Cryptanalysis on prime power RSA modulus of the form \(N = p^r q\)

Let \(N = p^r q\) be an RSA prime power modulus for \(r \geq 2\) and \(q < p < 2 q\). This paper propose three new attacks. In the first attack we consider the class of public exponents satisfying an equation \(e X - N Y = u p^r + \frac{q^r}{u} + Z\) for suitably small positive integer \(u\). Using continued fraction we show that \(\frac{Y}{X}\) can be recovered among the convergents of the continued fraction expansion of \(\frac{e}{N}\) and leads to the successful factorization of \(N p^r q\). Moreover we show that the number of such exponents is at least \(N^{\frac{r+3}{2(r+1)}-\varepsilon}\) where \(\varepsilon \geq 0\) is arbitrarily small for large \(N\). The second and third attacks works when \(k\) RSA public keys \((N_i,e_i)\) are such that there exist \(k\) relations of the shape \(e_i x - N_i y_i = p_i^r u + \frac{q_i^r}{u} + z_i\) or of the shape \(e_i x_i - N_i y = p_i^r u + \frac{q_i^r}{u} + z_i\) where the parameters \(x\), \(x_i\), \(y\), \(y_i\), \(z_i\) are suitably small in terms of the prime factors of the moduli. We apply the LLL algorithm, and show that our strategy enable us to simultaneously factor the \(k\) prime power RSA moduli \(N_i\).

IN SEARCH OF MOST COMPLEX REGULAR LANGUAGES

Sequences (Ln| n ≥ k), called streams, of regular languages Lnare considered, where k is some small positive integer, n is the state complexity of Ln, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: (1) the state complexity n of Ln, that is, the number of left quotients of Ln(used as a reference); (2) the state complexities of the left quotients of Ln; (3) the number of atoms of Ln; (4) the state complexities of the atoms of Ln; (5) the size of the syntactic semigroup of Ln; and the state complexities of the following operations: (6) the reverse of Ln; (7) the star of Ln; (8) union, intersection, difference and symmetric difference of Lmand Ln; and (9) the concatenation of Lmand Ln. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (Un(a, b, c) | n ≥ 3) is defined by the deterministic finite automaton with state set {0, 1, … , n−1}, initial state 0, set {n−1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the n states, b transposes states 0 and 1, and c maps state n − 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use Un(a, b, c) and Un(b, a, c), where the roles of a and b are interchanged in the second language. In this sense, Un(a, b, c) is a universal witness. This witness and its extensions also apply to a large number of combined regular operations.