A Lattice Attack on the McEliece Public Key Cryptosystem: Lattice Basis Reduction Algorithms in Cryptography

The McEliece cryptosystem was proposed by R.McEliecein 1978. In its original version it is based on Goppacodes. Given a public key matrix G and a codewordc=mG+e, we reduce the problem of recovering the errorvector e to the shortest lattice vector problem.Using Conway and Sloane's Construction A" weconstruct a basis of a lattice in which the norm ofthe shortest vector w.r.t. lp norm is equal to the lpnorm of the error vector e for p>log(t) where t isthe weight of the error vector e. To find suchshortest vector in our lattice we use the LLL andblock basis reduction algorithms for the lp normwhich guarantee only an approximation of the lengthof the shortest lattice vector. Our tests show thatthis attack method provides no positive results forGoppa codes of length more than 127."