Since you specified "without implying distrust", I think it's at the heart of the matter, and consider it worth noting that the concepts of personal integrity ("my word is my honor") and 'saving face' (avoiding humiliation) are central to the interpersonal issues related to how asking for a receipt can be interpreted as distrust or mistrust.
People all over the world are likely to believe that their own integrity is so high that there really is no need to issue a receipt for goods or money received. Asking for a receipt is thus often interpreted as questioning the trustworthiness of the individual, with all sorts of related emotional connotations, so that the interpersonal problem of perceived lack of trust becomes more serious than the practical financial matter of having a concrete record of the transaction.
The person who is asked to provide a receipt "loses face" if it appears that their trustworthiness is being queried. Rather more complicatedly, the person who asks for a receipt can also lose face by bringing up the whole issue of trust in an indelicate manner. It can create serious tensions between family members or friends, but in my experience is not usually so serious or emotive an issue between strangers.
This is such a widespread problem here in India, and asking someone for a receipt is typically interpreted as an insulting lack of trust, especially among people used to traditional ways of doing business. One pompous old gentleman actually collected a significant cash deposit on a land sale from my father and refused to give a receipt, saying
No question of giving receipt, my word is my honor. You can expect the land to be registered into your name within 2 days. If you doubt my integrity, you need not purchase my property.
That was 3 decades ago and luckily for us his word was indeed good as gold, but over the years I have almost always needed receipts mainly for my own sense of security. So while making various sizable purchases or payments I learned to achieve my aims by asking for a receipt in such a way that no mistrust was implied -- most commonly by suggesting that I needed it for my personal (or an organization's) financial records:
Could you please give me a receipt for that Rs.12,500 payment? I need it for my files, you see, when I calculate the annual financial statement.
I need to furnish a receipt to the company to prove for their files that I actually made this purchase in the expected manner...
I need a receipt for my financial records to claim an income tax deduction.
Someone I know would even go so far as to put the blame on his wife:
My wife absolutely insists on collecting and filing receipts to know where the money is disappearing every year!
Now, what is important is not that your whatever reason for requesting a receipt should be extremely credible, but that by expressing your need for a receipt with sincere goodwill, you do indirectly manage to convey to your friend that you absolutely do not mistrust him, while also being extremely careful not to mention trust anywhere in the conversation: personal loan situations are often complicated further by the 'debt of gratitude' element in that your friend was good enough to help you with a significantly large amount when you really needed the money, and actually trusted you to be able to repay it later; so if your friend appears reluctant to issue a receipt for whatever reason, you might consider not pressing him to do so, and that allows both of you to "save face" while completing the transaction.
Note: according to Psychology Today, "The phrase to "save face" has been around a long time. It's been part of English vernacular since the 19th century. The concept is a core social value in Asian cultures, among others. The meaning has remained stable across time. Saving Face signifies a desire -- or defines a strategy -- to avoid humiliation or embarrassment, to maintain dignity or preserve reputation."