Number theory contains an algebraic system of its own called the theory of congruences. The mathematical notion of congruences was introduced by Karl Friedrich Gauss in Disquisitiones (1801).

Definition[edit]

If and are two integers, and their difference is evenly divisible by , this can be written with the notation

This is expressed by the notation for a congruence

where the divisor is called the modulus of congruence. can equivalently be written as

where is an integer.

Note in the examples that for all cases in which , it is shown that . with this in mind, note that

Represents that is an even number.

Represents that is an odd number.

Examples[edit]

Properties of Congruences[edit]

All congruences (with fixed ) have the following properties in common

if and only if
If and then
implies that
Given there exists a unique such that

These properties represent an equivalence class, meaning that any integer is congruent modulo to one specific integer in the finite field .

Congruences as Remainders[edit]

If the modulus of an integer , then for every integer

which can be understood to mean is the remainder of divided by , or as a congruence

Two numbers that are incongruent modulo must have different remainders. Therefore, it can be seen that any congruence holds if and only if and are integers which have the same remainder when divided by .

Example[edit]

 is equivalent to
 implies
 is the remainder of  divided by 

The Algebra of Congruences[edit]

Suppose for this section we have two congruences, and . These congruences can be added or subtracted in the following manner

If these two congruences are multiplied together, the following congruence is obtained

or the special case where

Note: The above does not mean that there exists a division operation for congruences. The only possibility for simplifying the above is if and only if and are coprime. Mathematically, this is represented as

implies that if and only if

The set of equivalence classes defined above form a commutative ring, meaning the residue classes can be added, subtracted and multiplied, and that the operations are associative, commutative and have additive inverses.

Reducing Modulo m[edit]

Often, it is necessary to perform an operation on a congruence where , when what is desired is a new integer such that with the resultant being the least nonnegative residue modulo m of the congruence. Reducing a congruence modulo is based on the properties of congruences and is often required during exponentiation of a congruence.

Algorithm[edit]

Input: Integers  and  from  with 
Output: Integer  such that 

1. Let 
2.     
3.     
4. Output 

Example[edit]

Given 




Note that is the least nonnegative residue modulo

Exponentiation[edit]

Assume you begin with . Upon multiplying this congruence by itself the result is . Generalizing this result and assuming is a positive integer

Example[edit]





This simplifies to

 implies 
 implies 

Repeated Squaring Method[edit]

Sometimes it is useful to know the least nonnegative residue modulo of a number which has been exponentiated as . In order to find this number, we may use the repeated squaring method which works as follows:

1. Begin with 
2. Square  and  so that 
3. Reduce  modulo  to obtain 
4. Continue with steps 2 and 3 until  is obtained.
   Note that  is the integer where  would be just larger than the exponent desired
5. Add the successive exponents until you arrive at the desired exponent
6. Multiply all 's associated with the 's of the selected powers
7. Reduce the resulting  for the desired result

Example[edit]

To find :










Adding exponents:



Multiplying least nonnegative residues associated with these exponents:




Therefore: 


Inverse of a Congruence[edit]

Description[edit]

While finding the correct symmetric or asymmetric keys is required to encrypt a plaintext message, calculating the inverse of these keys is essential to successfully decrypt the resultant ciphertext. This can be seen in cryptosystems Ranging from a simple affine transformation

Where

To RSA public key encryption, where one of the deciphering (private) keys is

Definition[edit]

For the elements where , there exists such that . Thus, is said to be the inverse of , denoted where is the power of the integer for which .

Example[edit]
Find 

This is equivalent to saying 

First use the Euclidean algorithm to verify .
Next use the Extended Euclidean algorithm to discover the value of .
In this case, the value is .

Therefore, 

It is easily verified that 

Fermat's Little Theorem[edit]

Definition[edit]

Where is defined as prime, any integer will satisfy the following relation:

Example[edit]

When and

implies that

Conditions and Corollaries[edit]

An additional condition states that if is not divisible by , the following equation holds

Fermat's Little Theorem also has a corollary, which states that if is not divisible by and then

Euler's Generalization[edit]

If , then

Chinese Remainder Theorem[edit]

If one wants to solve a system of congruences with different moduli, it is possible to do so as follows:

A simultaneous solution exists if and only if with , and any two solutions are congruent to one another modulo .

The steps for finding the simultaneous solution using the Chinese Remainder theorem are as follows:

1. Compute
2. Compute for each of the different 's
3. Find the inverse of for each using the Extended Euclidean algorithm
4. Multiply out for each
5. Sum all
6. Compute to obtain the least nonnegative residue

Example[edit]

Given:













Using the Extended Euclidean algorithm:











Quadratic Residues[edit]

If is prime and , examining the nonzero elements of , it is sometimes important to know which of these are squares. If for some , there exists a square such that . Then all squares for can be calculated by where . is a quadratic residue modulo if there exists an such that . If no such exists, then is a quadratic non-residue modulo . is a quadratic residue modulo a prime if and only if .

Example[edit]

For the finite field , to find the squares , proceed as follows:



The values above are quadratic residues. The remaining (in this example) 9 values are known as quadratic nonresidues. the complete listing is given below.


Quadratic residues: 
Quadratic nonresidues: 

Legendre Symbol[edit]

The Legendre symbol denotes whether or not is a quadratic residue modulo the prime and is only defined for primes and integers . The Legendre of with respect to is represented by the symbol . Note that this does not mean divided by . has one of three values: .

Jacobi Symbol[edit]

The Jacobi symbol applies to all odd numbers where , then:

If is prime, then the Jacobi symbol equals the Legendre symbol (which is the basis for the Solovay-Strassen primality test).

Primality Testing[edit]

Description[edit]

In cryptography, using an algorithm to quickly and efficiently test whether a given number is prime is extremely important to the success of the cryptosystem. Several methods of primality testing exist (Fermat or Solovay-Strassen methods, for example), but the algorithm to be used for discussion in this section will be the Miller-Rabin (or Rabin-Miller) primality test. In its current form, the Miller-Rabin test is an unconditional probabilistic (Monte Carlo) algorithm. It will be shown how to convert Miller-Rabin into a deterministic (Las Vegas) algorithm.

Pseudoprimes[edit]

Remember that if is prime and , Fermat's Little Theorem states:

However, there are cases where can meet the above conditions and be nonprime. These classes of numbers are known as pseudoprimes.

is a pseudoprime to the base , with if and only if the least positive power of that is congruent to evenly divides .

If Fermat's Little Theorem holds for any that is an odd composite integer, then is referred to as a pseudoprime. This forms the basis of primality testing. By testing different 's, we can probabilistically become more certain of the primality of the number in question.

The following three conditions apply to odd composite integers:

I. If the least positive power of which is congruent to and divides which is the order of in , then is a pseudoprime.
II. If is a pseudoprime to base and , then is also a pseudoprime to and .
III. If fails , for any single base , then fails for at least half the bases .

An odd composite integer for which holds for every is known as a Carmichael Number.

Miller-Rabin Primality Test[edit]

Description[edit]

Examples[edit]

Factoring[edit]

The Rho Method[edit]

Description[edit]

Algorithm[edit]

Example[edit]

Fermat Factorization[edit]

Example[edit]

Random Number Generators[edit]

RNGs vs. PRNGs[edit]

ANSI X9.17 PRNG[edit]

Blum-Blum-Shub PRNG[edit]

RSA PRNG[edit]

Entropy Extractors[edit]

Whitening Functions[edit]

Large Integer Multiplication[edit]

Karatsuba Multiplication[edit]

Example[edit]

Furers Multiplication[edit]

Elliptic Curves[edit]

Description[edit]

As I Have Gone Alone in the, and with my treasures Bold, i can keep my secrets where and hint of riches new and old, Begin it where warm waters halt, and take it in the canyon down, not too far, but too far to walk, put in below the home of brown, from there it's no place for the meek, the end is ever drawing neigh, there'll be no paddle up your creek, just heavy loads and water high,

Definition[edit]

Properties[edit]

Summary[edit]