Efficient Dissection of Bicomposite Problems with Cryptanalytic Applications

In this paper, we show that a large class of diverse problems have a bicomposite structure which makes it possible to solve them with a new type of algorithm called dissection, which has much better time/memory tradeoffs than previously known algorithms. A typical example is the problem of finding the key of multiple encryption schemes with r independent n-bit keys. All the previous error-free attacks required time T and memory M satisfying $$\textit{TM} = 2^{rn}$$TM=2rn, and even if “false negatives” are allowed, no attack could achieve $$\textit{TM}<2^{3rn/4}$$TM<23rn/4. Our new technique yields the first algorithm which never errs and finds all the possible keys with a smaller product of $$\textit{TM}$$TM, such as $$T=2^{4n}$$T=24n time and $$M=2^{n}$$M=2n memory for breaking the sequential execution of $$\hbox {r}=7$$r=7 block ciphers. The improvement ratio we obtain increases in an unbounded way as r increases, and if we allow algorithms which can sometimes miss solutions, we can get even better tradeoffs by combining our dissection technique with parallel collision search. To demonstrate the generality of the new dissection technique, we show how to use it in a generic way in order to improve rebound attacks on hash functions and to solve with better time complexities (for small memory complexities) hard combinatorial search problems, such as the well-known knapsack problem.