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An advantage of a cryptographic accumulator and actually the reason to use them is that due to the quasi commutativity you can compute witnesses for membership of values in the accumulator where the accumulator and the witnesses are of constant size. Say you have a set $Y=\{y_1,y_2,y_3\}$ and compute the accumulator as $acc=f(f(f(x,y_1),y_2),y_3)$ you ...


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Given a set of (unhashed) Lamport signatures using the same key, an attacker can trivially forge a signature for any message whose $k$-th bit, for each $k$, is equal to the $k$-th bit of at least one of the signed messages. For example, let's say I know the Lamport signatures for the following 16-bit messages using the same key: $$ m_1 = 0001111101110001 ...



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