Design and Validation of Low-Power Secure and Dependable Elliptic Curve Cryptosystem

The elliptic curve cryptosystem (ECC) has been proven to be vulnerable to non-invasive side-channel analysis attacks, such as timing, power, visible light, electromagnetic emanation, and acoustic analysis attacks. In ECC, the scalar multiplication component is considered to be highly susceptible to side-channel attacks (SCAs) because it consumes the most power and leaks the most information. In this work, we design a robust asynchronous circuit for scalar multiplication that is resistant to state-of-the-art timing, power, and fault analysis attacks. We leverage the genetic algorithm with multi-objective fitness function to generate a standard Boolean logic-based combinational circuit for scalar multiplication. We transform this circuit into a multi-threshold dual-spacer dual-rail delay-insensitive logic (MTD3L) circuit. We then design point-addition and point-doubling circuits using the same procedure. Finally, we integrate these components together into a complete secure and dependable ECC processor. We design and validate the ECC processor using Xilinx ISE 14.7 and implement it in a Xilinx Kintex-7 field-programmable gate array (FPGA).

Generalized orthopair fuzzy matrices

A q-rung orthopair fuzzy matrix (q-ROFM), an extension of the Pythagorean fuzzy matrix (PFM) and intuitionistic fuzzy matrix (IFM), is very helpful in representing vague information that occurs in real-world circumstances. In this paper we define some algebraic operations, such as max-min, min-max, complement, algebraic sum, algebraic product, scalar multiplication \((nA)\), and exponentiation \((A^n)\). We also investigate the algebraic properties of these operations. Furthermore, we define two operators, namely the necessity and possibility to convert q-ROFMs into an ordinary fuzzy matrix, and discuss some of their basic algebraic properties. Finally, we define a new operation(@) on q-ROFMs and discuss distributive laws in the case where the operations of \(\oplus_{q}, \otimes_{q}, \wedge_{q}\) and \(\vee_{q}\) are combined each other.

Novel Key Recovery Attack on Secure ECDSA Implementation by Exploiting Collisions between Unknown Entries

In this paper, we propose a novel key recovery attack against secure ECDSA signature generation employing regular table-based scalar multiplication. Our attack exploits novel leakage, denoted by collision information, which can be constructed by iteratively determining whether two entries loaded from the table are the same or not through side-channel collision analysis. Without knowing the actual value of the table entries, an adversary can recover the private key of ECDSA by finding the condition for which several nonces are linearly dependent by exploiting only the collision information. We show that this condition can be satisfied practically with a reasonable number of digital signatures and corresponding traces. Furthermore, we also show that all entries in the pre-computation table can be recovered using the recovered private key and a sufficient number of digital signatures based on the collision information. As case studies, we find that fixed-base comb and T_SM scalar multiplication are vulnerable to our attack. Finally, we verify that our attack is a real threat by conducting an experiment with power consumption traces acquired during T_SM scalar multiplication operations on an ARM Cortex-M based microcontroller. We also provide the details for validation process.

Compression on the Twisted Jacobi Intersection

Formulas for doubling, differential addition and point recovery after compression were given for many standard models of elliptic curves, and allow for scalar multiplication after compression using the Montgomery ladder algorithm and point recovery on a curve after this multiplication. In this paper we give such formulas for the twisted Jacobi intersection au2 + v2 = 1, bu2 + w2 = 1. To our knowledge such formulas were not given for this model or for the Jacobi intersection. In projective coordinates these formulas have cost 2M +2S +6D for doubling and 5M + 2S + 6D for differential addition, where M; S; D are multiplication, squaring and multiplication by constants in a field, respectively, choosing suitable curve parameters cost of D may be small.

Presburger Arithmetic with algebraic scalar multiplications

We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger arithmetic ($\alpha$-PA). We show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than in PA. Indeed, when $\alpha$ is quadratic and $r\geq 4$, deciding $\alpha$-PA sentences with $r$ alternating quantifier blocks and at most $c\ r$ variables and inequalities requires space at least $K 2^{\cdot^{\cdot^{\cdot^{2^{C\ell(S)}}}}}$ (tower of height $r-3$), where the constants $c, K, C>0$ only depend on $\alpha$, and $\ell(S)$ is the length of the given $\alpha$-PA sentence $S$. Furthermore deciding $\exists^{6}\forall^{4}\exists^{11}$ $\alpha$-PA sentences with at most $k$ inequalities is PSPACE-hard, where $k$ is another constant depending only on~$\alpha$. When $\alpha$ is non-quadratic, already four alternating quantifier blocks suffice for undecidability of $\alpha$-PA sentences.