RSA Algorithm Functions (MBX)¶

RSA Notation¶

The following description uses PKCS #1 v2.1: RSA Cryptography Standard conventions:

n - RSA modulus
e - RSA public exponent
d - RSA private exponent, e*d = mod lambda(n), lambda(n) = LCM
(n, e) - RSA public key
a pair (n, d) - so-called 1-st representation of the RSA private key
p, q - two prime factors of the RSA modulus n, n = p*q
dP - the p’s CRT exponent, e*dP = 1 mod(p-1)
dQ - the q’s CRT exponent, e*dQ = 1 mod(q-1)
qInv - the CRT coefficient, q*qInv = 1 mod(p)
a quintuple (p, q, dP, dQ, qInv) - so-called 2-nd representation of the RSA private key

All the numbers above are positive integers.

Keep in mind the following assumptions:

Current implementation supports RSA-1024, RSA-2048, RSA-3072 and RSA-4096 (the number denotes size of RSA modulus in bits)
Public exponent is fixed, e=65537
No specific assumption relatively “d”, except bitsize(d) ~ bitsize(n) and d<n
Size of p and q in bits is approximately the same and equals bitsize(n)/2

y = x^emod n, x and y are plane- and ciphertext correspondingly

x = y^dmod n, y and x are cipher- and plaintext correspondingly

x1 = y^dPmod p

x2 = y^dQmod q

t = (x1-x2) * qInv mod p

x = x2 + q*t