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Outline

Title
Abstract
Method of Russian Peasants
Cryptanalysis
Rijndael's Cryptanalysis
Cryptanalysis and Factorization
Oblivious Data Transfer
References

MATHEMATICAL CRYPTOLOGY

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Abstract
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This work discusses mathematical cryptology, focusing on the principles of encryption and decryption processes. It categorizes cryptographic systems into symmetric and nonsymmetric types, elaborating on the characteristics of each, notably the key management issues associated with symmetric cryptography. The paper highlights the fundamental structures of cryptosystems, including their components and functionalities, and refers to key foundational works in the field.

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References (56)

  1. 's quantum computers are very small and have no practical meaning. Handling bigger quantum registers with quantum computers would however mean that procedures central for the safety of for example RSA and ELGAMAL, such as factorization and computing discrete logarithms modulo a prime, could be performed in polynomial time. Indeed, these problems are in the class BQP. This was shown by Peter Shor in 1994. Let's see Shor's factorization algorithm here. See the reference SHOR mentioned in Footnote 2. Shor's factorization algorithm is very similar to the exponent algorithm for cryptanalysis of RSA in Section 8.3. The mysterious algorithm A, that appeared there, is just replaced by a quantum algorithm. Of course, the number n to be factored can here have many more prime factors than just two. The "classical part" of the algorithm is the following when the input is the integer n ≥ 2: Shor's factorization algorithm: 1. Check whether n is a prime. If it is then return n and quit.
  2. Check whether n is a higher power of some integer, compare to the Agrawal-Kayal- Saxena algorithm in Section 7.4. If n = u t , where t ≥ 2, we continue by finding the prime factors of u from which we then easily obtain the factors of n. This part, as the previous one, is included only to take care of some "easy" situations quickly.
  3. Choose randomly a number w from the interval 1 ≤ w < n.
  4. Compute d = gcd(w, n) by the Euclidean algorithm.
  5. If d = 1, compute with the quantum computer a number r > 0 such that w r ≡ 1 mod n.
  6. If r is odd, go to #9.
  7. If r is even, set r ← r/2 and go to #7.
  8. Compute ω = (w r ,mod n) by the algorithm of Russian peasants.
  9. If ω ≡ 1 mod n, give up and quit.
  10. If ω ≡ 1 mod n, set ω ′ ← ω and ω ← (ω 2 ,mod n), and go to #11.
  11. Eventually we obtain a square root ω ′ of 1 modulo n such that ω ′ ≡ 1 mod n. If now ω ′ ≡ -1 mod n, give up and quit. Otherwise compute t = gcd(ω ′ -1, n) and continue from t and n/t. Note that because ω ′ + 1 ≡ 0 mod n and on the other hand ω ′2 -1 = (ω ′ + 1)(ω ′ -1) ≡ 0 mod n, some prime factor of n is a factor of ω ′ -1.
  12. So, #6 is left to be performed with the quantum computer. This can be done based on the fact that (w j ,mod n) is periodic with respect to j and a period r can be found by a quantum Fourier transformation. The procedure itself is the following: References
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