This paper presents new short decryption exponent attacks on RSA, which successfully leads to the factorization of RSA modulus $N=pq$ in polynomial time. The paper has two parts. In the first part, we report the usage of the small prime difference method of the form $|b^{2}p-a^{2}q|<N^{\gamma }$ where the ratio of $\frac{q}{p}$ is close to $\frac{b^2}{a^2}$ , which yields a bound $d<\frac{\sqrt{3}}{\sqrt{2}}{N}^{\frac{3}{4}-\gamma }$ from the convergents of the continued fraction expansion of $\frac{e}{N-\lceil \frac{a^2+b^2}{ab}\sqrt{N}\rceil +1}$ . The second part of the paper reports four cryptanalytic attacks on t instances of RSA moduli $N_s=p_s{q}_s$ for $s=1,2,\dots ,t$ where we use $N-\lceil \frac{a^2+b^2}{ab}\sqrt{N}\rceil +1$ as an approximation of $\varphi (N)$ satisfying generalized key equations of the shape $e_{s}d-k_s\varphi (N_s)=1$ , $e_s{d}_s-k\varphi (N_s)=1$ , $e_{s}d-k_s\varphi (N_s)=z_s$ , and $e_s{d}_s-k\varphi (N_s)=z_s$ for unknown positive integers $d,k_s,d_s,k_s$ , and $z_s$ , where we establish that t RSA moduli can be simultaneously factored in polynomial time using combinations of simultaneous Diophantine approximations and lattice basis reduction methods. In all the reported attacks, we have found an improved short secret exponent bound, which is considered to be better than some bounds as reported in the literature.

A cryptography system was developed previously based on Cipher Polygraphic Polyfunction transformations, $C_{i\times j}^{(t)}\equiv A_{i\times i}^t{P}_{i\times j}modN$ where $C_{i\times j}$ , $P_{i\times j}$ , $A_{i\times i}$ are cipher text, plain text, and encryption key, respectively. Whereas, $(t)$ is the number of transformations of plain text to cipher text. In this system, the parameters ( $A_{i\times i},(t)$ ) are kept in secret by a sender of messages. The security of this system, including its combination with the second order linear recurrence Lucas sequence (LUC) and the Ron Rivest, Adi Shamir and Leonard Adleman (RSA) method, until now is being upgraded by some researchers. The studies found that there is some type of self-invertible $A_{4\times 4}$ should be not chosen before transforming a plain text to cipher text in order to enhance the security of Cipher Tetragraphic Trifunction. This paper also seeks to obtain some patterns of self-invertible keys $A_{6\times 6}$ and subsequently examine their effect on the system of Cipher Hexagraphic Polyfunction transformation. For that purpose, we need to find some solutions $L_{3\times 3}$ for $L_{3\times 3}^2\equiv A_{3\times 3}modN$ when $A_{3\times 3}$ are diagonal and symmetric matrices and subsequently implement the key $L_{3\times 3}$ to get the pattern of $A_{6\times 6}$ .

An algorithm was developed for elliptic scalar multiplication (SM) on Koblitz curve where the multiplier of SM is in the form of Pseudo τ-adic Non-Adjacent Form (pseudoTNAF). This expansion is equivalent to τ-adic Non-Adjacent Form (TNAF) and Reduced τ-adic Non-Adjacent Form (RTNAF) that was produced by Solinas in the year 1997 and 2000 respectively. Some properties for the multiplier of SM was proposed in order to guess the secret message. For the same reason, the objective of this paper is to give five new properties for such multiplier.

Let E be an elliptic curve defined over $F_{2^m}$ and T denotes the Frobenius endomorphism from the set $E\left(F_{2^m}\right)$ to itself. The Koblitz curves are a special type of curves for which the T can be used for improving the performance of computing a scalar multiplication. In this paper, we give an alternative formula for T^m = r_m + s_mT where r_m and s_m are integers. We used the triangular and pyramid numbers, Nicomachus’s theorem and Faulhaber’s formula to produce the formula for T^m. Through this approach, we improve the technique to express T^m.

Scalar multiplication in elliptic curve cryptography is the most expensive and time-consuming operation. The elliptic curve cryptography attracted interest due to the development of modern technology since it could offer the equivalent high level of security with a reduced length of key. Therefore, improving elliptic curve scalar multiplication performance has always been the primary goal of cryptography. In this paper, a novel scalar multiplication algorithm based on the modified double and double add via elliptic net with Karatsuba method was proposed in order to enhance the efficiency of scalar multiplication. In the experimental results, the elliptic net equivalence sequence was applied to the Twisted Edwards curve together with safe curves of numsp384t1 and numsp512t1. At the point operational level, the proposed method reduced the cost of multiplication by 46.15% and 42.30% for double and double add, respectively, when compared to elliptic net using eight blocks method. The proposed double lowered the multiplication cost by 12.5% and the squaring cost by 20% when compared to elliptic net using ten temporary variables method. Following this, proposed double add cost reductions of 6.25% and 20% were obtained to multiplication and squaring. At the field operational level, in comparison to the binary method, the eight-block elliptic net method, and the elliptic net method with ten temporary variables for the 384 bits scenario, the developed scalar multiplication algorithm obtained cost reductions of 57.6%, 31.3%, and 13.2%, respectively. On 512 bits with similar comparison, the designed algorithm exhibited better performance by averages 59.2%, 31.0% and 13.2%. The results signified that the designed algorithm over prime field performed better at the point and field operational levels with larger scalar bit size.

This paper presents a new factoring technique on the modulus , where and are large prime numbers. Suppose there exists an integer satisfies the equation , for some unknown integer and is the Euler’s totient function. Our method exploits the term to be the closest integer to the unknown parameter . Hence we show that the unknown parameters and can be recovered from the list of the continued fractions expansion of Furthermore, we present an algorithm to compute the prime factors of in polynomial time after obtaining the correct tuple and.