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-despre RSA algorithm si incepem cu inceputul:

Key Generation Algorithm


Generate two large random primes, p and q, of approximately equal size such that their product n = pq is of the required bit length, e.g. 1024 bits

1. Compute n = pq and (φ) phi = (p-1)(q-1).
2. Choose an integer e, 1 < e < phi, such that gcd(e, phi) = 1

Compute the secret exponent d, 1 < d < phi, such that ed ≡ 1

1. The public key is (n, e) and the private key is (n, d). Keep all the values d, p, q and phi secret.

• n is known as the modulus.
• e is known as the public exponent or encryption exponent or just the exponent.
• d is known as the secret exponent or decryption exponent.

Encryption


Sender A does the following:-

1. Obtains the recipient B's public key (n, e).
2. Represents the plaintext message as a positive integer m

1. Computes the ciphertext c = me mod n.
2. Sends the ciphertext c to B.

Decryption

Recipient B does the following:-

1. Uses his private key (n, d) to compute m = cd mod n.
2. Extracts the plaintext from the message representative m.

Digital signing

Sender A does the following:-

1. Creates a message digest of the information to be sent.
2. Represents this digest as an integer m between 0 and n-1.
3. Uses her private key (n, d) to compute the signature s = md mod n.
4. Sends this signature s to the recipient, B.

Signature verification

Recipient B does the following:-

1. Uses sender A's public key (n, e) to compute integer v = se mod n.
2. Extracts the message digest from this integer.
3. Independently computes the message digest of the information that has been signed.
4. If both message digests are identical, the signature is valid.

Summary of RSA

• n = pq, where p and q are distinct primes.
• phi, φ = (p-1)(q-1)
• e < n such that gcd(e, phi)=1
• d = e-1 mod phi.
• c = me mod n, 1<m<n.
• m = cd mod n.

Key length

When we talk about the key length of an RSA key, we are referring to the length of the modulus, n, in bits. The minimum recommended key length for a secure RSA transmission is currently 1024 bits. A key length of 512 bits is now no longer considered secure, although cracking it is still not a trivial task for the likes of you and me. The longer your information is needed to be kept secure, the longer the key you should use. Keep up to date with the latest recommendations in the security journals.
There is small one area of confusion in defining the key length. One convention is that the key length is the position of the most significant bit in n that has value '1', where the least significant bit is at position 1. Equivalently, key length = ceiling(log2(n+1)). The other convention, sometimes used, is that the key length is the number of bytes needed to store n multiplied by eight, i.e. ceiling(log256(n+1)).



The key used in the RSA Example paper s an example. In hex form the modulus is 0A 66 79 1D C6 98 81 68 DE 7A B7 74 19 BB 7F B0
C0 01 C6 27 10 27 00 75 14 29 42 E1 9A 8D 8C 51
D0 53 B3 E3 78 2A 1D E5 DC 5A F4 EB E9 94 68 17
01 14 A1 DF E6 7C DC 9A 9A F5 5D 65 56 20 BB AB
The most significant byte 0x0A in binary is 00001010'B. The most significant bit is at position 508, so its key length is 508 bits. On the other hand, this value needs 64 bytes to store it, so the key length could also be referred to by some as 64 x 8 = 512 bits. We prefer the former method. You can get into difficulties with the X9.31 method for signatures if you use the latter convention.
Computational Efficiency and the Chinese Remainder Theorem (CRT)

Key generation is only carried out occasionally and so computational efficiency is less of an issue.
The calculation a = be mod n is known as modular exponentiation and one efficient method to carry this out on a computer is the binary left-to-right method. To solve y = x^e mod n let e be represented in base 2 as
e = e(k-1)e(k-2)...e(1)e(0)
where e(k-1) is the most significant non-zero bit and bit e(0) the least.
set y = x
for bit j = k - 2 downto 0
begin
y = y * y mod n /* square */
if e(j) == 1 then
y = y * x mod n /* multiply */
end
return y
The time to carry out modular exponentation increases with the number of bits set to one in the exponent e. For encryption, an appropriate choice of e can reduce the computational effort required to carry out the computation of c = me mod n. Popular choices like 3, 17 and 65537 are all primes with only two bits set: 3 = 0011'B, 17 = 0x11, 65537 = 0x10001.
The bits in the decryption exponent d, however, will not be so convenient and so decryption using the standard method of modular exponentiation will take longer than encryption. Don't make the mistake of trying to contrive a small value for d; it is not secure.
An alternative method of representing the private key uses the Chinese Remainder Theorem (CRT). The private key is represented as a quintuple (p, q, dP, dQ, and qInv), where p and q are prime factors of n, dP and dQ are known as the CRT exponents, and qInv is the CRT coefficient. The CRT method of decryption is four times faster overall than calculating m = cd mod n. For more details, see [PKCS1]. The extra values for the private key are:-
dP = (1/e) mod (p-1)
dQ = (1/e) mod (q-1)
qInv = (1/q) mod p where p > q
These are pre-computed and saved along with p and q as the private key. To compute the message m given c do the following:-
m1 = c^dP mod p
m2 = c^dQ mod q
h = qInv(m1 - m2) mod p
m = m2 + hq
Even though there are more steps in this procedure, the modular exponentation to be carried out uses much shorter exponents and so it is less expensive overall.

Someone has pointed out that most large integer packages will fail when computing h if m1 < m2. This can be easily fixed by computing h = qInv(m1 + p - m2) mod p or, alternatively, as we do it in our BigDigits implementation of RSA,
if (bdCompare(m1, m2) < 0)
bdAdd(m1, m1, p);
bdSubtract(m1, m1, m2);
/* Let h = qInv ( m_1 - m_2 ) mod p. */
bdModMult(h, qInv, m1, p);