Cryptanalytic Attacks on RSA by Song Y. Yan

By Song Y. Yan

RSA is a public-key cryptographic method, and is the main well-known and widely-used cryptographic approach in contemporary electronic international. Cryptanalytic assaults on RSA, a certified publication, covers just about all significant identified cryptanalytic assaults and defenses of the RSA cryptographic approach and its variations.

Since RSA relies seriously on computational complexity conception and quantity thought, historical past info on complexity conception and quantity conception is gifted first. this can be via an account of the RSA cryptographic approach and its variants.

Cryptanalytic assaults on RSAis designed for a qualified viewers composed of practitioners and researchers in undefined. This booklet is usually compatible as a reference or secondary textual content publication for complex point scholars in laptop science.

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3 Efficient Number-Theoretic Algorithms In this section, we study some well-known and widely used efficient numbertheoretic algorithms, including Euclid’s algorithm, continued fraction algo- 16 1. 11. The First Page of Karp’s Paper rithm, modular exponentiations, the Chinese Remainder Theorem, point additions of elliptic curves, and primality testing algorithms. 3 Efficient Number-Theoretic Algorithms 17 ax ≡ b (mod n) and the linear Diophantine equation ax − by = c. First notice that ax ≡ 1 ( mod n) is a special case of ax ≡ b ( mod n).

To see how similar they are, let us use both methods to compute gcd(195, 135). 195 = 135 · 1 + 60 135 = 60 · 2 + 15 60 = 15 · 4 + 0 195 − 135 = 60 135 − 60 = 75 75 − 60 = 15 60 − 15 = 45 45 − 15 = 30 30 − 15 = 15 One may conclude from the above example that Euclid’s mutual division algorithm is faster than the Chinese mutual subtraction algorithm. One could also conclude that Euclid’s mutual division algorithm is an improvement of the Chinese mutual subtraction algorithm. However, the Chinese mutual subtraction algorithm still has its merits and advantages, which Euclid’s mutual division algorithm does not have or is difficult to have; two of the advantages are as follows.

Of course here we are only interested in the value of x, not y. 3 as follows. q 407667 48 4 2 1 19 x1 1 0 1 -48 193 -434 627 ↑ x y1 0 1 -407667 19568017 -78679735 176927487 -255607222 r1 5033464705 12347 256 59 20 19 1 ↑ d Thus, x = 627 is a solution to the congruence equation 5033464705x ≡ 1 (mod 12345). This can be verified quickly to be true, since 5033464705 · 627 ≡ 1 (mod 12345). How about if gcd(a, N ) > 1 when solving ax ≡ b ( mod N )? For example, find the x in 5033464705x ≡ 7 ( mod 12345), where gcd(5033464705, 12345) = 5 > 1.

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