The Mathematics behind RSA
In RSA, we have two large primes p and q, amodulus N = pq, an encryption exponent e anda decryption exponent d that satisfy ed = 1 mod (p - 1)(q - 1). The public key is the pair (N,e) and the private key is d.
To encrypt a message M, compute
C = Me mod N.
We want to show
M = Cd mod N,
i.e., that we can decrypt by raising the ciphertext C to the dpower and reducing the result modulo N. But first we must takea slight mathematical detour.
Two positive integers m and n are said to be relativelyprime if they have no common factors other than 1. For example, though both 10 and 9 are composite numbers, they arerelatively prime, since they have no factor (other than 1) incommon.
For a positive integer n, define φ(n) to be the numberof integers less than n that are relatively prime with n.For example, φ(12) = 4, since only 11, 7, 5 and 1 are less than 12 and relatively prime to 12, while φ(7) = 6. In fact, for any prime number p we have φ(p) = p - 1.
Suppose the prime factorization of n is given by
n = p1k1p2k2 ... prkr
Then it can be shown that
φ(n) = n (1 - 1/p1)(1 - 1/p2) ... (1 - 1/pr)
Note that for the RSA modulus N = pq this result implies
φ(N) = (p - 1)(q - 1)
The final mathematical result we need is Fermat'sLittle Theorem. This theorem is usually stated as
Fermat's Little Theorem:If p is prime and p does not divide x, thenxp - 1 = 1 mod p
However, a generalization of Fermat's Little Theorem(sometimes known as Euler's Theorem) is more directlyapplicable to RSA. This theorem states that
Euler's Theorem: If x is relatively prime to n thenxφ(n) = 1 mod n
Now back to RSA decryption. We want to show that
M = Cd = (Me)d = Med mod N.
Recall that ed = 1 mod (p - 1)(q - 1). Also, since N = pq, as noted above, we have
φ(N) = (p - 1)(q - 1)
and it follows that
ed = 1 mod φ(N).
Then by the definition of "mod",there is some k such that ed - 1 = kφ(N).We now have
Med = M(ed - 1) + 1= M Med - 1= M Mkφ(N) mod N
Finally, Fermat's Little Theorem (in the form of Euler's Theorem)can be applied to yield the desired result
Med = M (Mk)φ(N) =M mod N = M.
FAQs
The security of RSA relies on the practical difficulty of factoring the product of two large prime numbers, the "factoring problem". Breaking RSA encryption is known as the RSA problem. Whether it is as difficult as the factoring problem is an open question.
What is the formula for RSA cipher? ›
In RSA, we have two large primes p and q, a modulus N = pq, an encryption exponent e and a decryption exponent d that satisfy ed = 1 mod (p - 1)(q - 1). The public key is the pair (N,e) and the private key is d. C = Me mod N.
What is the logic behind RSA algorithm? ›
RSA is a type of asymmetric encryption, which uses two different but linked keys. In RSA cryptography, both the public and the private keys can encrypt a message. The opposite key from the one used to encrypt a message is used to decrypt it.
What is the RSA secret key challenge? ›
The RSA Secret-Key Challenge was a series of cryptographic contests organised by RSA Laboratories with the intent of helping to demonstrate the relative security of different encryption algorithms.
Why is RSA difficult to crack? ›
The security resilience in RSA is achieved because of the inherent difficulty in factorizing very large numbers into their constituent prime factors. As an example, consider n=77, which can be easily factorized into p=11 and q=7. This factorization is easy because of the small magnitudes involved.
Is RSA as hard as factoring? ›
By the above method, the RSA problem is at least as easy as factoring, but it might well be easier. Indeed, there is strong evidence pointing to this conclusion: that a method to break the RSA method cannot be converted necessarily into a method for factoring large semiprimes.
How is RSA calculated? ›
Steps in RSA Algorithm
- Choose two large prime numbers (p and q)
- Calculate n = p*q and z = (p-1)(q-1)
- Choose a number e where 1 < e < z.
- Calculate d = e-1mod(p-1)(q-1)
- You can bundle private key pair as (n,d)
- You can bundle public key pair as (n,e)
These are some real-world examples that demonstrate the usage of RSA encryption in practice: Securing email messages in email providers. Encrypting messages in messaging apps and chat rooms. Securing P2P data transfer.
What is the math formula for encryption? ›
The encryption process follows the formula ( C \equiv M^e \mod n ), ensuring that even if the ciphertext is intercepted, it remains indecipherable without the corresponding private key.
What is the mathematical foundation of RSA algorithm? ›
Choice of N
- Randomly generate two large prime numbers p and q of size 2048 bits each.
- Compute N=p.q and φ(N) = (p-1). (q-1)
- Choose a number e coprime with φ(N)
- Using Euclid extended algorithm, compute d the inverse of e modulo φ(N)
- Store (e, N) as the public key, and (d, p, q, N) as the private key.
RSA is a popular and secure cryptographic algorithm that encrypts and decrypts data. It provides a secure method for transmitting sensitive data over the Internet. While RSA has some vulnerabilities, it is still utilized for various applications, like digital signatures to authenticate the source of a message.
How does encryption work mathematically? ›
The mathematics behind symmetric encryption algorithms, such as the Advanced Encryption Standard (AES), involve operations like substitution, permutation, and modular arithmetic. These mathematical operations make it extremely challenging for unauthorized parties to decipher the ciphertext without knowing the key.
How to decrypt RSA code? ›
The encrypted message appears in the lower box. To decrypt a message, enter valid modulus N below. Enter decryption key d and encrypted message C in the table on the right, then click the Decrypt button. The decrypted message appears in the lower box.
What is the best RSA key? ›
In most cases, 2048-bit keys are secure and generally recommended. If you need higher security, such as for critical infrastructure or storing sensitive data, consider using longer keys (3072 or 4096 bits). Periodically review and update key lengths as technology changes to protect against new threats.
How long would it take to crack RSA key? ›
Time Required for 10^40 Operations:
So, even with the assumed computational capacity of Google's data centers, it would take approximately 19.8 quadrillion years to crack RSA-2048 using brute force. This is an astronomical time frame, far longer than the current age of the universe (which is about 13.8 billion years).
What does the RSA algorithm depend on? ›
The security of the RSA algorithm heavily relies on large, difficult-to-factor prime numbers used for the key generation process. Factoring the product of two large prime numbers is more difficult when the key length is higher. The key length should be increased as computing power increases.
What is the RSA algorithm based on Euler? ›
The RSA cryptosystem is based on the generation of two random primes, p and q, of equal bit-size and the generation of random exponents, d and e satisfying ed ≡ 1 (mod φ(N)) where φ(N)=(p − 1)(q − 1) is Euler's totient function. The RSA modulus N is the product N = pq.
What mathematical operation is central to the RSA algorithm? ›
Prime Numbers and Modular Arithmetic
Prime numbers are central to many cryptographic algorithms. The difficulty of factoring large composite numbers into their prime components serves as the basis for the security of RSA encryption.
Does RSA use Chinese remainder theorem? ›
Chinese Remainder Theorem in RSA-CRT
In RSA-CRT, it is a common practice to employ the Chinese Remainder Theo- rem during decryption. It results in a decryption much faster than modular exponen- tiation.
