The RSA algorithm is named after Ron Rivest, Adi Shamir and Len Adleman, who invented it in 1. The basic technique was first discovered in 1. Clifford Cocks [COCK7. CESG (part of the British GCHQ) but this was a secret until 1. The patent taken out by RSA Labs has expired. The RSA cryptosystem is the most widely- used public key cryptography algorithm in the world.
It can be used to encrypt a message without the need to exchange a secret key separately. The RSA algorithm can be used for both public key encryption and digital signatures. Its security is based on the difficulty of factoring large integers. Party A can send an encrypted message to party B without any prior exchange of secret keys. A just uses B's public key to encrypt the message and B decrypts it using the private key, which only he knows. RSA can also be used to sign a message. A can sign a message using their private key and B can verify it using A's public key. We look into the mathematics behind the algorithm on our RSA Theory page. Contents. Key Generation Algorithm. This is the original algorithm. Generate two large random primes, p and q, of approximately equal size such that. Compute n = pq and (phi) φ = (p- 1)(q- 1). See note 6]. Choose an integer e. See note 2]. Compute the secret exponent d, 1 < d < phi, such that. See note 3]. The public key is (n, e) and the private key (d, p, q). Hallo SoftAsm, i need ur help could u send me the Keygen of “PDF2XL Enterprise OCR 6.5.4.1 Activation Key” to [email protected]. No-registration upload of files up to 250MB. Not available in some countries. Keep all the values d, p, q and phi secret. We prefer sometimes to write the private key as (n, d) because you need the value of n when using d. Other times we might write the key pair as ((N, e), d).]. A practical key generation algorithm. Incorporating the advice given in the notes below. RSA key pair is given below. Typical bit lengths are k = 1. You will not go far wrong if you choose e as 6. Algorithm: Generate an RSA key pair. INPUT: Required modulus bit length, k. OUTPUT: An RSA key pair ((N,e), d) where N is the modulus, the product of two primes (N=pq) not exceeding k bits in length. Select a value of e from {3, 5, 1. N ← pq. L ← (p- 1)(q- 1)d ← modinv(e, L)return(N, e, d). The function genprime(b) returns a prime of exactly b bits, with the bth bit set to 1. Note that the operation k/2 is integer division giving the integer quotient with no fraction. If you've chosen e = 6. The final value of N may have a bit length slightly short of the target k. This actually does not matter too much (providing the message m is always < N). If this is the case, then just repeat the entire algorithm until you get one. It should not take too many goes. Alternatively, use the trick setting the two highest bits in the prime candidates described in note 1. Sender A does the following: -. Obtains the recipient B's public key (n, e). Represents the plaintext message as a positive integer m, 1 < m < n. Computes the ciphertext c = me mod n. Sends the ciphertext c to B. Decryption. Recipient B does the following: -. Uses his private key (n, d) to compute m = cd mod n. Extracts the plaintext from the message representative m. Digital signing. Sender A does the following: -. Creates a message digest of the information to be sent. Represents this digest as an integer m between 1 and n- 1. See note 5]. Uses her private key (n, d) to compute the signature s = md mod n. Sends this signature s to the recipient, B. Signature verification. Recipient B does the following: -. Uses sender A's public key (n, e) to compute integer v = se mod n. Extracts the message digest from this integer. Independently computes the message digest of the information that has been signed. If both message digests are identical, the signature is valid. Notes on practical applications. To generate the primes p and q. Rabin- Miller test). This is p. Repeat for q starting with a random integer of length k- k/2. If p< q, swop p and q (this only matters if you intend using the CRT form of the private key). In the extremely unlikely event that p = q, check your random number generator. Alternatively, instead of incrementing by 2, just generate another random number each time. There are stricter rules in ANSI X9. There is much argument about this topic. It is probably better just to use a longer key length. In practice, common choices for e are 3, 5, 1. These particular values are chosen because they are primes and make the modular exponentiation operation faster, having only two bits of value 1. Aside: These five numbers are the first five Fermat numbers, referred to as F0 to F4. Just be careful, these first five Fermat numbers are prime ("Fermat primes"), but the numbers F5 and above are not prime. For example, F5 = 4. The usual choice for e is F4 = 6. Also, having chosen e, it is simpler to test whether gcd(e, p- 1)=1 and gcd(e, q- 1)=1 while generating. Values of p or q that fail this test can be rejected there and then. Even better: if e is an odd prime then you can do the less- expensive test. Why is that? If e is prime then gcd(p- 1, e) > 1 if and only if p- 1 is a multiple of e. That is, if p - 1 ≡ 0 (mod e) or p ≡ 1 (mod e). Hence gcd(p- 1, e) = 1 ⇔ p mod e ≠ 1. To compute the value for d. Extended Euclidean Algorithm to calculate. This is known as modular inversion. Note that this is not integer division. The modular inverse d is defined as the integer value such that ed = 1 mod phi. It only exists if e and phi have no common factors. When representing the plaintext octets as the representative integer m. If m = 0 or 1 or n- 1 there is no security as the ciphertext has the same value. For more details on how to represent the plaintext octets as a suitable representative integer m. PKCS#1 Schemes below or the reference itself [PKCS1]. It is important to make sure that m < n. This is usually done by making sure the first octet of m is equal to 0x. Decryption and signing are identical as far as the mathematics is concerned as both use the private key. Similarly, encryption and verification both use the same mathematical operation with the public key. That is, mathematically, for m < n. However, note these important differences in implementation: -The signature is derived from a message digest of the original information. The recipient will need to follow exactly the same process to derive the. The recommended methods for deriving the representative integers are different. The original definition of RSA uses the Euler totient function. More recent standards use the Charmichael function λ(n) = lcm(p- 1, q- 1) instead. The value of d' computed by d' = e- 1 mod λ(n). Both d and d' will decrypt a message me mod n and both will give the same signature value. To compute λ(n), use the relation. You might ask if there is a way to find the factors of n given just d and e. This is possible. Summary of RSAn = pq, where p and q are distinct primes. For more on the theory and mathematics behind the algorithm, see the RSA Theory page. 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 1. A key length of 5. 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 one small 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'. Equivalently, key length = ceiling(log. The other convention, sometimes used, is that the key length is the number of bytes needed to store n multiplied by eight. The key used in the RSA Example paper [KALI9. In hex form the modulus is. A 6. 6 7. 9 1. D C6 9. DE 7. A B7 7. 4 1. BB 7. F B0. C0 0. C6 2. 7 1. 0 2. 7 0. E1 9. A 8. D 8. C 5. D0 5. 3 B3 E3 7. 8 2. A 1. D E5 DC 5. A F4 EB E9 9. A1 DF E6 7. C DC 9. A 9. A F5 5. D 6. BB AB. The most significant byte 0x. A in binary is 0. B. The most significant bit is at position 5. On the other hand, this value needs 6. We prefer the former method. You can get into difficulties with the X9. Minimum key lengths. The following table is taken from NIST's Recommendation for Key Management [NIST- 8. It shows the recommended comparable key sizes for symmetrical block ciphers (AES and Triple DES) and the RSA algorithm. That is, the key length you would need to use to have comparable security. Symmetric key algorithm. Comparable RSA key length. Comparable hash function. Bits of security. TDEA*1. 02. 4SHA- 1. TDEA2. 04. 8SHA- 2. AES- 1. 28. 30. 72. SHA- 2. 56. 12. 8AES- 1. SHA- 3. 84. 19. 2AES- 2. SHA- 5. 12. 25. 6. Nexiq USB- Link 2 | RP1. J2. 53. 4 | Pass.
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