Isomorphism and generation of montgomery-form elliptic curves suitable for cryptosystems

Elliptic Curves over Fp Suitable for Cryptosystems

Montgomery-Suitable Cryptosystems

Montgomery’s algorithm [8], hereafter denoted Mn(·, ·), is a process for computing Mn(A,B) = ABN mod n where N is a constant factor depending only on n. Usually, AB mod n is obtained by Mn(Mn(A,B), N −2 mod n) but in this article, we introduce an alternative approach consisting in pre-integrating N into cryptographic keys so that a single Mn(·, ·) will replace directly each modular multiplicati...

Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications

We show that the elliptic curve cryptosystems based on the Montgomery-form E : BY 2 = X+AX+X are immune to the timingattacks by using our technique of randomized projective coordinates, while Montgomery originally introduced this type of curves for speeding up the Pollard and Elliptic Curve Methods of integer factorization [Math. Comp. Vol.48, No.177, (1987) pp.243-264]. However, it should be n...

Multi-Dimensional Montgomery Ladders for Elliptic Curves

Montgomery’s ladder algorithm for elliptic curve scalar multiplication uses only the xcoordinates of points. Avoiding calculation of the y-coordinates saves time for certain curves. Montgomery introduced his method to accelerate Lenstra’s elliptic curve method for integer factoring. Bernstein extended Montgomery’s ladder algorithm by computing integer combinations of two points, thus accelerati...

Special Polynomial Families for Generating More Suitable Elliptic Curves for Pairing-Based Cryptosystems

Constructing non-supersingular elliptic curves for pairing-based cryptosystems have attracted much attention in recent years. The best previous technique builds curves with ρ = lg(q)/lg(r) ≈ 1 (k = 12) and ρ = lg(q)/lg(r) ≈ 1.25 (k = 24). When k > 12, most of the previous works address the question by representing r(x) as a cyclotomic polynomial. In this paper, we propose a new method to find m...