Volume 4, Issue 1

Bilinear pairings derived from supersingular elliptic curves of embedding degrees 4 and 6 over finite fields 𝔽 2 m and 𝔽 3 m , respectively, have been used to implement pairing-based cryptographic protocols. The pairing values lie in certain prime-order subgroups of the cyclotomic subgroups of orders 2 2 m + 1 and 3 2 m – 3 m + 1, respectively, of the multiplicative groups and . It was previously known how to compress the pairing values over characteristic two fields by a factor of 2, and the pairing values over characteristic three fields by a factor of 6. In this paper, we show how the pairing values over characteristic two fields can be compressed by a factor of 4. Moreover, we present and compare several algorithms for performing exponentiation in the prime-order subgroups using the compressed representations. In particular, in the case where the base is fixed, we expect to gain at least a 54% speed up over the fastest previously known exponentiation algorithm that uses factor-6 compressed representations.

Every graph has a canonical finite abelian group attached to it. This group has appeared in the literature under a variety of names including the sandpile group, critical group, Jacobian group, and Picard group. The construction of this group closely mirrors the construction of the Jacobian variety of an algebraic curve. Motivated by this analogy, it was recently suggested by Norman Biggs that the critical group of a finite graph is a good candidate for doing discrete logarithm based cryptography. In this paper, we study a bilinear pairing on this group and show how to compute it. Then we use this pairing to find the discrete logarithm efficiently, thus showing that the associated cryptographic schemes are not secure. Our approach resembles the MOV attack on elliptic curves.

In this work we re-examine two common modulus attacks on RSA. First, we show that Guo's continued fraction attack works much better in practice than previously expected. Given three instances of RSA with a commonmodulus N and private exponents each smaller than N 0.33 , the attack can factor the modulus about 93% of the time in practice. The success rate of the attack can be increased up to almost 100% by including a relatively small exhaustive search. Next, we consider Howgrave-Graham and Seifert's lattice-based attack and show that a second necessary condition for the attack exists that limits the bounds (beyond the original bounds) once n ≥ 7 instances of RSA are used. In particular, by construction, the attack is limited to private exponents at most N 0.5– ε , given sufficiently many instances, instead of the original bound of N 1– ε . In addition, we also consider the effectiveness of the attacks when mounted against multi-prime RSA and Takagi's variant of RSA. For multi-prime RSA, we show three (or more) instances with a common modulus and private exponents smaller than N 1/3– ε is unsafe. For Takagi's scheme, we show that three or more instances with a common modulus N = p t q is unsafe when all the private exponents are smaller than N 2/(3( t +1))– ε . The results, for both variants, is obtained using Guo's method and are almost always successful with the inclusion of a small exhaustive search. When only two instances are available, Howgrave-Graham and Seifert's attack can be successfully mounted on multiprime RSA, with r primes in the modulus, when the private exponents are both smaller than N (3+ r )/7 r – ε .