Strong fully homomorphic white-box and method for using same

A method of computing an algorithm (m,S) having i operations with input m and secret S, comprising:

• defining a white-box fully-homomorphic key generation function (P,p)←Gen(1w) with public-key P and private-key p that selects random prime numbers p, q, s∈W of similar size, wherein:
• B={0,1}b is the domain of order b, of the algorithm , where b>=8;
• W={0,1}w is a white-box domain of order w, for w>>b;
• p>2b is a white-box fully-homomorphic private key;
• N=pq;
• k=s(p−1);
• P=(N,k) is a white-box fully-homomorphic public key;
• defining a white-box fully-homomorphic encoding function Enc(P,m)=mrk+1(mod N) that generates a random integer r∈W, then performs an encoding of the input m∈B;
• defining a white-box fully-homomorphic decoding function Dec(p,c):=c(mod p) that decodes c by computing c modulo p;
• defining i transform key pairs (Ti, ti), at least on part from P and p, wherein Ti is the ith transform public key and ti is the ith transform private key;
• selecting prime numbers ti, and qi∈W of similar size to p, with ti>2b;
• computing Ni=pip−1q2 such that Ni and N are pairwise co-prime;
• computing ei=pip−1−1 where Ti=(Ni, ei);
• accepting an encoded input c0, where c0=Enc(P,m);
• generate an encoded output c′ by performing, for each of the i operations:
• accepting an encoded transform public key Ti=Enc(Tt,S);
• performing the ith operation on the encoded input ci−1 and the encoded transform public key in modulo Ni to obtain an encoded output ci;
• reencoding ci with transform Ti without any interim decoding operation;
• decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p, c′), such that m′=(m,S).