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To randomly guess a single key from a 128-bit key space has a chance of 1 divided by the number of elements or $\frac{1} {2^{128}}$ where $2^{128}$ is the number of keys possible. To get ballpark figures to convert between base 2 exponents and base 10 exponents you can use the following trick: Because $2^{10} = 1024 \approx 10^3$ you can easily count the ...

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See https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29 Under Key Generation: Compute n = pq. n is used as the modulus for both the public and private keys. Its length, usually expressed in bits, is the key length.

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The size of the public key depends on the elliptic curve used. Any private key will produce a point on the curve, which is the same size – approximately 256 bits for 256-bit curves, for example. Random numbers from a small range could be insecure, however. The secure way to generate the private key is to take it from the range $[1, l-1]$, where $l$ is the ...

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