Continued Fractions

Continued Fractions

Continued fractions are a way of representing a number as a sum of an integer and a fraction.

Mathematically, a continued fraction is a representation

$$a_{0} + \frac{b_{0}}{ a_{1} + \frac{b_{1}}{ a_{2} + \frac{b_{2}}{ \ddots }}}$$

$a_{i}, b_{i}$are complex numbers. The continued fraction with $b_{i} = 1\ \forall i$ is called a simple continued fraction and continued fractions with finite number of $a_{i}$ are called finite continued fractions.

Consider example rational numbers,

$$\frac{17}{11} = 1 + \frac{6}{11} \\[10pt] \frac{11}{6} = 1 + \frac{5}{6} \\[10pt] \frac{6}{5} = 1 + \frac{1}{5} \\[10pt] \frac{5}{1} = 5 + 0$$

the continued fractions could be written as

$$\frac{5}{1} =5 \\[10pt] \frac{6}{5} = 1 + \frac{1}{5} \\[10pt] \frac{11}{6} = 1 + \frac{5}{6} = 1 + \frac{1}{\frac{6}{5}} = 1 + \frac{1}{1 + \frac{1}{5}} \\[10pt] \frac{17}{11} = 1 + \frac{6}{11} = 1 + \frac{1}{\frac{11}{6}} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{5}}}$$

Notation

$$a_{0} + \frac{1}{ a_{1} + \frac{1}{ a_{2} + \frac{1}{ \ddots }}}$$

A simple continued fraction is represented as a list of coefficients($a_{i}$) i.e

$$x = [a_{0};\ a_{1},\ a_{2},\ a_{3},\ a_{4},\ a_{5},\ a_{6},\ \ldots]$$

for the above example

$$\frac{17}{11} = [1;\ 1,\ 1,\ 5]\ \ ,\frac{11}{6} = [1;\ 1,\ 5]\ \ ,\frac{6}{5} = [1; 5]\ \ ,\frac{5}{1} = [5;]$$

Computation of simple continued fractions

Given a number $x$, the coefficients($a_{i}$) in its continued fraction representation can be calculated recursively using

$$x_{0} = x \\[4pt] a_{i} = \lfloor x_{i} \rfloor \\[4pt] x_{i+1} = \frac{1}{x_{i} - a_{i}}$$

The above notation might not be obvious. Observing the structure of continued fraction with few coefficients will make them more evident:

$$x_{0} = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2}}},\ \ \ x_{1} = a_{1} + \frac{1}{a_{2}}, \ \ \ x_{2} = a_{2} \\[10pt] x_{i} = a_{i} + \frac{1}{x_{i+1}} \\[10pt] x_{i+1} = \frac{1}{x_{i} - a_{i}}$$

SageMath provides functions continued_fraction and continued_fraction_list to work with continued fractions. Below is presented a simple implementation of continued_fractions.

def continued_fraction_list(xi):
    ai = floor(xi)
    if xi == ai: # last coefficient
        return [ai]
    return [ai] + continued_fraction_list(1/(x - ai))

Convergents of continued fraction

The $k^{th}$convergent of a continued fraction$x = [a_{0}; a_{1},\ a_{2},\ a_{3},\ a_{4},\ldots]$is the numerical value or approximation calculated using the first$k - 1$coefficients of the continued fraction. The first$k$convergents are

$$\frac{a_{0}}{1},\ \ \ a_{0} + \frac{1}{a_{1}}, \ \ \ a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2}}}, \ \ldots,\ a_{0} + \frac{1}{a_{1} + \frac{\ddots} {a_{k-2} + \frac{1}{a_{k-1}}}}$$

One of the immediate applications of the convergents is that they give rational approximations given the continued fraction of a number. This allows finding rational approximations to irrational numbers.

Convergents of continued fractions can be calculated in sage

sage: cf = continued_fraction(17/11)
sage: convergents = cf.convergents()
sage: cf
[1; 1, 1, 5]
sage: convergents
[1, 2, 3/2, 17/11]

Continued fractions have many other applications. One such applicable in cryptology is based on Legendre's theorem in diophantine approximations.

Theorem: if$\mid x - \frac{a}{b} \mid < \frac{1}{b^{2}}$, then$\frac{a}{b}$is a convergent of$x$.

Wiener's attack on the RSA cryptosystem works by proving that under certain conditions, an equation of the form$\mid x - \frac{a}{b} \mid$could be derived where$x$is entirely made up of public information and$\frac{a}{b}$is made up of private information. Under assumed conditions, the inequality$\mid x - \frac{a}{b} \mid < \frac{1}{b^{2}}$is statisfied, and the value$\frac{a}{b}$(private information) is calculated from convergents of$x$(public information), consequently breaking the RSA cryptosystem.