Functions that are only second-preimage resistant? Are there functions that satisfy the second-preimage resistance property without satisfying the others (preimage and collision resistance) ? 

I'm looking about such functions that compress data (exit the identity function).
Take a function $H:\mathbb S\to\{0,1\}^k$ where $\mathbb S$ is a large finite subset of $\{0,1\}^*$, such that $H$ "compress data" [however this is defined], and $H$ is [conjectured] collision-resistant [thus second-preimage-resistant] and first-preimage-resistant; e.g., SHA-512, for $k=512$. Let $«0»$ and $«1»$ be two public distinct elements of $\mathbb S$. Define $H':\mathbb S\to\{0,1\}^k$ by
$$H'(M)= \begin{cases}H(«1»)&amp;\text{if }M\text{ is }«0»\\H(M)&amp;\text{otherwise}\end{cases}$$

This $H'$ is compressing, and not collision-resistant [argument: $H'(«0»)=H'(«1»)$]. However $H'$ remains preimage-resistant, both first and second [argument: preimage resistance is defined for random message, thus the minor change we made to $H$ in order to build $H'$ does not matter, for odds that a random choice of message hits $«0»$ or $«1»$ are negligible].

Hence $H'$ is second-preimage resistant, but not collision resistant. Which answers the question by the affirmative [I'm reading "without satisfying the others" as "not (satisfying the others)"].

Note: I conjecture without proof that the answer is negative when reading "without satisfying any of the others".



Informal definitions: a function $F$ is


collision-resistant when a [computationally bounded] adversary can't exhibit any $(a,b)$ with $a\ne b$ and $F(a)=F(b)$;
first-preimage-resistant when, given $f$ determined as $F(a)$ for an unknown random $a$, a [computationally bounded] adversary can't [with sizable odds] exhibit any $b$ with $F(b)=f$;
second-preimage-resistant when, given a random $a$, a [computationally bounded] adversary can't [with sizable odds] exhibit any $b$ with $a\ne b$ and $F(a)=F(b)$.


Note: these definitions are a compromise between simplicity and precision, but lacking about what random $a$, sizable odds and computationally bounded means. However they can be made formal by making $F$ a parameterized function family; and adding some appropriate parameterized upper bound for the size of that random $a$ when the input domain $\mathbb S$ is unbounded.