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Cryptanalysis of RSA and Its Variants by M. Jason Hinek

By M. Jason Hinek

Thirty years after RSA used to be first publicized, it continues to be an energetic examine zone. even supposing a number of stable surveys exist, they're both a bit of outmoded or purely specialize in one form of assault. delivering an up to date examine this box, Cryptanalysis of RSA and Its editions provides the simplest recognized mathematical assaults on RSA and its major variations, together with CRT-RSA, multi-prime RSA, and multi-power RSA.

Divided into 3 elements, the publication first introduces RSA and experiences the mathematical heritage wanted for almost all of assaults defined within the rest of the textual content. It then brings jointly all the most well liked mathematical assaults on RSA and its versions. for every assault provided, the writer incorporates a mathematical evidence if attainable or a mathematical justification for assaults that depend upon assumptions. For the assaults that can't be confirmed, he offers experimental proof to demonstrate their functional effectiveness.

Focusing on mathematical assaults that take advantage of the constitution of RSA and particular parameter offerings, this publication presents an up to date choice of the main recognized assaults, in addition to info of the assaults. It allows an realizing of the cryptanalysis of public-key cryptosystems, functions of lattice foundation relief, and the safety of RSA and its variants.

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Additional resources for Cryptanalysis of RSA and Its Variants

Example text

When this occurs, it is possible to increase the bounds obtained for the method. For example, in the above example, let X, Z, W be bounds for x, z, w, respectively, and suppose X > Z, W . That is, we assume that x is the largest of x, z, w. Notice that we can multiply the vector-matrix equation, from the right, with the diagonal matrix ⎡ ⎤ 1 0 0 0 ⎦, D = ⎣ 0 X/Z 0 0 X/W to obtain a new equation xB = xBD = vD = v given by ⎡ 1 (x, z, y) ⎣0 0 0 X/Z 0 ⎤ AX/W CX/W ⎦ = (x, zX/Z, wZ/W ). BX/W Here, the new target vector v is a vector in a new lattice L generated by 30 Cryptanalysis of RSA and Its Variants the rows of the new basis matrix B .

A simplification of the attack by Judy Moore2 is as follows. Suppose an adversary is given a ciphertext c = me mod N and wants to compute m. Selecting a random x ∈ ZN , the adversary asks for the plaintext of the ciphertext c0 = cxe mod N . Since the requested plaintext m0 satisfies d m0 = c0 d mod N = (cxe ) mod N = cd xed mod N = mx mod N, the adversary, given m0 , can simply compute m = m0 x−1 mod N to recover the desired plaintext. Another attack that uses the homomorphic property of RSA is by Boneh, Joux and Nguyen [33].

5 ) 7: until b1 < b2 Output: b1 , b2 which is Gaussian reduced An important class of reduced bases, for our purposes, are Lov´ aszreduced or, more commonly called, LLL-reduced bases. Let b1 , . . , bm be 26 Cryptanalysis of RSA and Its Variants a basis for a lattice L and let b∗1 , . . 2). The basis b1 , . . 4) 2 for 1 < i ≤ n. 5) or equivalently b∗i 2 ≥ Notice that the vectors b∗i + μi,i−1 b∗i−1 and b∗i−1 are the projections of bi and bi−1 , respectively, on the orthogonal complement of the span of {b1 , .

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