Rényi-∞ divergence machinery for oracle-bounded reductions - #1076
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| (* Single Iface clone — both top-level access target (consumers use | ||
| [RDO.Iface.Oracle]/[RDO.Iface.Adv]) and BPS substitution target. *) | ||
| clone import BPS_Iface as Iface with | ||
| type out_t = out_t, |
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Use <- to avoid proliferation.
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| declare axiom A_bound1 : | ||
| hoare[ A(BPS1.Count(Sampler1)).main : | ||
| BPS1.Count.n = 0 ==> BPS1.Count.n <= N ]. |
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Here you probably want a quantification over the initial value of the counter, so the result can be applied generally where there is a preamble that cannot trigger distinguishing behaviour. (Did we discuss this already? If so, ignore this.)
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| }. | ||
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| module type Adv (O : Oracle) = { | ||
| proc main(p : param_t) : bool |
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We could generalise this to arbitrary return types. This would require replace all the raw res events in probabilities below with a "for all event over res (and perhaps the game's globals)".
Not super clear what value this would add, but the presampling argument certainly doesn't rely on the event being the game's output only.
If you choose to generalise, be careful: if you include the globals in scope, you will need to post-sample unused queries in the lazy version.
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Adds a Rényi-∞ (max-divergence) toolkit for probability-preservation
arguments in bounded-query reductions — e.g. rejection-sampling hops
where each of N oracle queries costs a multiplicative factor M.
- theories/distributions/RDiv.ec: the dominance predicate
[dominated M d1 d2] and the divergence [rdiv_inf], probability
preservation (mu d1 E <= M * mu d2 E), and closure under the standard
constructions (dmap, dlet, dprod, dlist, djoin(map), dexcepted,
drestrict, dcond, dfst/dsnd, dopt, dfold, dfun), each in an
explicit-bound and a tight (rdiv_inf) form. An abstract
[Distinguisher] theory lifts preservation to adversaries
([adv_rdiv_inf], [adv_rdiv_inf_le], pre-composed variants), reusing
SDist.GenDist for the Pr-to-distribution transport.
- theories/distributions/BoundedPreSample.ec: for an adversary making
at most N oracle queries, answering queries by fresh sampling is
Pr-equal, per event, to pre-sampling a list of N draws
([eq_pr_fresh_ref_ev]); proved by a counting/up-to-bad step plus
eager sampling via PROM.
- theories/distributions/RDivOracle.ec: the user-facing bound
[rdiv_bound_sampler],
Pr[Game1(A) : E res] <= M ^ N * Pr[Game2(A) : E res],
for any N-query-bounded A against per-parameter samplers d1/d2 that
are uniformly M-dominated on the support of the parameter
distribution. A second theory [RDivOracleValid] packages the common
"valid parameters only" restriction (conditioning the parameter
distribution), stating validity as an event conjunct — no additive
slack and no M >= 1 assumption — with a corollary paying the
invalid-parameter mass additively.
- tests/rdiv-instantiate.ec: smoke instantiation realizing all
parameter axioms of both oracle theories at a trivial instance.
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The Rényi-∞ (max-divergence / dominance) machinery for bounded-query reductions: