![Choose two distinct prime numbers p and q.
For security purposes, the integers p and q should be chosen at random, and should be of similar bit-length. Prime integers can be efficiently found using a primality test.
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.
Compute ?(n) = ?(p)?(q) = (p ? 1)(q ? 1) = n - (p + q -1), where ? is Euler's totient function.
Choose an integer e such that 1 < e < ?(n) and gcd(e, ?(n)) = 1; i.e., e and ?(n) are coprime.
e is released as the public key exponent.
e having a short bit-length and small Hamming weight results in more efficient encryption – most commonly 216 + 1 = 65,537. However, much smaller values of e (such as 3) have been shown to be less secure in some settings.[5]
Determine d as d ? e?1 (mod ?(n)); i.e., d is the multiplicative inverse of e (modulo ?(n)).
This is more clearly stated as: solve for d given d?e ? 1 (mod ?(n))
This is often computed using the extended Euclidean algorithm. Using the pseudocode in the Modular integers section, inputs a and n correspond to e and ?(n), respectively.
d is kept as the private key exponent.](https://i.imgur.com/YAYW1sC.png "Choose two distinct prime numbers p and q.
For security purposes, the integers p and q should be chosen at random, and should be of similar bit-length. Prime integers can be efficiently found using a primality test.
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.
Compute ?(n) = ?(p)?(q) = (p ? 1)(q ? 1) = n - (p + q -1), where ? is Euler's totient function.
Choose an integer e such that 1 < e < ?(n) and gcd(e, ?(n)) = 1; i.e., e and ?(n) are coprime.
e is released as the public key exponent.
e having a short bit-length and small Hamming weight results in more efficient encryption – most commonly 216 + 1 = 65,537. However, much smaller values of e (such as 3) have been shown to be less secure in some settings.[5]
Determine d as d ? e?1 (mod ?(n)); i.e., d is the multiplicative inverse of e (modulo ?(n)).
This is more clearly stated as: solve for d given d?e ? 1 (mod ?(n))
This is often computed using the extended Euclidean algorithm. Using the pseudocode in the Modular integers section, inputs a and n correspond to e and ?(n), respectively.
d is kept as the private key exponent.")
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Choose two distinct prime numbers p and q.
For security purposes, the integers p and q should be chosen at random, and should be of similar bit-length. Prime integers can be efficiently found using a primality test.
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.
Compute ?(n) = ?(p)?(q) = (p ? 1)(q ? 1) = n - (p + q -1), where ? is Euler's totient function.
Choose an integer e such that 1 < e < ?(n) and gcd(e, ?(n)) = 1; i.e., e and ?(n) are coprime.
e is released as the public key exponent.
e having a short bit-length and small Hamming weight results in more efficient encryption – most commonly 216 + 1 = 65,537. However, much smaller values of e (such as 3) have been shown to be less secure in some settings.[5]
Determine d as d ? e?1 (mod ?(n)); i.e., d is the multiplicative inverse of e (modulo ?(n)).
This is more clearly stated as: solve for d given d?e ? 1 (mod ?(n))
This is often computed using the extended Euclidean algorithm. Using the pseudocode in the Modular integers section, inputs a and n correspond to e and ?(n), respectively.
d is kept as the private key exponent.