User:Metalim/crack


 * So far computer speed has doubled about once per 1.5 years. (See Moore's law.) This means that each 1.5 years one more bit of key strength is possible to crack. This means that the 16 extra bits of strength is worth about 16×1.5 = 24 years later cracking. 

That's true only for key that can be cracked in years.

Example 1:
 * Let's imagine x-bit key can be cracked in 1 minute. Now, let's add 1 bit to the key: (x+1)-bit key will be cracked in 2 minutes, not 1.5 years + 1 minute.

Example 2:
 * Let x-bit key can be cracked in 15 years with computation power growing 2x every 1.5 years ($$P(t) = 2^{\frac{t}{1.5}}$$, where t = time in years). Let's find how many "years" (in terms of current computers) will be "calculated" in t real years:


 * $$\int 2^{\frac{x}{1.5}}\,\mathrm{d}x = \frac{1.5}{\ln 2}\cdot2^{\frac{x}{1.5}}$$


 * $$N(t) = \int_{0}^{t} 2^{\frac{x}{1.5}}\,\mathrm{d}x = \frac{1.5}{\ln 2}\cdot\left(2^{\frac{t}{1.5}}-2^{\frac{0}{1.5}}\right) = \frac{1.5}{\ln 2}\cdot\left(2^{\frac{t}{1.5}}-1\right)$$


 * For example, if key can be cracked in 15 years with computer speed growing, current computers will crack it in:


 * $$\frac{1.5}{\ln 2}\cdot\left(2^{\frac{15}{1.5}}-1\right) = \frac{1.5}{\ln 2}\cdot(1024-1) = 2213.8\,years$$ (impressive, isn't it?)


 * Now let's add 1 bit to the key. It requires 2x calculations(but not computation power!). Let's calculate how fast it can be cracked:


 * $$N\left(t_{2}\right)=2\cdot N(t_{1})$$


 * $$\frac{1.5}{\ln 2}\cdot\left(2^{\frac{t_{2}}{1.5}}-1\right) = 2\cdot\frac{1.5}{\ln 2}\cdot\left(2^{\frac{t_{1}}{1.5}}-1\right)$$


 * $$2^{\frac{t_{2}}{1.5}} = 2\cdot 2^{\frac{t_{1}}{1.5}}-1$$


 * $$\frac{t_{2}}{1.5} = log_{2}(2\cdot 2^{\frac{2t_{1}}{3}}-1)$$


 * $$t_{2}(t_{1}) = 1.5\cdot log_{2}(2\cdot 2^{\frac{2t_{1}}{3}}-1)$$


 * Examples:


 * $$t_{2}(1\, month) = t_{2}(\frac{1}{12}) = 1.5\cdot log_{2}\left(2^{\frac{38}{36}}-1\right) =\frac{1.96}{12}= almost\, 2\, months$$.
 * $$t_{2}(1) = 1.5\cdot log_{2}\left(2^{\frac{5}{3}}-1\right) =1.681\, years$$.
 * $$t_{2}(3) = 1.5\cdot log_{2}\left(2^{3}-1\right) =4.211\, years$$.
 * $$t_{2}(6) = 1.5\cdot log_{2}\left(2^{5}-1\right) =7.431\, years$$.
 * $$t_{2}(15) = 1.5\cdot log_{2}\left(2\cdot 2^{\frac{2\cdot 15}{3}}-1\right) = 1.5\cdot log_{2}(2^{11}-1) =16.498943\, years$$.

Example 3:
 * Let x-bit key can be cracked in 100 years with constant computation power. Let's calculate how fast it can be cracked with computation power growing 2x every 1.5 years:

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